— African---— Fractals---M O D E R N C O M P U T IN G A N D IN D IG E N O U S D E S IG N RO N EG LA SH C o n ten ts A cknow ledgm ents ix PART I I n tr o d u c tio n CHAPTER J I n t r o d u c t i o n t o f r a c ta l g e o m e t r y CHAPTER 2 F r a c t a ls in A f r i c a n s e t t l e m e n t a r c h i t e c t u r e CHAPTER 3 F ra c t a ls in c r o s s - c u l tu r a l c o m p a r i s o n CHAPTER 4 I n t e n t i o n a n d i n v e n t i o n in d e s i g n P A R T II A fr ic a n fr a c ta l 7 n a t h e m a t i c s CHAPTER 5 G e o m e tric a lg o rith m s CHAPTER 6 S calin g CHAPTER 7 N u m e r i c s y s te m s CHAPTER 8 R ecursion CHAPTER 9 I n f in it y CHAPTER IO C o m p lex ity 3 20 39 49 61 71 86 / 109 147 151 C ontents v iii PART III Im plication s a p t e r 11 c h a p t e r 12 c h a p t e r 13 F ra c t a ls in t h e E u r o p e a n h i s t o r y o f m a t h e m a t i c s c h a p t e r 14 F u tu r e s fo r A f r i c a n f r a c t a l s Ch a p p e n d ix T h e o r e t i c a l fr a m e w o r k s i n c u l t u r a l s t u d i e s o f k n o w l e d g e 1 79 T h e p o li ti c s o f A f r i c a n f r a c t a l s 192 203 216 M easu rin g th e fractal d im e n s io n o f A fric a n s e ttle m e n t a r c h ite c tu r e N o te s 235 R eferences Index 253 243 231 •Acknowledgments T h a n k s to g o first t o m y wife, N a n c y C a m p b e l l , w h o h a s t o l e r a t e d m y o b se ssio n s w i t h g ra c e , a n d to E v e ly n , A l b e r t , a n d J o a n n e E glash, w h o in s p ire d m a n y o f t h e m . I a m g ra te fu l for t h e a ss ista n c e o f m y p rofesso rs a t U C S C : R a l p h A b r a h a m , S t e v e C a t o n , J a m e s C lifford, D o n n a H a ra w a y , G o t t f r i e d M ayer-Kress, K e n N o rris, H ein zO u o P e ii g e n , C a r o ly n M a r t i n S h a w , a n d P a tr ic i a Z av ella. E q u ally i m p o r t a n t were m y fello w U C S C g r a d u a t e s t u d e n t s , in p a r t i c u l a r D a v i d B a in , J u l i a n B le e c k e r, P e t e r B r o a d w e l l , K irb y B u n a s , C l a u d i a C a s t e n a d a , G i o v a n n a D i C h i r o , J o e D u m it , V i n c e n t e Diaz, P a u l E d w ard s, L i n d a G a r c i a , J e n n i fe r G o n z a le s , C h r i s G ray, J o h n H a r ti g a n , S h a r o n H elsel, L aura K an g, L o rrain e Kenny, M a t t h e w K obbe, A n g ie Rosga, W a r r e n S ack, M e g S a tt e r t h w a i te , S a n d y S to n e , M a r ita S tu r k e n , B e m t W a h l, a n d S a r a h W il li a m s . T h a n k s a ls o t o B illie H a r r i s , M i r a n d a H a y s, R e b e c c a Lyle, K e n M a r t i n , S h e i l a P e u s e , A d o l p h S m i t h , J o s h u a W e i n s t e i n , a n d P a u l Yi. R e s e a r c h f u n d i n g fr o m t h e I n s t i t u t e for I n t e r c u l t u r a l S t u d i e s a n d t h e Fulb r i g h t p r o g r a m m a d e p o s s ib le m y f i e ld w o rk in w e st a n d c e n t r a l A f r i c a . A s c h a p ­ t e r 10 m a k e s clear, 1 o w e m u c h to m y S e n e g a le s e colleag ues, C h r i s t i n e S t n a D ia tt a a n d N f a l l y B a d ia n e . A l s o o f g r e a t h e l p in S e n e g a ! w e r e A b d o u l i Ba, R e a l B asso, C h a r l e s B e c k e r, K o l a d o C is s e , I b n o u D 'tag n e, P a t h e D i a g n e , S o u l e y m a n e B a c h i r D i a g n e , M o u s s e D io p , W a ly C o l y F ay e, M a x , M a r i e - L o u i s e M o r e a u , M a r g o t N d ia y e , V ic to r Sagna, O u s in a n S e n , F a to u Sow, Yoro Sylla, S a k ir T h a i m , a n d R ie n e Acknowledgments T o je . From th e W e s t A fric a n R e se a rc h C e n te r I received th e e x p e rt a d v ic e of A m e r i c a n p r o f e s s o rs E i l e e n J u l i e n a n d J a n i s M a y s . T h a n k s a l s o t o S h a m i r a J o h n s o n , P a u l a n d B e ts e y H a r n e y , J a n e H a l e , 'L i s a M c N e e , a n d Liz M e r m i n . I a m a ls o g r a t e f u l t o Is siak a Is a a c D r a b o a n d t h e b r i l l i a n t C a n a d i a n p h o ­ t o g r a p h y t e a m , M i c h e l e t D i d i , in B u r k i n a F a s o . T h a n k s a l s o t o A m a d o u C o u li b a ly , K alifa K o n £ a n d A b d o u l a y e S y lla in M a li. In C a m e r o o n I r e c e i v e d t h e g e n e r o s i t y o f I r e k e B e ssik e , N g w a E m m a n u e l , N o i f e M e b o u b o , t h e la t e E n g e l ­ b e r t M v e n g , a n d E d w a r d N j o c k . M y w o rk in B e n i n w o u ld n o t h a v e b e e n p ossib le w i t h o u t t h e a s s is t a n c e o f T o n y H u t c h i n s o n ; t h a n k s a ls o t o K n k e A l f r e d , N a t h e l i R o b e r t s , a n d M a r t i n e d e S o u s a for t h e i r e x p e r t i s e in v o d u n . I n G h a n a M i c h a e l O r la n s k y graciously in t r o d u c e d m e t o t h e m a n y c u l t u r a l res o u rces av aila b le . M a n y o f t h e lo c a l folks 1 s p o k e t o in w e st a n d c e n t r a l A fric a , w h i l e e x t e n d i n g g r e a t g e n ­ e ro s ity a n d e n t h u s i a s m , a s k e d t h a t t h e i r n a m e s r e m a i n u n r e c o r d e d , a n d I t h a n k t h e m as w ell. O n my r e tu r n to th e U n ite d S ta te s 1 receiv ed a fello w sh ip fro m th e C e n ­ te r fo r t h e H u m a n i t i e s a t O r e g o n S t a t e U n i v e r s ity , w h i c h a ls o o ff e re d t h e o p p o r ­ t u n i t y t o w o rk w i t h a n t h r o p o l o g i s t s J o a n G r o s s , D a v i d G r o s s , a n d C o r t S m i t h , as well as K a m a u S a d ik i from t h e P o r t l a n d B lack E d u c a t i o n a l C e n t e r . T h a n k s also to M i c h a e l R o b e r s o n fo r b is g e o m e t r y a d v ic e , a n d D a v i d a n d B a r b a r a T h o m a s ( n o w m a t h t e a c h e r s a t H e n d e r s o n v i l l e H i g h , N o r t h C a r o l i n a ) for i n v e s t i g a t i n g o w a ri p a tc e r n s . A t w o m o n t h f e l l o w s h i p a t t h e U n i v e r s i t y o f O r e g o n g o t m e th ro u g h th e su m m er, and in to my c u rr e n t p o sitio n at T h e O h io S ta te U n iv e r ­ sity. H e r e I h a v e b e e n t h a n k f u l fo r h e l p fro m P a t t i B r o s n a n , W a y n e C a r l s o n , Ja c q u e lin e C h a n d a , C y n th i a D illa rd , D avid H o r n , L in d sa y Jo n e s , O k e c h u k w u O d i t a , E g o n d u R o s e m a r y O n y e j e k w e , R o b e r t R a n s o m , D a n R eff, R o s e K a p i a n , C a r o l y n S i m p s o n , D a a ’i y a h I S a l e e m , J e n n i f e r T e r r y , C y n t h i a T y s o n , a n d M a n ju la W ald ro n . T h e r e are also m a n y co lleg u es, g e o g ra p h ic a lly s c a t te r e d , w h o s e fe e d b a c k h as b e e n i n v a l u a b l e . In p a r t i c u l a r 1 w o u ld li k e t o t h a n k M a d e l e i n e A k r i c h , Ja c k A l e x a n d e r , M a r y J o A r n o l d i , G e o r g e A r t h u r , M a r c i a A s c h e r , J i m B a rta , S il v io B e d i n i , T Q B e rg , J e a n - P a u l B o u r d i e r , G e o f B o w k e r , M i c h a e l T . B r o w n , P a t C a p l i n , B ria n C a se y , J e n n i f e r C r o i s s a n t , D o n C r o w e , J i m C r u t c h f i e l d , U b i r a t a n D 'A m b r o s i o , R o n a l d Bell, O s e i D a r k w a , M a r i a n n e d e L a e t, G a r y L e e D o w n e y , M u n r o e E agles, A r t u r o E sc o b a r, F l o r e n c e F a s a n e l li , J a m e s F e r n a n d e z , M a r i l y n F ra n k e n s te in , R a y v o n F o u c h e , P au lu s G e rd e s, C h o n a t G e tz , G lo ria G ilm e r, D a v id H a k k e n , T u r tl e H e a r t , D e b o r a h H e a t h , D a v id H ess, S t e f a n H e l m r e i c h , D arian H e n d ric k s , D a v id H u g h e s , S a n d y Jones, E sm aeli K n te h , R o g e r P. K o v a c h , G e ls a K n ijn i k , B r u n o L ato u r, M u r r a y Leaf, Bea L u m p k i n , R o b i n M a c k a y , C a r o ! M alloy, B en o it M a n d e lb ro t, M ik e M a rin a c c i, J o a n n a M asin g iia, Lynn M c G e e , Jam es Ac/cnowledgnien is M o rro w , D a v id M o s im e g e , B ria n M M u r p h y , D i a n a B aird N ’D iay e, N a n c y N o o te r , K a r e n N o r w o o d , S p u r g e o n E k u n d a y o P a r k e r , C li ffo rd P ic k o v e r , P a t r i c i a P o o le , A r t h u r P o w e ll, D e a n P r e b l e , D a n R e g a n , J i'm R a uff, S a l R e s tiv o , P ierre R o n d e a u , J o h n R osew all, R u d y R u c k e r, N o r a S a b e lli, J n ro n S a m p s o n , D o u g S c h u le r , P a tr ic k ( R i c k ) S c o t t , R o b S h a w , E n i d S c h i l d k r o u t , D a v i d W i l l i a m s o n S h a ffe r, L a rry S h ir le y , D e n n i s S m i t h , G e o r g e S p ie s , S u s a n L e ig h S ta r , P a u l S to l le r , P e t e r T a y ­ lor, A g n e s T u s k a , G a r y V a n W y k , D o n n e l l W a l t o n , D M W a r r e n , D o r o t h y W a s h ­ b u r n , H e l e n W a t s o n - V e r r a n , M a r k W. W essels, P a tr ic i a S. W i l s o n , a n d C l a u d i a Z aslavsky. L a s t a n d n o t least, t h a n k s to m y e d it o r s a t R u tg e r s , D o r e e n V a l e n t i n e a n d M a r t h a H e lle r . PART -Introductio n -----------------— CHAPTER In troduction------------------------------------------to--------------------------- :------------------- — fractal-----------------------------------------------—geometry-----------------:---------------------------- F r a c t a l g e o m e t r y h a s e m e r g e d as o n e o f t h e m o s t e x c i t i n g f r o n t i e r s in t h e f u s io n b e t w e e n m a t h e m a t i c s a n d i n f o r m a t i o n t e c h n o l o g y . F r a c t a ls c a n b e s e e n in m a n y o f t h e s w ir lin g p a t t e r n s p r o d u c e d by c o m p u t e r g r a p h i c s , a n d t h e y h a v e b e c o m e a n i m p o r t a n t n e w t o o l fo r m o d e l i n g i n bio lo g y , geology, a n d o t h e r n a t ­ u ral s c i e n c e s . W h i l e f r a c t a l g e o m e t r y c a n i n d e e d t a k e u r i n t o t h e far r e a c h e s . . o f h i g h - r e c h s c i e n c e , its p a t t e r n s a re su r p ris in g ly c o m m o n in t r a d i t i o n a l A f r i c a n d e s i g n s , a n d s o m e o f its b a s i c c o n c e p t s a r e f u n d a m e n t a l t o A f r i c a n k n o w l e d g e s y s te m s . T h i s b o o k w ill p r o v i d e a n easy i n t r o d u c t i o n t o f r a c t a l g e o m e t r y for p e o p l e w i t h o u t a n y m a t h e m a t i c s b a c k g r o u n d , a n d it w ill s h o w h o w t h e s e s a m e c a t e g o r i e s o f g e o m e t r i c p a t t e r n , c a l c u l a t i o n , a n d t h e o r y a r e e x p r e s s e d in A f r ic a n cu ltu res. M a t h e m a t i c s a n d culture For m a n y y ears a n t h r o p o l o g i s t s h a v e o b s e r v e d t h a t t h e p a t t e r n s p r o d u c e d in d i f ­ fe re n t c u ltu re s c a n b e ch a ra c te riz e d by specific design therr.es. In E u ro p e a n d A m e r ­ ic a, fo r e x a m p l e , we o f t e n see c it ie s la id o u t in a grid p a t t e r n o f s t r a i g h t s t r e e t s a n d r i g h t - a n g l e c o r n e r s . A n o t h e r g rid , t h e C a r t e s i a n c o o r d i n a t e s y s te m , h a s lo ng b e e n a f o u n d a t i o n for t h e m a t h e m a t i c s used in th e s e societies. In m a n y w orks Introductton o f C h i n e s e a r t w e find h e x a g o n s u s e d w i t h e x t r a o r d i n a r y g e o m e t r i c p r e c i s i o n — a c h o i c e t h a t m i g h t s e e m a r b i t r a r y w e r e it n o t fo r t h e i m p o r t a n c e o f t h e n u m ­ b e r six in t h e h e x a g r a m s o f th e i r f o r tu n e te llin g system ( t h e I Ching), in th e a n a to m y c h a r t s for a c u p u n c t u r e (liu-qi o r “six s p irits ” ), a n d e v e n in C h i n e s e a r c h i t e c t u r e . 1 S h a p e a n d n u m b e r a r e n o t o n l y t h e u n i v e r s a l ru le s o f m e a s u r e m e n t a n d logic; t h e y a r e a ls o c u l t u r a l to o l s t h a t c a n b e u se d for e x p r e s s i n g p a r t i c u l a r so c ia l id eas a n d l i n k i n g d i f f e r e n t a r e a s o f life. T h e y a re , as C l a u d e L e v i - S t r a u s s w o u ld p u t it, “g o o d to t h i n k w i t h . ” D e s i g n t h e m e s a r e lik e t h r e a d s r u n n i n g t h r o u g h t h e s o c ia l fa b r ic ; t h e y are less a c o m m a n d i n g f o r c e t h a n s o m e t h i n g w e c o m m a n d , w e a v i n g th e s e s t r a n d s i n t o m a n y d i f f e r e n t p a t t e r n s o f m e a n i n g . T h e a n c i e n t C h i n e s e e m p i r e s , for e x a m p l e , u s e d a b a s e - i o c o u n t i n g s y s te m , a n d t h e y e v e n b e g a n t h e first u n i v e r ­ sal m e t r i c s y s te m .^ S o t h e f r e q u e n t u se o f t h e n u m b e r 6 0 i n C h i n e s e k n o w l e d g e s y s te m s c a n b e l i n k e d t o t h e c o m b i n a t i o n o f t h i s o fficia l b a se 10 n o t a t i o n w i t h t h e i r sacred n u m b e r six. I n s o m e A m e r i c a n cities we find t h a t t h e s tre e ts a re n u m ­ b e r e d lik e C a r t e s i a n c o o r d i n a t e s , b u t in o t h e r s t h e y a r e n a m e d a f t e r h is to r i c a l figures, a n d still o t h e r s c o m b i n e t h e tw o . T h e s e c i t y d i f f e r e n c e s ty p i c a ll y c o r r e ­ s p o n d t o d i f f e r e n t s o c i a t m e a n i n g s — a n e m p h a s i s o n h is t o r y v e rs u s efficiency, for exam ple. S u p p o s e t h a t v i s i t o r s fr o m a n o t h e r w o r ld w e r e to v ie w t h e g rid o f a n A m e r i c a n city. For a c ity w i t h n u m b e r e d stre ets , t h e v is ito rs (a s s u m in g th e y c o u l d r e a d o u r n u m b e r s ) c o u l d safely c o n c l u d e t h a t A m e r i c a n s m a d e u s e o f a c o o r d i ■n a t e s t r u c t u r e . B u t d o t h e s e A m e r i c a n s a c t u a l l y u n d e r s t a n d c o o r d i n a t e m a t h e ­ m a tic s ? C a n th e y use a c o o r d i n a t e grid t o g r a p h e q u a ti o n s ? j u s t h o w s o p h i s ti c a te d is t h e i r m a t h e m a t i c a l u n d e r s t a n d i n g ? I n t h e f o l l o w i n g c h a p t e r , w e w ill find o u r ­ selves in a sim ilar p o sitio n , for A fric a n s e t t l e m e n t a r c h i te c t u r e is filled w ith r e m a r k ­ a b l e e x a m p l e s o f f r a c t a l s t r u c t u r e . D id p r e c o l o n i a l A f r i c a n s a c t u a l l y u n d e r s t a n d a n d a p p l y f r a c ta l g e o m e t r y ? A s I w ill e x p l a i n in t h i s c h a p t e r , f r a c t a l s a r e c h a r a c t e r i z e d b y t h e r e p e t i ­ t i o n o f s i m i l a r p a t t e r n s a t e v e r - d i m i n i s h i n g sc a le s. T r a d i t i o n a l A f r i c a n s e t t l e ­ m e n t s t y p i c a l l y s h o w t h i s “s e l f - s i m i l a r ” c h a r a c t e r i s t i c : c i r c l e s o f c i r c l e s o f c i r c u l a r d w e l l i n g s , r e c t a n g u l a r w a lls e n c l o s i n g e v e r - s m a l l e r r e c t a n g l e s , a n d s t r e e t s in w h i c h b r o a d a v e n u e s b r a n c h d o w n t o t i n y f o o t p a t h s w i t h s t r i k i n g g e o ­ m e t r i c r e p e t i t i o n . T h e f r a c ta l s t r u c t u r e w ill b e e a s ily i d e n t i f i e d w h e n w e c o m ­ p a r e a e r i a l v ie w s o f t h e s e A f r i c a n v illa g e s a n d c i t i e s w i t h c o r r e s p o n d i n g f r a c ta l graphics sim u latio n s. . . W h a t a r e w e t o m a k e o f t h i s c o m p a r i s o n ? L e t ’s p u t o u r s e l v e s b a c k in th e s h o e s o f t h e v is ito rs fro m a n o t h e r p l a n e t . H a v i n g b e a m e d d o w n t o a n A m e r i c a n s e t t l e m e n t n a m e d “C o rv a llis , O r e g o n , " th e y d is c o v e r t h a t t h e s t re e ts a r e n o t n u m - F ractal geom etry b e r e d , b u t r a t h e r t i t l e d w i t h w h a t a p p e a r t o b e a r b i t r a r y n a m e s : W a s h i n g t o n , Jef­ ferson, A d a m s , a n d so o n . A t first th e y m i g h t c o n c l u d e t h a t th e r e is n o t h i n g m a t h e ­ m a t i c a l a b o u t it. By u n d e r s t a n d i n g a b i t m o r e a b o u t t h e c u l t u r a l m e a n i n g , h o w e v e r* a m a t h e m a t i c a l p a t t e r n d o e s e m e r g e : t h e s e are n a m e s in h is to r i c a l s u c ­ c e s s io n . I t m i g h t b e o n l y o r d e r i n g in t e r m s o f p o s i t i o n in a series ( a n “o r d i n a l ” n u m b e r ) , b u t t h e r e is s o m e k i n d o f c o o r d i n a t e s y s te m a t w o r k a f t e r all. A f r i c a n d esigns h a v e to b e a p p ro a c h e d in th e s a m e way. W e c a n n o t ju st assume t h a t A fric a n f r a c ta ls s h o w a n u n d e r s t a n d i n g o f f r a c ta l g e o m e t r y , n o r c a n w e d ism iss t h a t p o s ­ sibility. W e n e e d to lis te n t o w h a t t h e d e s i g n e r s a n d users o f th e s e s tru c tu r e s h a v e to say a b o u t it. W h a t a p p e a r s to b e a n u n c o n s c i o u s o r a c c i d e n t a l p a t t e r n m i g h t actu a lly h a v e a n in te n tio n a l m a t h e m a tic a l c o m p o n e n t. O v e r a l l , t h e p r e s e n c e o f m a t h e m a t i c s in c u l t u r e c a n b e t h o u g h t o f in t e r m s o f a s p e c t r u m fr o m u n i n t e n t i o n a l t o s e l f - c o n s c io u s . A t o n e e x t r e m e is th e e m e r g e n c e o f c o m p l e t e l y u n c o n s c i o u s s t r u c t u r e s . T e r m i t e m o u n d s , for e x a m p l e , are e x c e l l e n t fra c ta ls ( t h e y h a v e c h a m b e r s w i t h i n c h a m b e r s w i t h i n c h a m b e r s ) b u t n o o n e w o u ld c l a i m t h a t te r m i t e s u n d e r s t a n d m a t h e m a t i c s . I n t h e s a m e way, p a t t e r n s a p p e a r in t h e g ro u p d y n a m i c s o f large h u m a n p o p u l a t i o n s , b u t th e s e are g e n e r a l ly n o t p a t t e r n s o f w h i c h a n y i n d i v i d u a l is a w a r e . U n c o n s c i o u s s t r u c t u r e s d o n o t c o u n t as m a t h e m a t i c a l k n o w l e d g e , e v e n t h o u g h w e c a n u se m a t h e m a t i c s t o d e s c r ib e t h e m . M o v i n g a l o n g th i s s p e c t r u m t o w a r d t h e m o r e i n t e n t i o n a l , w e n e x t find e x a m p le s o f d e c o ra tiv e designs w h ic h , a lt h o u g h consciously created, h a v e n o e x p l i c i t k n o w l e d g e a t t a c h e d t o t h e m . I t is p o s s ib le , for e x a m p l e , t h a t a n a r t i s t w h o d o e s n o t k n o w w h a t t h e w o r d “h e x a g o n ” m e a n s c o u l d s till d ra w o n e w i t h g r e a t p r e c i s i o n . T h i s w o u ld b e a c o n s c i o u s d e s i g n , b u t t h e k n o w l e d g e is s t ric tly im p li c it .'1 I n t h e n e x t s t e p a l o n g o u r s p e c t r u m , p e o p l e m a k e rh e s e c o m p o n e n t s e x p l i c i t — t h e y h a v e n a m e s for t h e p a t t e r n s t h e y o b s e r v e in s h a p e s a n d n u m b e r s . T a k i n g t h e i n t e n t i o n s p e c t r u m o n e m o r e s t e p , w e h a v e ru les fo r h o w th e s e p a t ­ t e r n s c a n b e c o m b i n e d . H e r e we c a n f i n d “a p p l i e d m a t h e m a t i c s . ” O f c o u rs e t h e r e is a w o rld o f d i f f e r e n c e b e t w e e n t h e a p p li e d m a t h o f a m o d e r n e n g i n e e r a n d t h e a p p li e d m a t h o f a s h o p k e e p e r — w h e t h e r o r n o t s o m e t h i n g is i n t e n t i o n a l tells u s n o t h i n g a b o u t its c o m p l e x it y . F in ally we m o v e to “ p u re m a th e m a tic s ," as fo u n d in th e a b s tra c t th e o rie s o f m o d e r n a c a d e m i c m a t h e m a t i c i a n s . P u r e m a t h c a n a l s o b e v e ry s i m p l e — for e x a m p l e , t h e d i s t i n c t i o n b e t w e e n o r d i n a l n u m b e r s (first, s e c o n d , t h i r d ) a n d c a r ­ d i n a l n u m b e r s ( o n e , tw o , t h r e e ) is a n e x a m p l e o f p u r e m a t h . B u t it w o u l d n o t b e e n o u g h f o r p e o p l e in a s o c i e t y s i m p l y t o u se e x a m p l e s o f b o t h ty p es; th e y w o u l d h a v e t o h a v e w o rd s for t h e s e tw o c a t e g o r i e s a n d e x p l i c i t l y r e f l e c t o n a c o m p a r i s o n o f t h e i r p r o p e r t i e s b e f o r e w e w o u l d say t h a t t h e y h a v e a t h e o r y o f In tro d u c tio n 6 th e d is tin c tio n b e tw e e n o rd in a l a nd c a rd in a l n u m b e rs. W h il e app lied m a t h e ­ m a t i c s m a k e s u s e o f ru le s, p u r e m a t h t e l ls us w h y t h e y w o r k — a n d h o w t o f i n d .new o n es. ' ' ' T h i s b o o k b e g i n s by m o v i n g a l o n g t h e s p e c t r u m ju s t d e s c r i b e d . W e w ill s t a r t b y s h o w i n g t h a t A f r i c a n fra c ta ls a re n o t s i m p l y d u e t o u n c o n s c i o u s a c tiv ity . W e will t h e n lo o k a t e x a m p l e s w h e r e t h e y a re c o n s c i o u s b u t i m p l i c i t d e s i g n s , f o llo w e d b y e x a m p l e s in w h i c h A f r i c a n s h a v e d e v i s e d e x p l i c i t ru les fo r g e n e r a t i n g t h e s e p a t t e r n s , a n d fin ally t o e x a m p l e s o f a b s t r a c t t h e o r y in t h e s e i n d i g e n o u s k n o w l ­ ed g e systems. T h e re a s o n for ta k in g s u c h a c a u ti o u s r o u t e c a n b e e xp ressed in te r m s o f w h a t p h i l o s o p h e r K arl P o p p e r c a l l e d “falsifiability." P o p p e r p o i n t e d o u t t h a t e v e r y o n e h a s t h e u r g e t o c o n f i r m t h e i r f a v o r i t e t h e o r i e s ; a n d so w e h a v e t o t a k e p r e c a u t i o n s n o t to li m i t o u r a t t e n t i o n t o s u c c e s s — a t h e o r y is o n l y g o o d if you tr y t o t e s t it fo r failu re . If w e o n ly u s e e x a m p l e s w h e r e A f r i c a n k n o w l e d g e sy s­ t e m s su c c e s sfu lly m a t c h e d f r a c ta l g e o m e t r y , w e w o u ld n o t k n o w its l i m i t a t i o n s . T h e r e are i n d e e d gaps w h e r e t h e fam ily o f t h e o r i e s a n d p r a c t i c e s c e n t e r e d a r o u n d fractal g eo m etry in h i g h - t e c h m a t h e m a t i c s h a s n o c o u n t e r p a r t in tr a d it io n a l Africa. A l t h o u g h s u c h g a p s a re s i g n if i c a n t, t h e y d o n o t i n v a l i d a t e t h e c o m p a r i s o n , b u t r a t h e r p r o v i d e t h e n e c e s s a ry q u a l i f i c a t i o n s t o a c c u r a t e l y c h a r a c t e r i z e t h e i n d i g e ­ n o u s f r a c ta l g e o m e t r y o f A fric a . O ve rv ie w o f th e text F o l l o w i n g t h e i n t r o d u c t i o n t o f r a c ta l g e o m e t r y i n t h e n e x t s e c t i o n , i n c h a p t e r 2 w e w ill e x p l o r e f r a c ta ls in A f r i c a n s e t t l e m e n t s . I t w ill b e c o m e c l e a r t h a t t h e e x p l a n a t i o n o f u n c o n s c i o u s g r o u p a c t i v i t y d o e s n o t fit t h i s c a se . W h e n w e t a l k to th e in d ig e n o u s a rc h ite c ts , th e y are q u ite e x p lic it a b o u t th o s e sa m e fractal f e a t u r e s we o b s e r v e , a n c l u s e s e v e r a l o f t h e b a s i c c o n c e p t s o f f r a c ta l g e o m e t r y in d is c u ssin g th e i r m a te ria l d esigns a n d a ss o c ia te d k n o w le d g e system s. T e rm ite s m ay m a k e fractal a rc h ite c tu re s , b u t th e y d o n o t p a in t a b s tr a c t m o d e ls o f th e s t r u c t u r e o n its- w a lls o r c r e a t e s y m b o ls fo r its g e o m e t r i c p r o p e r t i e s . W h i l e t h e s e i n t r o d u c t o r y e x a m p l e s w o n ’t s e t t l e a ll t h e q u e s t i o n s , w e w ill a t le a s t h a v e e s t a b ­ l i s h e d t h a t t h e s e a r c h i t e c t u r a l d e s i g n s s h o u l d b e e x p l a i n e d by s o m e t h i n g m o r e t h a n u n i n t e n t i o n a l so c ia l d y n a m i c s . I n c h a p t e r 3 w e will e x a m i n e a n o t h e r e x p l a n a tio n : p e r h a p s fractal s e t t l e m e n t ' p a t t e r n s a re n o t u n i q u e to A fric a , a n d w e h a v e sim p ly o b s e r v e d a c o m m o n c h a r a c ­ t e r i s t i c o f all n o n - W e s t e r n a r c h i t e c t u r e s . H e r e t h e c o n c e p t o f d e s i g n t h e m e s b e c o m e im p o rta n t. A n th ro p o lo g is ts h a v e fo u n d th a t th e design th e m e s fo u n d in e a c h c u l t u r e a r e fairly d i s t i n c t — t h a t is, d e s p i t e t h e a r t i s t i c d i v e r s i t y w i t h i n F ractal geometr} e a c h s o c ie ty , m o s t o f t h e c u l t u r e ’s p a t t e r n s c a n b e c h a r a c t e r i z e d b y s p e c ific g e o ­ m e t r i c p r a c t i c e s . W e w ill see t h a t a l t h o u g h f r a c t a l d e s i g n s d o o c c u r o u t s i d e o f A f r i c a ( C e l t i c k n o t s , U k r a i n i a n eggs, a n d M a o r i^ r a f te r s h a v e s o m e e x c e l l e n t e x a m p l e s ) , t h e y a r e n o t e v e r y w h e r e . T h e i- r s t r o n g p r e v a l e n c e in A f r i c a ( a n d in A f r i c a n - i n f l u e n c e d s o u t h e r n I n d i a ) is q u i t e s p e c i fic . C h a p t e r 4 r e t u r n s to th i s e x p l o r a t i o n w i t h f r a c ta l s in A f r i c a n e s t h e t i c d e s i g n . T h e s e e x a m p l e s are i m p o r t a n t for t w o r e a s o n s . F irst, t h e y r e m i n d us t h a t we c a n n o t a s s u m e e x p l i c i t , f o r m a l k n o w l e d g e s i m p l y o n t h e basis o f a p a t t e r n . I n c o n t r a s t to t h e fra c ta l p a t t e r n s o f A f r i c a n s e t t l e m e n t a r c h i t e c t u r e , t h e s e a e s ­ t h e t i c f r a c ta ls, a c c o r d i n g to t h e a r t i s a n s , w e r e m a d e “j u s t b e c a u s e it lo o k s p r e t t y t h a t w ay.” T h e y a re n e i t h e r fo r m a l s y s te m s ( n o ru les to t h e g a m e ) n o r d o t h e a r t i ­ s a n s ’ r e p o r t e x p l i c i t t h i n k i n g (“ I d o n ’t k n o w h o w o r why, it j u s t c a m e to m e ” ). S e c o n d , th e y p ro v i d e o n e p ossib le r o u t e by w h i c h a p a r t i c u l a r s e t o f m a t h e m a t i c a l c o n c e p t s c a m e to b e s p r e a d o v e r a n e n o r m o u s c o n t i n e n t . T r a d e n e t w o r k s c o u l d h a v e d iffu se d t h e fra c ta l a e s t h e t i c a c ro s s A f r i c a , r e i n f o r c i n g a d e s i g n t h e m e t h a t m a y h a v e b e e n s c a t te r e d a b o u t in o t h e r a r e a s o f life. O f co u rs e, s u c h o r i g in sto ries a r e n e v e r c e r t a i n , a n d all t o o easy t o i n v e n t . P a r t 11 o f th is b oo k , s ta rtin g w i t h c h a p t e r 5, p re s e n ts th e e x p lic it d esign m e t h ­ o d s a n d s y m b o li c sy stem s t h a t d e m o n s t r a t e f r a c ta l g e o m e t r y as a n A f r i c a n k n o w l ­ e d g e s y s te m . A s in t h e i n t r o d u c t i o n t o f r a c t a l s in t h e first c h a p t e r , I. w ill a s s u m e th e re a d e r has n o m a th e m a tic s b ac k g ro u n d a n d p rovide a n in tro d u c tio n to any n e w c o n c e p t s a lo n g w i t h t h e A f r i c a n v e rs io n s . W e will see t h a t n o t o n ly in a r c h i ­ t e c t u r e , b u t i n t r a d i t i o n a l h a t r s t y l i n g , t e x t i l e s , a n d s c u l p t u r e , in p a i n t i n g , c a r v ­ in g , a n d i n e r a l w o r k , in r e l i g i o n , g a m e s , a n d p r a c t i c a l c ra f t, i n q u a n t i t a t i v e t e c h n i q u e s a n d s y m b o li c sy s te m s , A f r i c a n s h a v e used t h e p a t t e r n s a n d a b s t r a c t c o n c e p t s o f f r a c ta l geo m e rry . C h a p t e r 10, t h e la st in p a r t n , is t h e r e s u l t o f m y c o l l a b o r a t i o n w i t h a n A f r i c a n p h y s i c is t , P ro f e sso r C h r i s t i a n S i n a D i a t t a . A s p o n s o r for t h e F u l b r i g h t f e l l o w s h i p t h a t e n a b l e d m y f i e ld w o rk in w e s t a n d c e n t r a l A fric a , D r. D i a t t a t o o k t h e i d e a o f i n d i g e n o u s f r a c ta l s a n d r a n w i t h it, m o v i n g us in d i r e c t i o n s t h a t 1 w o u l d n e v e r h a v e c o n s i d e r e d o n m y o w n , a n d s till h a v e y e t t o e x p l o r e fully. In t h e t h i r d a n d final p a r t o f t h i s b o o k w e w ill e x a m i n e t h e c o n s e q u e n c e s o f A f r i c a n f r a c ta l g e o m e t r y : g i v e n t h a t i t d o e s e x is t, w h a t a re its s o c ia l i m p l i c a ­ tio n s ? C h a p t e r 1 1 w ill briefly r e v i e w p r e v i o u s s t u d i e s o f A f r i c a n k n o w l e d g e sy s­ te m s . W e w ill see t h a t a l t h o u g h s e v e r a l r e s e a r c h e r s h a v e p r o p o s e d ideas r e l a t e d t o t h e f r a c ta l c o n c e p t — H e n r y L o u is G a t e s ’s “ r e p e t i t i o n w i t h r e v i s i o n , ” L e o p o l d S e n g h o r ' s ‘“ d y n a m i c s y m m e try ," a n d W i l l i a m Fag g’s " e x p o n e n t i a l m o r p h o l o g y ” are all g oo d ex am ples— th e r e h a v e b e e n specific o bstacles t h a t p rev en te d anthrop olo gists a n d o t h e r s fr o m t a k i n g u p th e s e c o n c e p t s in t e r m s o f A f r i c a n m a t h e m a t i c s . Introduction 8 C h a p t e r i 2 c o v e r s t h e p o l i t i c a l c o n s e q u e n c e s o f A f r i c a n f r a c ta l s . O n t h e o n e h a n d , we w ill find t h e r e is n o e v i d e n c e t h a t g e o m e t r i c f o r m h a s a n y in h e r­ e n t s o c ia l m e a n i n g . I n s e t t l e m e n t d e s i g n , for e x a m p l e , p e o p l e r e p o r t b o t h o p p r e s ­ siv e a n d H b e ra to r y s o c ia l e x p e r i e n c e s w i t h f r a c t a l a r c h i t e c t u r e s . F r a c t a b v e r s u s n o n f r a c t a l ( “E u c l i d e a n " ) g e o m e t r y d o e s n o t im p ly g o o d v e r s u s b a d . O n t h e o t h e r h a n d , p e o p l e d o i n v e s t a b s t r a c t fo r m s w i t h p a r t i c u l a r lo c a l m e a n i n g s . T o tak e a c o n tro v e rsia l ex a m p le , re c e n t U .S . su p rem e c o u rt d e c is io n s d e c la re d th a t v o t i n g d is tr i c ts c a n n o t h a v e “b iz a r r e ” o r “h i g h l y i r r e g u la r ” s h a p e s , a n d s e v e r a l o f th e s e fractal c o n to u r s h a v e b e e n re p la c e d by th e s tr a ig h t lines o f E u c lid e a n fo r m . If f r a c ta l s e t t l e m e n t p a t t e r n s a r e t r a d i t i o n a l for p e o p l e o f A f r i c a n d e s c e n t , a n d E u c l i d e a n s e t t l e m e n t p a t t e r n s fo r E u r o p e a n s , is i t e t h n o c e n t r i c t o in s ist o n o n l y E u c l i d e a n v o t i n g d i s t r i c t lines? C h a p t e r 13 w ill e x a m i n e t h e c u l t u r a l h i s t o r y o f f r a c t a l g e o m e t r y a n d its m a t h e m a t i c a l p re c u r s o rs in E u r o p e . W e w ill s e e t h a t t h e g a p s a r e n o t o n e - s i d e d : j u s t as A f r i c a n s w e r e m i s s i n g c e r t a i n m a t h e m a t i c a l i d e a s i n t h e i r v e r s i o n o f f r a c t a l g e o m e t r y , E u r o p e a n s w e r e e q u a l l y a f f e c t e d by t h e i r o w n c u l t u r a l v ie w s a n d h a v e b e e n s lo w to a d o p t s o m e o f t h e m a t h e m a t i c a l c o n c e p t s t h a t w e r e l o n g c h a m p i o n e d b y A f r i c a n s . I n d e e d , t h e r e is s t r i k i n g e v i d e n c e t h a t s o m e o f t h e s o u r c e s o f m a t h e m a t i c a l i n s p i r a t i o n for E u r o p e a n f r a c t a l s w e r e o f A f r i c a n o r i g i n . T h e f i n a l c h a p t e r w ill m o v e f o r w a r d in t i m e , h i g h l i g h t i n g t h e c o n ­ te m p o ra r y v e rs io n s o f fr a c ta l d e s ig n t h a t h a v e b e e n p ro p o s e d by A f r i c a n a r c h i t e c t s in S e n e g a l , M a l i , a n d Z a m b i a , a n d o t h e r i l l u s t r a t i o n s o f p o s s ib le f r a c ­ ta l futures. B u t to u n d e r s t a n d a ll th i s , w e m u s t first v is it t h e f r a c t a l p ast. A h i s t o r i c a l i n tr o d u c t io n to f r a c t a l g e o m e tr y T h e w o r k o f G e o r g C a n t o r ( 1 8 4 5 - 1 9 1 8 ) , w h i c h p r o d u c e d t h e first f r a c ta l , t h e C a n t o r set (fig. 1.1), p r o v e d to b e t h e b e g i n n i n g o f a n e w o u t l o o k o n infinity. In f in ­ ity h a d l o n g b e e n c o n s i d e r e d s u s p e c t by m a t h e m a t i c i a n s . H o w c a n w e c l a i m to b e u s i n g o n l y e x a c t , e x p l i c i t ru le s if w e h a v e a s y m b o l t h a t v a g u e l y m e a n s “ t h e n u m b e r yo u w o u ld g e t if y o u c o u n t e d f o r e v e r" ? S o m a n y m a t h e m a t i c i a n s , s t a r t ­ in g w i t h A r i s t o t l e , h a d j u s t b a n n e d it o u t r i g h t . C a n t o r s h o w e d t h a t i t w as p o s ­ s i b le t o k e e p t r a c k o f f h e n u m b e r o f e l e m e n t s in a n i n f i n i t e s e t , a n d d i d s o in a d e c e p t i v e l y s i m p l e fa s h io n . S t a r t i n g w i t h a s in g le s t r a i g h t li n e , C a n t o r e r a s e d t h e m i d d l e t h i r d , le a v i n g t w o lin es. H e t h e n c a r r i e d o u t t h e s a m e o p e r a t i o n o n t h o s e t w o lin e s , e r a s i n g t h e i r m i d d l e s a n d l e a v i n g fo u r l i n e s . I n o t h e r w o r d s , h e u s e d a s o r t o f f e e d b a c k lo o p , w i t h t h e e n d re s u lt o f i o n e s t a g e b r o u g h t b a c k as t h e s t a r t i n g p o i n t for t h e n e x t . T h i s t e c h n i q u e is c a l l e d “ r e c u r s i o n . " C a n t o r s h o w e d Fractal geom etry t h a t if t h i s re c u r s iv e c o n s t r u c t i o n w a s c o n t i n u e d fo r e v e r, it w o u l d c r e a t e a n i n f i n i t e n u m b e r o f lin e s , a n d y e t w o u l d h a v e ze ro l e n g t h . N o t o n ly d i d C a n t o r r e i n t r o d u c e i n f i n i t y - a s a p r o p e r o b j e c t o f m a t h e ­ m a t i c a l stu d y , b u t h is r e c u r s i v e c o n s t r u c t i o n c o u l d b e u s e d as a m o d e l for o t h e r " p a t h o l o g i c a l c u rv e s ," s u c h as t h a t c r e a t e d b y H e l g e v o n K o c h in 1 9 0 4 (figs. 1.2, 7 .3). T h e m a t h e m a t i c a l p r o p e r t i e s o f t h e s e fig u re s w e r e e q u a l l y p e r p l e x i n g . S m a l l p o rtio n s lo o k e d ju s t like t h e w h o le , a n d th e s e reflections w ere re p e a te d d o w n t o in f i n i t e s i m a l sc a le s. H o w c o u l d w e m e a s u r e t h e l e n g t h o f t h e K o c h c u r v e ? If FIGURE 1.1 T h e C a n t o r se t In 1877 Georg Cantor came up with the idea of repeatedly subdividing a line to illustrate the concept of an infinite set. This looping technique is called recursion. By specifying that the recursion continues forever, C antor was able to define an infinite set. 9 F I G U R E 1. 2 T h e K o c h c u rv e Helge v o n Koch used the same k ind of recursive loop as C antor, b u t he added lines instead of erasing them . He began w ith a triangular shape made of four line^, th e “seed." He t h e n replaced each o f the lines w ith a reduced version o f the original seed shape. b a F I G U R E 1 .3 K o c h curve v a r ia tio n s B^ iflt There is nothing special about the particular shape Koch first used. For example, we can make similar shapes that are more flat or more spiked losing variations on the seed shape (a). Nor is there anything special about tridngles— any shape can'undergo this recursive replacement process. Mathematician Giuseppe Peano, for example, experimented with rectangular seed shapes such as those in (b). Introduction w e h o l d u p a s i x - i n c h r u l e r t o t h e c u r v e (fig. 1.4) w e g e t six i n c h e s , b u t o f c o u r s e t h a t m isse s all t h e n o o k s a n d c r a n n i e s . If w e u s e a s m a l l e r ru ler, w e g e t g r e a t e r l e n g t h , a n d w i t h a s m a l l e r o n e e v e n g r e a t e r l e n g t h , a n d so oix,to i n f i n ­ ity. O b v i o u s l y t h i s is n o t a v e ry usefu l w ay t o c h a r a c t e r i z e o n e o f t h e s e c u r v e s . A n e w w ay o f t h i n k i n g a b o u t m e a s u r e m e n t w as n e e d e d . T h e a n s w e r w as to p l o t th e s e d if fe re n t m ea s u res o f ru le r size versus l e n g t h , a n d see h o w fast w e g a in l e n g t h as w e s h r i n k t h e r u l e r (fig. 1 .5 ). T h i s r a t e ( t h e s l o p e ) t e l ls us j u s t h o w c r i n k l e d Ik * o r t o r t u o u s t h e c u r v e is. F o r e x t r e m e l y c r i n k l e d c u r v e s , t h e p l o t w ill s h o w t h a t w e r a p i d ly g a i n l e n g t h as w e s h r i n k t h e ru ler, s o it w ill h a v e a s t e e p s l o p e . For r e l a ti v e ly s m o o t h c u r v e s , yo u d o n ’t g a in m u c h l e n g t h as y o u s h r i n k t h e ru l e r size, so t h e p l o t h a s a s h a l l o w slop e. T o m a t h e m a t i c i a n s t h i s s l o p e w a s m o r e t h a n j u s t a p r a c t i c a l w ay t o c h a r ­ acte riz e crin k le s. R e c a l l t h a t w h e n we first tr ie d to m e a s u r e t h e l e n g t h o f t h e K o c h c u r v e , w e f o u n d t h a t its l e n g t h w as p o t e n t i a l l y in f in i te . Y et t h i s i n f in i te l e n g t h fits i n t o a b o u n d e d sp ace . M a t h e m a t i c i a n F elix H a u s d o r f f ( 1 8 6 8 - 1 9 4 2 ) fo u n d t h a t t h i s p a r a d o x c o u l d b e r e s o l v e d if w e t h o u g h t o f t h e p a t h o l o g i c a l c u r v e s as s o m e ­ h o w t a k i n g u p m o r e t h a n o n e d i m e n s i o n , as all n o r m a l lin e s d o , b u t less t h a n tw o d im e n s i o n s , as flat sh a p e s iike squares a n d c ircle s do. I n H a u s d o r f f ’s view, t h e K o c h c u r v e h a s a f r a c t i o n a l d i m e n s i o n , a p p r o x i m a t e l y 1.3, w h i c h is t h e s l o p e o f o u r r u l e r - v e r s u s - l e n g t h p l p t . B e in g p u r e m a t h e m a t i c i a n s , t h e y w e r e f a s c i n a t e d w i t h t h i s id e a o f a f r a c t i o n a l d i m e n s i o n a n d n e v e r t h o u g h t a b o u t p u t t i n g 'it--to p r a c ­ t i c a l use. T h e c o n c e p t u a l l e a p to p ra c t ic a l a p p l i c a t i o n w as c r e a t e d by B e n o i t M a n d e l ­ b r o t (b. 1924), w h o h a p p e n e d u p o n a stu dy o f lo n g - te rm river flu c tu a tio n s by British c iv il s e r v a n t H . E. H u r s t . H u r s t . h a d f o u n d t h a t t h e y e a rly flo o d s o f riv e rs d id n o t h a v e a n y o n e a v e r a g e , b u t r a t h e r v a r i e d o v e r m a n y d i f f e r e n t s c a l e s — t h e r e w e re flood y ears, flood d e c a d e s , e v e n flood c e n t u r i e s . H e c o n c l u d e d t h a t t h e o n l y way t o c h a r a c t e r i z e t h i s t e m p o r a l w ig g lin e ss w as t o p l o t t h e a m o u n t o f f l u c t u a t i o n a t e a c h s c a l e a n d use t h e s l o p e o f t h i s li n e . M a n d e l b r o t re a l iz e d t h a t t h i s w as e q u i v a l e n t t o t h ^ k i n d o f s c a l in g m e a s u r e t h a t h a d b e e n , u se d for C a n t o r ’s p a t h o ­ lo g ic a l c u r v e s . A s h e b e g a n to a p p l y c o m p u t e r g r a p h i c s (figs. 1.6, 1 .7 ), h e f o u n d t h a t t h e s e s h a p e s w e r e n o t p a t h o l o g i c a l a t all, b u t r a t h e r v e ry c o m m o n t h r o u g h ­ o u t t h e n a tu r a l w o rld . M o u ntain* ran g es h a d p e a k s w ith in peak s, trees h a d b r a n c h e s m a d e o f b r a n c h e s , c lo u d s we,re puffs w i t h i n p u f f s ^ - e V e n h is o w n b o d y w a s full o f re c u r s iv e s c a l i n g s t r u c t u r e s . T h e f r a c t a l s i m u l a t i o n s fo r n a t u r a l o b j e c t s i n fig u r e 1.7 w e r e c r e a t e d j u s t lik e t h e C a n t o r set, K o c h c u r v e , a n d o t h e r e x a m p l e s w e h a v e a lr e a d y se e n , w i t h a s e e d s h a p e t h a t u n d e r g o e s re c u r s iv e r e p l a c e m e n t . T h e o n l y d i f f e r e n c e is t h a t som e o f th ese s im u la tio n s require th a t c e rta in lines in th e seed sh a p e do n o t get 6 inches 6 inches FIGURE I . 4 M e a s u r in g th e le n g th o f fr a c ta l c u r v e s m for T he new curves of Rancor, Koch, an d others represented a problem in m easurem ent theory. T he length of the curve d epen ds o n th e size of the ruler. As we shrink the ruler dow n, the length approaches infinity. t OJ > U O •£ So c “O 4u> D « e sm aller ruler s i z e >• FIGURE 1 . 5 A b e t t e r w a y to m e a s u r e f r a c t a l c u r v e s Our experiment in shrinking rulers wasn’t a total waste. In fact, it turns out that if you keep track of how the measured length changes with ruler size, you get a very good way of characterizing the curve. A relatively smooth fractal won’t increase length very quickly with shrinking ruler size, but very crinkled fractals will, (a) This smooth Koch curve doesn’t add much length with shrinking ruler size, so the plot shows only a small rise, (b) Since a small ruler can get into all the nooks and crannies, this more crinkled Koch curve shows a steeper rise in measured length with a shrinking ruler, (c) An extremely tortuous Koch curve has a very sreep slope for its plot. N o te f o r m ath sticklers: T h e s e figures are p lo tted o n 0 logarithm ic graph. F ractal geom etry 15 r e p l a c e d . T h i s is i l l u s t r a t e d fo r t h e l u n g m o d e l a t t h e b o t t o m o f fig u re 1.7. T h e l i n e s t h a t g e t r e p l a c e d in e a c h i t e r a t i o n a r e c a l l e d “a c t i v e l i n e s . ” T h o s e t h a t d o n o t get replaced are called “passive lines.” W e w ill bemusing t h e d is ti n c ti o n b e tw e e n a c t i v e a n d p a s s iv e l i n e s in s i m u l a t i o n s fo p - A f r ic a n d e s i g n s as well. M a n d e l b r o t c o i n e d t h e t e r m " f r a c t a l ” fo r t h i s n e w g e o m e t r y , a n d it is n o w u se d in e v e ry s c i e n ti fic d i s c i p l i n e fr o m a s t r o p h y s i c s t o zoology. I t is o n e o f th e m o s t p o w erfu l to o ls for t h e c r e a t i o n o f n e w t e c h n o l o g i e s as well as a re v o l u ti o n a ry a p p r o a c h t o t h e a n a ly s is o f t h e n a t u r a l w o r l d . I n m e d i c i n e , fo r e x a m p l e , fra c ta l South Africa Fractal dimension *» 1 .0 0 Smooth Koch curve Fractal dimension = 1.1 Great Britain Fractal dimension = 1 .2 5 Rough Koch curve Fractal dimension = 1.3 Norway Fractal dimension = 1.5 2 T o r t u o u s Koch curve Fractal dimension « 1.5 FIGURE 1.6 M e a s u r i n g n a t u r e u d t h f r a c t a l g e o m e tr y Although the curves of Cantor and others were introduced as abstractions without physical meaning, Benoit Mandelbrot realized chat their scaling measure, which he called “fractal dimension,” could be put to practical use in characterizing irregular shapes in nature. The classic example is the measurement, of coastlines. Even though it is a very crude model, we can see how the variations of the roughness in the Koch curve are similar to the variations in these coasts. Note that tire fractal dimension is our plot slope from figure a.5; the coastlines were measured in the same way. acacia tree clouds shell fern This vertical line is passive. These two horizontal lines (gray) are the active lines that will be replaced by a reduced version of this seed shape- we see that only the active lines were replaced; the passive line remains the same. Now there are three passive lines (center) and four active lines (the ends). By the eighth iteration we can sec the similarity to the scaling structure of the human lungs. . F I G U R E 1-7 S i m u l a t i n g n a t u r e w i t h f r a c t a l g e o m e tr y In his experiments with computer graphics, Mandelbrot found that fractal shapes abound in nature, where continual processes such as biological growth, geological change, and atmospheric turbulence result in a wide variety of recursive scaling structures (a). T he recursive construction of these natural shapes is basically the same as that of the other fractal shapes we have seen so far. In some examples, like the lung model (b), certain lines of the original seed shape do not participate in the replacement step; they are called "passive lines.” T he ones which do go through replacement are called "active lines." Each step is referred to as an “iteration." Fractal geom etry d i m e n s i o n c a n be used as a d iag n o stic tool. A h e a l t h y lu n g h a s a h ig h fractal d i m e n ­ s i o n , b u t w h e n b l a c k l u n g d is e a s e b e g i n s it lo ses s o m e o f t h e fine b r a n c h i n g — a c o n d i t i o n t h a t c a n be d e t e c t e d by m e a s u r i n g t h e f r a c ta l d i m e n s i o n o f t h e X ray. F o r t h i s r e a s o n , B e n o i t M a n d e l b r o t w as r e c e n t l y n a m e d a n h o n o r a r y m e m b e r o f th e F re n c h C o al M in ers U n io n . O f c o u rs e , n o r e v o l u t i o n is w i t h o u t its c o u n t e r r e v o l u t i o n a r i e s . I t w as n o t l o n g b e f o r e s o m e s c i e n t i s t s s t a r t e d o b j e c t i n g t h a t M a n d e l b r o t w as ig n o r i n g th e p r e s e n c e o f t h e n a t u r a l o b j e c t s t h a t c o u l d b e d e s c r i b e d by E u c l i d e a n g e o m e try , s u c h as c ry s ta ls o r eggs. I t ’s tr u e t h a t n o t a ll o f n a t u r e is f r a c ta l— a n d th i s w ill b e a n i m p o r t a n t p o i n t f o r us t o k e e p in m i n d . S o m e w r i t e r s h a v e m i s t a k e n l y a t t e m p t e d t o p o r t r a y A f r i c a n s as “m o r e n a t u r a l ”— a d a n g e r o u s a n d m i s l e a d i n g c l a i m , e v e n w h e n m a d e by w e l l - m e a n i n g r o m a n t i c s . S i n c e fra c ta ls are a s s o c ia te d w i t h n a t u r e , a b o o k a b o u t “ A f r i c a n f r a c t a l s ” c o u l d b e m i s i n t e r p r e t e d as s u p p o r t fo r s u c h r o m a n t i c o rg a n i c is t s . P o i n t i n g o u t t h a t s o m e E u c l i d e a n s h a p e s e x i s t in t h e r e a l m o f n a t u r e m a k e s i t e a s i e r t o u n d e r s t a n d t h a t A f r i c a n f r a c ta ls a re fr o m t h e artific ia l r e a l m o f c u l t u r e . B e f o r e m o v i n g o n t o t h e s e A f r i c a n d e s ig n s , l e t ’s re v i e w t h e b a sic c h a r a c t e r i s t i c s o f f r a c ta l g e o m e t r y . F i v e e s s e n t ia l c o m p o n e n t s o f f r a c ta l g e o m e t r y RECURSION W e h a v e s e e n t h a t fra c ta ls are g e n e r a t e d b y a c i r c u l a r p ro c e s s , a lo o p in w h i c h t h e o u t p u t a t o n e sta g e b e c o m e s t h e i n p u t fo r t h e n e x t . R e s u lt s a re r e p e a t e d l y r e t u r n e d , so t h a t t h e sam e o p e r a t i o n c a n b e c arrie d o u t again . T h i s is o ft e n referred t o as “ re c u r s io n ,” a very p o w e rfu l c o n c e p t . L a t e r w e will d is tin g u is h b e t w e e n th r e e • u i f f c . e n i ty p e s o f r e c u r s io n , b u t f o r n o w j u s t t h i n k o f i t in te r m s o f t h i s ic e r a tiv e f e e d b a c k lo o p . W e ’ve a lr e a d y s e e n h o w i t e r a t i o n w o r k s t o c r e a t e t h e C a n t o r s et a n d t h e K o c h cu rv e . A l t h o u g h w e c a n c r e a t e a m a t h e m a t i c a l a b s t r a c t io n in w h i c h t h e r e c u r s io n c o n t i n u e s forever, t h e r e a re also cases w h e r e t h e r e c u r s io n w ill “b o t ­ t o m o u t . ” In o u r g e n e r a t i o n o f t h e K o c h c u r v e , for e x a m p l e , we q u i t o n c e th e lines g e t t o o s m a ll to p r i n t . I n fa c t, a n y p h y s i c a l l y e x i s t i n g o b j e c t w ill o n l y b e fractalw i t h i n a p a r t i c u l a r r a n g e o f scales. SCALING If yo u lo o k a t t h e c o a s t l i n e o f a c o n t i n e n t — ta k e th e P acific side o f N o r t h A m e r ­ ica fo r i n s t a n c e — y ou will see a ja g g e d s h a p e , a n d if y o u lo o k a t a s m a ll p i e c e o f t h a t c o a s t l i n e — say, C a l i f o r n i a — w e c o n t i n u e t o see s i m i l a r ja g g e d n e s s . I n fa ct, a s i m i l a r ja g g e d c u r v e c a n b e s e e n s t a n d i n g o n a cliff o v e r l o o k i n g a ro ck y C a l i ­ fo r n ia sh o r e , o r e v e n s t a n d i n g o n t h a t s h o r e lo o k i n g a t o n e rock. O f co urse, t h a t ’s Intro d u ctio n i8 o n ly ro u g h ly sim ilar, a n d i t ’s o n l y g o o d for a c e r t a i n r a n g e o f scales, b u t it is a s t o n ­ is h in g t o rea lize h o w w e ll t h i s w o r k s for m a n y n a t u r a l f e a t u r e s . It is t h i s " s c a l ­ in g ” p ro p e r ty o f n a tu r e t h a t a llo w s fra c ta l g e o m e try to b e so e ffe c tiv e for m o d e l in g . T o h a v e a " s c a li n g s h a p e ” m e a n s t h a t t h e r e a r e s i m i l a r p a t t e r n s a t d if ­ f e r e n t s c a le s w i t h i n t h e r a n g e u n d e r c o n s i d e r a t i o n . E n l a r g i n g a t i n y s e c t i o n w ill p r o d u c e a p a t t e r n t h a t lo o k s s i m i l a r t o t h e w h o l e p i c t u r e , a n d s h r i n k i n g d o w n t h e w h o l e w ill g iv e us s o m e t h i n g t h a t lo o k s li k e a t i n y p a r t . SELF-SIMILARITY J u s t h o w s i m i l a r d o t h e s e p a t t e r n s h a v e t o b e t o q u a li fy as a fr a c ta l ? M a t h e ­ m a t ic i a n s d is tin g u is h b e t w e e n s t a tis tic a l self-sim ilarity, as in t h e c a s e o f t h e c o a s t ­ l i n e , a n d e x a c t s e l f - s i m i l a r i t y , as i n t h e c a s e o f t h e K o c h c u r v e . I n e x a c t s e l f - s im i la r it y w e n e e d t o b e a b l e t o s h o w a p r e c i s e r e p l i c a o f t h e w h o l e in a t l e a s t s o m e o f its p a r t s . I n t h e K o c h c u r v e a p r e c i s e r e p l i c a o f t h e w h o l e c o u l d b e f o u n d w i t h i n a n y s e c t i o n o f t h e f r a c t a l ( " s t r i c t l y s e l f - s i m i l a r ” ), b u t t h i s i s n ’t t r u e fo r all f r a c ta l s . T h e b r a n c h i n g f r a c t a l s u s e d t o m o d e l t h e lu n g s a n d a c a c i a tr e e (fig. 1 .7), fo r e x a m p l e , h a v e p a r t s (e .g ., t h e s t e m ) t h a t d o n o t c o n t a i n a tin y im a g e o f t h e w h o l e . U n l i k e t h e K o c h c u r v e , t h e y w e r e n o t g e n e r a t e d by r e p l a c ­ in g e v e r y l i n e in t h e s e e d s h a p e w i t h a m i n i a t u r e v e r s i o n o f t h e s e e d ; i n s t e a d , w e u s e d s o m e p a s s iv e li n e s t h a t w e r e j u s t c a r r i e d t h o u g h t h e i t e r a t i o n s w i t h o u t c h a n g e , in a d d i t i o n t o a c t i v e l i n e s t h a t c r e a t e d a g r o w i n g t i p b y t h e u s u a l recu rsiv e re p la c e m e n t. INFINITY S i n c e f r a c ta ls c a n b e l i m i t e d t o a fi n it e r a n g e o f sca les, i t m a y s e e m li k e in f in ity is ju st a h is to rical artifact, a t best a H o ly G r a i l w h o se q u e s t allo w e d m a t h e m a t i c i a n s s e r e n d ip it o u s l y t o s t u m b l e acro ss fractals. It is t h i s k i n d o f o m i s s i o n t h a t h a s m a d e m a n y p u r e m a t h e m a t i c i a n s r a t h e r n o n p l u s s e d a b o u t t h e w h o l e f r a c ta l affair, a n d in s o m e c ases d o w n r i g h t h o s t i l e (cf. K r a n t z 1 9 8 9 ) . T h e r e is n o w ay to c o n ­ n e c t fractals t o t h e idea o f d i m e n s i o n w i t h o u t u s i n g in fin ity , a n d for m a n y m a t h ­ e m a t i c i a n s t h a t is t h e i r c r u c i a l ro le. FRACTIONAL DIMENSION H o w c a n it be t h a t t h e K o c h c u r v e , o r a n y m e m b e r o f its f r a c t a l fam ily, h a s in f i­ n i t e l e n g t h i n a fi n it e b o u n d a r y ? W e a r e u s e d to t h i n k i n g o f d i m e n s i o n as o n l y w h o l e n u m b e r s — t h e o n e - d i m e n s i o n a l li n e , t h e t w o - d i m e n s i o n a l p l a n e — b u t t h e t h e o r y o f m e a s u r e m e n t t h a t g o v e r n s fracta ls allow s d i m e n s i o n s t o be frac tion s. C o n s id e r , for e x a m p l e , t h e i n c r e a s i n g d i m e n s i o n o f t h e K o c h c u rv e s in figure 1.6. A b o v e t h e t o p , we c o u ld g o as c lo s e as we like t o a o n e - d i m e n s i o n a l lin e. B elo w F ractal geometry t h e b o t t o m , we c o u ld m a k e t h e c u r v e s o j a g g e d t h a t it s t a r t s to fill i n tw o d im e n sio n a l areas of th e p la n e . In b e tw e e n , w e n e e d a n in -b e tw e e n d im e n sio n . L o o k in g fo r f r a c ta ls in A f r ic a n c u l t u re A s w e e x a m i n e A f r i c a n d e s i g n s a n d k n o w l e d g e sy s tem s , t h e s e five e s s e n t i a l c o m p o n e n t s will b e a u se fu l w ay t o k e e p t r a c k o f w h a t d o e s o r d o e s n o t m a t c h f r a c ta l g e o m e try . S i n c e s c a l i n g a n d s e lf- s im i la r it y a r e d e s c r i p t i v e c h a r a c t e r i s t i c s , o u r first s te p will b e to lo ok for th e s e p r o p e r tie s in A f r i c a n designs. O n c e we e s t a b ­ lish t h a t t h e m e , we c a n ask w h e t h e r o r n o t th e s e c o n c e p t s h a v e b e e n in t e n ti o n a ll y a p p li e d , a n d s t a r t to lo o k for t h e o t h e r t h r e e e s s e n t i a l c o m p o n e n t s . W e w ill n o w t u r n t o A f r i c a n a r c h i t e c t u r e , w h e r e w e find s o m e o f t h e c l e a r e s t i l l u s t r a t i o n s o f i n d i g e n o u s se l f - s im i la r d esig n s. 19 CHAPTER -* * » ■ , : 2 _ in--------------------------------------------------—A frican-----------------------------------------------settlem ent--------------------------------------------architecture— ^------------------------- -------------- A rc h ite c tu r e o fte n p ro v id es e x c e lle n t e x a m p le s o f c u ltu ra l d e sig n th em es, b e c a u s e a n y t h i n g t h a t is g o i n g t o b e so m u c h a p a r t o f o u r l i v e s — a s t r u c t u r e t h a t m a k e s u p o u r b u i l t e n v i r o n m e n t , o n e in w h i c h w e w ill liv e , w o r k o r p la y — is likely t o h a v e its d e s i g n i n f o r m e d by o u r s o c i a l c o n c e p t s . T a k e re l ig i o u s a r c h i ­ t e c t u r e for e x a m p l e . S e v e i a l c h u r c h e s h a v e b e e n u u i l t u s i n g a t r i a n g u l a r f l o o r p la n to sym bolize th e C h r i s t i a n tr in ity ; o th e r s h a v e used a cross s h a p e . T h e R o m a n P a n t h e o n was d iv id e d in to th r e e v e rtic a l lev els: th e b o t t o m w ith s e v e n n i c h e s r e p r e s e n t i n g t h e h e a v e n l y b o d i e s , t h e m i d d l e w i t h t h e i 2 z o d ia c signs, a n d o n to p a h e m i s p h e r e s y m b o liz in g th e o r d e r o f t h e c o s m o s as a w h o l e . 1 B u t w e d o n ’t n e e d to lo o k t o g r a n d i o s e m o n u m e n t s ; e v e n t h e m o s t m u n ­ d a n e s h a c k w ill i n v o l v e g e o m e t r i c d e c i s i o n s — s h o u l d it b e s q u a r e o r o b l o n g ? p i t c h e d r o o f o r f la t? f a c e n o r t h o r w e s t ? — a n d so c u l t u r e w ill p l a y a r o l e h e r e a s w ell. A t first g l a n c e A f r i c a n a r c h i t e c t u r e m i g h t s e e m so v a r i e d t h a t o n e w o u ld c o n c l u d e its s t r u c t u r e s h a v e n o t h i n g in c o m m o n . A l t h o u g h t h e r e is g r e a t d i v e r ­ sity a m o n g t h e m a n y c u l t u r e s o f A f r i c a , e x a m p l e s o f f r a c ta l a r c h i t e c t u r e c a n be f o u n d in e v e r y c o r n e r o f t h e A f r i c a n c o n t i n e n t . N o t all a r c h i t e c t u r e in A f r i c a is f r a c ta l— f r a c ta l g e o m e t r y is n o t t h e o n l y m a t h e m a t i c s u s e d in A f r i c a —-hu t its r e p e a t e d p r e s e n c e a m o n g s u c h a w id e v a r i e t y o f s h a p e s is q u i t e s t r ik i n g . Fractals in A fr ic a n se ttle m en t architecture In e a c h c a s e p r e s e n t e d h e r e w e w ill c o m p a r e t h e a e r i a l p h o t o o r a r c h i t e c ­ t u r a l d i a g r a m o f a s e t t l e m e n t t o a c o m p u t e r - g e n e r a t e d f r a c ta l m o d e l . T h e f r a c ­ ta l s i m u l a t i o n w ill m a k e t h e s e l f - s im i la r a s p e c t s o f t h e p h y s i c a l s t r u c t u r e m o r e e v i d e n t , a n d in s o m e cases it will e v e n h e l p us u n d e r s t a n d t h e lo cal c u ltu ra l m e a n ­ in g o f t h e a r c h i t e c t u r e . S i n c e t h e A f r i c a n d e s i g n e r s u s e d t e c h n i q u e s lik e i t e r a ­ t i o n in b u i l d in g th e s e s t ru c tu r e s , o u r v ir tu a l c o n s t r u c t i o n t h r o u g h frac tal g rap h ics w ill g iv e us a c h a n c e to s e e h o w t h e p a t t e r n s e m e r g e t h r o u g h t h i s p ro cess. R e c t a n g u l a r f r a c t a l s in s e t t l e m e n t a r c h i t e c t u r e I f y ou fly o v e r t h e n o r t h e r n p a r t o f C a m e r o o n , h e a d i n g t o w a r d L a k e C h a d a lo n g t h e L o g o n e R iv e r , y ou will s e e s o m e t h i n g lik e figure 2 .1 a. T h i s a erial p h o t o s h o w s t h e c ity o f L o g o n e - B ir n i in C a m e r o o n . T h e K o t o k o p e o p le , w h o f o u n d e d th is city c e n t u r i e s a g o , u se t h e lo c a l c la y t o c r e a t e h u g e r e c t a n g u l a r b u i l d i n g c o m p l e x e s . T h e la r g e st o f t h e s e b u il d in g s , in t h e u p p e r c e n t e r o f t h e p h o t o , is t h e p a l a c e o f t h e ch ief, o r “M i a r r e ” (fig. 2 .1 b ). E a c h c o m p l e x is c r e a t e d by a pro cess o f t e n c alle d “a r c h i te c t u r e by a c c r e t io n , ” in th is case a d d in g r e c t a n g u la r en closu res t o p reex istin g r e c t a n g l e s . S i n c e n e w e n c l o s u r e s o f t e n i n c o r p o r a t e t h e w a lls o f t w o Qt m o r e . o f t h e o ld o n e s , e n c l o s u r e s t e n d to g e t l a r g e r a n d la r g e r as yo u g o o u t w a r d f r o m t h e c e n te r . T h e e n d re s u lt is t h e c o m p l e x o f r e c t a n g l e s w i t h i n r e c t a n g l e s w i t h i n r e c ­ t a n g l e s t h a t we see in t h e p h o t o . S i n c e t h i s a r c h i t e c t u r e c a n be d e s c r ib e d in te r m s o f s elf-s im ilar s c a l in g — it m a k e s use o f t h e sa m e p a t t e r n a t sev eral differe nt scales— it is easy to sim u la te using a c o m p u te r- g e n e ra te d fractal, as we see in figures 2 . i c - e . T h e seed shape o f th e m o d el is a r e c t a n g le , b u t e a c h sid e is m a d e u p o f b o t h a c t i v e lin e s (gray) a n d passive lines (b la c k ). A f t e r . t h e first it e r a t io n we see h o w a sm all v e rs io n o f th e original re c ta n g le is re p r o d u c e d by e a c h o f t h e a c ti v e lines. O n e m o r e it e r a t io n gives a ra n g e o f scales t h a t is a b o u t t h e sa m e as t h a t o f t h e p a la c e ; t h i s is e n la r g e d in figure 2 . i e . D u r i n g m y v isit t o L o g o n e - B i r n i in t h e s u m m e r o f 1 9 93 , t h e M i a r r e k in d l y a llo w e d m e t o c l i m b o n t o t h e p a la c e r o o f a n d ta k e th e p h o t o s h o w n in figure 2 . if. 1 asked sev eral o f th e K o to k o m e n a b o u t th e v a r i a ti o n in scale o f t h e i r a rc h ite c tu re . T h e y e x p l a i n e d it in t e r m s o f a c o m b i n a t i o n o f p a t r i l o c a l h o u s e h o l d e x p a n s i o n , a n d t h e h i s t o r i c n e e d for d e f e n s e . " A m a n w o u l d lik e h i s s o n s t o live n e x t to h i m , " t h e y s a id , “a n d so w e b u i l d by a d d i n g w alls t o t h e f a t h e r ’s h o u s e . ” I n t h e p ast, i n v a s i o n s b y n o r t h e r n m a r a u d e r s w e re c o m m o n , a n d so a la rg e r d e f e n s iv e w a ll w a s a ls o n e e d e d . S o m e t i m e s t h e a s s e m b ly o f f a m i l i e s w o u ld o u t g r o w th i s d e f e n s iv e e n c l o s u r e , a n d so th e y w o u ld t u r n t h a t w all i n t o h o u s i n g , a n d b u ild a n e v e n la rg e r e n c l o s u r e a r o u n d it. T h e s e s c a l i n g a d d i t i o n s c r e a t e d t h e t r a d i t i o n o f s e l f - s im i la r s h a p e s we s till see to d a y , a l t h o u g h t h e p o p u l a t i o n is far b e lo w t h e a. A n aerial view of the city of Logone-Birni in Cameroon. The largest building complex, in the center, is the palace of the chief. b. A closer view of the palace. T he smallest rectangles, in the center, are the royal chambers. Pho to co urtesy M usee de I ' H o m m e , Paris. c. Seed shape for the fractal simulation'of the palace. T he active lines, in gray, will be replaced by a scaleddown replica of the entire seed. e. Enlargment of the third iteration. d. First three iterations of the fractal simulation. FIGURE 2.1 L o g o n e -B irn i (figure c o n t i n u e s ) f. Photo by the author taken from the roof of the palace. g. The guti, the royal insignia, painted on the palace walls. 13y permission Le chemin de la lumiere h. T he spiral path taken by visitors to the throne. B y permission o f L e b e u f 1 969. of Lebeuf 1969. F I G U R E 2 . 1 (continued.) In s id e L o g o n e-B irn i In tro d u c tio n o r i g i n a l 1 8 0 ,0 0 0 e s t i m a t e d Tor L o g o n e - B i r n i ’s p e a k in t h e n i n e t e e n t h c e n tu r y . A t t h a t t i m e t h e r e w a s a g i g a n t i c w a ll, a b o u t 10 f e e t t h i c k , t h a t e n c l o s e d t h e p erim eter o f th e e n tire settlem en t. T h e w o m e n I s p o k e w i t h w e r e m u c h less i n t e r e s t e d i n e i t h e r p a t r i l i n e a g e o r m ilita r y h is to ry ; t h e i r re s p o n s e s c o n c e r n i n g a r c h i t e c t u r a l s c a l in g w e r e prim arily a b o u t t h e c o n t r a s t b e t w e e n t h e ra w e x t e r i o r w alls a n d t h e s t u n n i n g w a t e r p r o o f fi n is h t h e y c r e a t e d f o r c o u r t y a r d s a n d i n t e r i o r r o o m s . T h i s b e g a n b y s m o o t h i n g w e t w a lls f l a t . w i t h s p e c i a l s t o n e s , a p p l y i n g a r e s in c r e a t e d f r o m a p l a n t e x t r a c t , a n d t h e n a d d i n g . b e a u t i f u l l y a u s t e r e d e c o r a t i v e lin e s . T h e m o s t i m p o r t a n t o f t h e s e d e c o r a t i v e d r a w in g s is t h e guri, a royal insig nia (fig. 2.1 g). T h e c e n t r a l m o t i f o f t h e g u ti s h o w s a re c t a n g le in s id e a r e c t a n g l e inside a r e c t a n g l e ; it is a k i n d o f a b s t r a c t m o d e l t h a t t h e K o t o k o t h e m s e l v e s h a v e c r e ­ a t e d . T h e r e a s o n f o r c h o o s i n g s c a l i n g r e c t a n g l e s as a s y m b o l o f r o y a l t y b e c o m e s c le a r w h e n w e lo o k a t t h e passage t h a t o n e m u s t t a k e to v is it t h e M i a r r e (fig. 2 . i h ) . T h e p assag e as a w h o l e is a r e c t a n g u l a r spiral. E a c h t i m e y o u e n t e r a s m a ll e r scale, y o u a re r e q u i r e d t o b e h a v e m o r e p o lite ly . By t h e t i m e y o u a r r i v e a t t h e t h r o n e y o u a r e s h o e l e s s a n d s p e a k w i t h a v e r y c u l t u r e d f o r m a l i t y . 2 T h u s t h e f r a c ta l s c a l i n g o f t h e a r c h i t e c t u r e is n o t s i m p l y t h e re s u lt o f u n c o n s c i o u s s o c i a l d y n a m ­ ics; it is a su b je c t o f a b s t r a c t r e p r e s e n t a t i o n , anti e v e n a p r a c t ic a l t e c h n i q u e a p p lied to social ran k in g . T o th e w est n ear th e N ig e ria n b order th e landscape o f C a m e ro o n becom es m u c h g r e e n e r ; t h i s is t h e f e r tile h i g h g ra s sla n d s r e g i o n o f t h e B a m i le k e . T h e y to o h a v e a f r a c t a l s e t t l e m e n t a r c h i t e c t u r e b a s e d o n r e c t a n g l e s (fig. 2 . 2 a ) , b u t i t h a s n o c u l t u r a l r e l a t i o n to t h a t o f t h e K o t o k o . R a t h e r t h a n t h e t h i c k c la y o f L o g o n e B ir n i , t h e s e h o u s e s a n d t h e a t t a c h e d e n c l o s u r e s a r e b u i l t f r o m b a m b o o , w h i c h , is v e r y s t r o n g a n d w id e ly a v a i l a b l e . A n d t h e r e w a s n o m e n t i o n o f k i n s h i p , d e f e n s e , o r p o l i t i c s w h e n I a s k e d a b o u t t h e a r c h i t e c t u r e ; h e r e 1 w a s t o l d it is p a t ­ t e r n s o f a g r i c u l t u r a l p r o d u c t i o n t h a t u n d e r l i e t h e s c a l in g . T h e g r a s s l a n d so il a n d c l i m a t e a re e x c e l l e n t for f a r m i n g , a n d t h e g a r d e n s n e a r t h e B a m i l e k e h o u s e s t y p ­ ic ally g ro w a d o z e n d if fe re n t p l a n t s all in a s in g le s p a c e , w i t h e a c h - t a k i n g its c h a r ­ a c t e r i s t i c v e r t i c a l p l a c e . B u t t h i s is l a b o r i n t e n s i v e , a n d s o m o r e d i s p e r s e d p l a n t i n g s — ro w s o f c o r n a n d g r o u n d - n u t — a r e u s e d in t h e w i d e r s p a c e s f a r t h e r fr o m t h e h o u s e . S i n c e t h e s a m e b a m b o o m e s h c o n s t r u c t i o n is u s e d fo r h o u s e s , h o u s e e n c l o s u r e s , a n d e n c l o s u r e s o f e n c l o s u r e s , t h e r e s u l t is a s e l f - s im i la r a r c h i ­ tectu re. U n lik e th e d efen siv e la b y rin th o f K o to k o a rc h ite c tu re , w h e re th ere w e re o n l y a few w e l l - p r o t e c t e d e n tr y w a y s , t h e f a r m i n g a c t i v i t i e s r e q u i r e a lo t o f m o v e m e n t b e t w e e n e n c l o s u r e s , so a t all s c a l e s we s e e g o o d - s iz e d o p e n i n g s . T h e f r a c t a l s i m u l a t i o n in figures 2 . 2 b , c s h o w s h o w t h i s s c a l i n g s t r u c t u r e c a n b e m o d ­ e l e d u s i n g a n o p e n s q u a r e as t h e s e e d s h a p e . FIGURE 2.2 B a m ile k e se ttlem en t (a) Plan of Bamileke settlement from about i960, (b) Fractal simulation of Bamileke architecture. In the first iteration (“seed shape”), the two active lines are shown in gray, (c) Enlarged view of fourth iteration. ( a , Begum 1952; reprinted with permission f r o m o r s t o m ) . 26 introduction C irc u la r fra cta ls in s e ttle m e n t a rc h ite c tu r e M u c h o f s o u t h e r n A f r i c a is m a d e u p o f a r i d p l a i n s w h e r e h e r d s o f c a t t l e - a n d o t h e r l i v e s t o c k a re ra is e d . R i n g - s h a p e d l i v e s t o c k p e n s , o n e fo r e a c h e x t e n d e d fam ily,^ c a n b e s e e n in t h e a erial p h o t o in figure 2.3a, a B a-ila s e t t l e m e n t in s o u t h e r n Z a m ­ bia. A d ia g r a m o f a n o t h e r B a -ila s e t t l e m e n t (fig. 2 .3 d ) m a k e s th e s e liv e sto c k e n c l o ­ sures ( “k ra a ls ” ) m o r e clear. T o w a r d t h e b a c k o f e a c h p e n w e find t h e fam ily liv in g q u a r t e r s , a n d t o w a r d t h e f r o n t is t h e g a t e d e n t r a n c e fo r l e t t i n g l i v e s t o c k in a n d o u t. For th is r e a s o n t h e f r o n t e n t r a n c e is a s s o c ia te d w i t h lo w s t a tu s ( u n c l e a n , a n i ­ m a l s ) , a n d t h e b a c k e n d w i t h h i g h s t a t u s ( c l e a n , p eop le).'* T h i s g r a d i e n t o f s t a ­ t u s is re f le c te d by t h e size g r a d i e n t in t h e a r c h i t e c t u r e : t h e f r o n t is o n l y f e n c i n g , as w e g o t o w a r d t h e b a c k s m a l l e r b u i l d i n g s (f o r s t o r a g e ) a p p e a r , a n d t o w a r d t h e v e ry b a c k e n d a r e t h e la r g e r h o u s e s . T h e t w o g e o m e t r i c e l e m e n t s o f t h i s s t r u c ­ t u r e — a r i n g s h a p e o v e r a l l , a n d a s t a t u s g r a d i e n t i n c r e a s i n g w i t h size fr o m f r o n t t o b a c k — e c h o e s t h r o u g h o u t e v e r y s c a l e o f t h e B a -il a s e t t l e m e n t . T h e s e t t l e m e n t as a w h o l e h a s t h e s a m e s h a p e : it is a r i n g o f rin gs. T h e s e t ­ t l e m e n t , like t h e li v e s t o c k p e n , h a s a f r o n t / b a c k s o c i a l d i s t i n c t i o n : t h e e n t r a n c e is lo w s t a t u s , a n d t h e b a c k e n d is h i g h s t a tu s . A t t h e s e t t l e m e n t e n t r a n c e t h e r e a re n o fam ily e n c l o s u r e s a t a ll fo r t h e first 2 0 y a r d s o r s o , b u t t h e f a r t h e r b ack, w e go, t h e la r g e r t h e fa m ily e n c l o s u r e s b e c o m e . A t t h e b a c k e n d o f t h e i n t e r i o r o f t h e s e t t l e m e n t , w e see a s m a ll e r d e t a c h e d r i n g o f h o u s e s , l i k e a s e t t l e m e n t w i t h i n t h e s e t t l e m e n t . T h i s is t h e c h i e f ’s e x t e n d e d family. A t t h e b a c k o f t h e i n t e r i o r o f tine c h i e f ’s e x t e n d e d fa m il y rin g , t h e c h i e f h a s h i s o w n h o u s e . A n d if w e w e r e t o v ie w a s i n g l e h o u s e f r o m a b o v e , w e w o u ld see t h a t it is a r i n g w i t h a s p e c i a l p l a c e a t t h e b a r k o f t h e i n t e r i o r : t h e h o u s e h o l d altar. S i n c e w e h a v e a s i m i l a r s t r u c t u r e a t all sc a le s, t h i s a r c h i t e c t u r e is e a sy to m o d e l w i t h fractals. F ig u re 2 . 3 b s h o w s t h e first t h r e e i t e r a t i o n s . W e b e g i n w i t h a s e e d s h a p e t h a t c o u l d be t h e o v e r h e a d v i e w o f a s i n g l e h o u s e . T h i s is c r e a t e d by a c t i v e li n e s t h a t m a k e u p t h e r i n g - s h a p e d w a lls, as w e ll as a n a c t i v e l i n e a t t h e p o s i t i o n o f t h e a l t a r a t t h e b a c k o f t h e i n t e r i o r . T h e o n l y p a s s iv e li n e s a r e those a d ja c e n t to th e e n tr a n c e . In th e n e x t ite ra tio n , we h a v e a sh a p e th a t c o u ld be t h e o v e r h e a d v ie w o f a fa m ily e n c l o s u r e . A t t h e e n t r a n c e t o t h e f a m ily e n c l o ­ su re w e h a v e o n l y f e n c i n g , b u t as w e go t o w a r d t h e b a c k w e h a v e b u i l d i n g s o f i n c r e a s i n g size. S i n c e t h e s e e d s h a p e u se d o n l y p a s s iv e l i n e s n e a r t h e e n t r a n c e a n d in c r e a s i n g ly la rg e r lin e s t o w a r d t h e b a c k , t h i s i t e r a t i o n o f o u r s i m u l a t i o n h a s t h e s a m e size g r a d i e n t t h a t t h e real fam ily e n c l o s u r e sh o w s. Fin ally , t h e t h i r d i t e r ­ a tio n p ro v id es a stru ctu re t h a t c o u ld b e th e o v e rh e a d v iew o f th e w h o le s e ttle ­ m e n t . A t th e e n t r a n c e to th e s e t t l e m e n t w e h a v e o n ly fe n c in g , b u t as we go to w a rd FIGURE 2 .3 B a -ila ( a) A e r i a l p h o t o o f B a - i l a s e t t l e m e n t b e f o r e 1 9 4 4 - ( b ) F r a c t a l g e n e r a t i o n o f B a - i l a s i m u l a t i o n . N o t e t h a t t h e s e e d s h a p e h a s o n l y a c t i v e l i n e s ( g r a y ) e x c e p t fo r t h o s e n e a r t h e o p e n i n g ( b l a c k ) , (a, A m e r i c a n Geograph ic Institute.) In tro d u c tio n t h e b a c k w e h a v e e n c l o s u r e s o f i n c r e a s i n g size. A g a i n , b y h a v i n g t h e s e e d s h a p e use o n l y p a s s iv e li n e s n e a r t h e e n t r a n c e a n d i n c r e a s i n g l y l a r g e r li n e s t o w a r d th e b a c k , th is i t e r a t i o n o f o u r s i m u l a t i o n h a s t h e sam e size g r a d i e n t t h a t the'T eal s ettle m e n t sh ow s. I n e v e r v i s i t e d t h e B a - il a m y self; m o s t o f m y i n f o r m a t i o n c o m e s fr o m t h e c la s s ic e t h n o g r a p h y b y E d w i n S m i t h a n d A n d r e w D a l e , p u b l i s h e d in 1 92 0. W h i l e t h e i r c o l o n i a l a n d m i s s i o n a r y m o t i v a t i o n s d o n o t i n s p i r e m u c h tr u s t, t h e y o f t e n s h o w e d a s t r o n g c o m m i t m e n t t o w a r d u n d e r s t a n d i n g t h e B a - il a p o i n t o f v ie w for s o c i a l s t r u c t u r e . T h e i r a n a l y s i s o f B a - il a s e t t l e m e n t a r c h i t e c t u r e p o i n t s o u t f r a c t a l a t t r i b u t e s . T h e y t o o n o t e d t h e s c a l i n g o f h o u s e size, fr o m t h o s e less t h a n 12 f e e t w i d e n e a r t h e e n t r a n c e , t o h o u s e s m o r e t h a n 4 0 f e e t w id e a t t h e b a c k , a n d e x p l a i n e d it as a s o c ia l s t a t u s g r a d i e n t ; “ t h e r e b e i n g a w o r l d o f d if fe re n c e b e t w e e n t h e s m a ll h o v e l o f a careless n o b o d y a n d t h e s p a c i o u s d w e lli n g o f a c h i e f ’ ( S m i t h a n d D a l e 1 9 6 8 , 1 1 4 ). It is in S m i t h ’s d i s c u s s i o n o f re lig io u s b eliefs, h o w e v e r , t h a t t h e m o s t s t r i k ­ in g f e a t u r e o f t h e B a - i l a ’s f r a c ta l a r c h i t e c t u r e is i l l u m i n a t e d . U n l i k e m o s t m i s ­ s i o n a r ie s o f h i s ti m e , S m i t h w as a s t r o n g p r o p o n e n t o f r e s p e c t fo r lo c a l re lig io n s. H e w as n o r e l a t i v i s t — u n d e r s t a n d i n g a n d r e s p e c t w ere' s t r a t e g i e s f o r c o n v e r ­ s i o n — b u t h i s d e l i g h t in t h e in d i g e n o u s s p i rit u a l s t r e n g t h c o m e s ac ro s s c le a r ly in h i s w r it in g s a n d p r o v i d e d h i m w i t h i n s i g h t i n t o t h e s u b t l e r e l a t i o n o f t h e so c ia l, s a c r e d , a n d p h y s i c a l s t r u c t u r e o f t h e B a - il a a r c h i t e c t u r a l p l a n . In this village th e r e are a b o u t 25 0 h u ts , b uilt m ostly o n th e edge o f a circle four h u n d r e d yards in d ia m e te r. Inside th is circle th e r e is a sub sid iary o n e o c c u p ie d by th e c h ie f, his family, a n d c a tt le . It is a village in itself, a n d th e form o f it in th e p la n is th e form o f th e g reater n u m b e r o f Ba-ila villages, w h ic h d o n o t a tta in to th e d im e n sio n s o f S h a l o b a ’s cap ital. T h e o p e n space in t h e c e n te r of t h e v il­ lage is also broken by a second subsidiary village, in w h ic h reside im p o rta n t m e m ­ bers o f th e c h i e f ’s family, a n d also by th r e e o r four m i n i a t u r e h u t s su r ro u n d e d by a fence: th e s e are th e mantlu a mizhivno ( “th e m a n e s ’ h u t s ”) w h e re offerings are m a d e to th e ancestral spirits. T h u s early d o we see traces o f th e nll-pervnding religious c o n sc io u sn e ss o f th e Ba-ila. ( S m i t h a n d D ale 1968, 1 13) I n t h e first i t e r a t i o n o f t h e c o m p u t e r - g e n e r a t e d m o d e l t h e r e is a d e t a c h e d a c t i v e li n e in s id e t h e r i n g , a t t h e e n d o p p o s i t e t h e e n t r a n c e . T h i s w a s m o t i v a t e d by t h e r i n g c o m p r i s i n g t h e c h i e f ’s fam ily, b u t it also d e s c r i b e s t h e l o c a t i o n o f t h e s a c r e d a l t a r w i t h i n e a c h h o u s e . A s a l o g i c ia n w o u ld p u t it, t h e c h i e f ’s f a m ily rin g is t o t h e w h o l e s e t t l e m e n t as t h e a l t a r is t o t h e h o u s e . It is n o t a s t a t u s g r a d i e n t , as we saw w i t h t h e f r o n t - b a c k axis, b u t r a t h e r a r e c u r r i n g f u n c t i o n a l ro le b e t w e e n d i f f e r e n t sca les: “T h e w o r d a p p l i e d t o t h e c h i e f 's r e l a t i o n t o h is p e o p l e is latlela: in t h e e x t r a c t s g iv e n a b o v e w e t r a n s l a t e it ‘to r u l e , 1 b u t it h a s t h i s o n l y as a sec- Fractals in A fr ic a n se ttle m en t architecture o n d a r y m e a n i n g . K u le la is p r i m a r i ly t o n u r s e , t o c h e r i s h ; it is t h e w o r d a p p li e d t o a w o m a n c a r i n g for h e r c h i l d . T h e c h i e f is t h e f a t h e r o f t h e c o m m u n i t y ; t h e y a re h is c h i l d r e n , a n d w h a t h e d o e s is lela t h e m ” ( S m i t h a n d D a le 1 9 6 8 , 3 0 7 ) . T h i s r e l a t i o n s h i p is e c h o e d t h r o u g h o u t ' f a m i l y a n d s p i r i t u a l tie s a t all scales', a n d is s t r u c t u r a l l y m a p p e d t h r o u g h t h e se l f - s im i la r a r c h i t e c t u r e . T h e n e s t i n g o f c i r c u l a r s h a p e s — a n c e s t r a l m i n i a t u r e s t o c h i e f ’s h o u s e r i n g t o c h i e f ’s e x t e n d e d fam ily r i n g to che g re a t o u t e r ring— was n o t a s t a tu s g r a d i e n t , as we saw f o r t h e e n c l o s u r e v a r i a t i o n fr o m f r o n t t o b a c k , b u t s u c c e s s iv e i t e r a t i o n s o f lela. A very d if fe re n t c ir c u la r fracta l a r c h i t e c t u r e c a n b e s e e n in t h e fam ou s s t o n e b u i l d i n g s in t h e M a n d a r a M o u n t a i n s o f C a m e r o o n . T h e v a r i o u s e t h n i c g ro u p s o f t h i s a r e a h a v e t h e i r o w n s e p a r a t e n a m e s , b u t c o l l e c t i v e l y a re o f t e n re f e rre d t o as K ird i, t h e F u l a n i w o rd for “ p a g a n , ” b e c a u s e o f t h e i r s t r o n g r e s i s t a n c e a g a i n s t c o n v e r s i o n t o Is la m . T h e i r b u i l d i n g s a r e c r e a t e d fr o m t h e s t o n e r u b b l e t h a t c o m m o n ly covers th e M a n d a r a m o u n t a in terrain. M u c h o f th e s to n e h as n a tu ra l f r a c t u r e l i n e s t h a t t e n d t o s p l i t i n t o t h i c k f l a t s h e e t s , so t h e s e r e a d y - m a d e b ric k s— a l o n g w i t h d e f e n s i v e n e e d s — h e l p e d t o in s p ire t h e c o n s t r u c t i o n o f t h e i r h u g e castlelike c o m p lex es. But r a th e r th a n b e in g th e E u c lid e a n sh apes o f E u ro ­ p e a n c a s t le s , t h i s A f r i c a n a r c h i t e c t u r e is fr a c ta l. ’ F ig u re 2 .4 a s h o w s t h e b u i l d i n g c o m p l e x o f t h e c h i e f o f M o k o u l e k , o n e o f t h e M o f o u s e t t l e m e n t s . A n a r c h i t e c t u r a l d i a g r a m o f M o k o u l e k , d r a w n by F re n c h , researchers from th e o r s t o m scie n c e in s titu te , show s its o v erall stru c tu re (fig. 2.4b). It is prim arily c o m p o s e d o f th r e e s t o n e e n c lo s u re s ( t h e large circles), e a c h o f w h i c h s u r r o u n d s ti g h t l y s p i r a le d g ra n a r ie s ( s m a ll c i r c le s ) . T h e s e e d s h a p e for t h e s i m ­ u l a t i o n r e q u i r e s a c ir c le , m a d e o f p a s s iv e li n e s , a n d t w o d i f f e r e n t sets o f a c t i v e lin e s (fig. 2.4 c ) . I n s id e t h e c ir c le is a s c a l i n g s e q u e n c e o f s m a ll a c t i v e lines; th e s e w ill b e c o m e t h e g r a n a r ie s . O u t s i d e t h e c ir c le t h e r e is a la r g e a c t i v e lin e ; t h i s w ill r e p l i c a t e t h e e n c l o s u r e filled w i t h g ra n a r ie s . By t h e f o u r t h i t e r a t i o n w e h a v e c r e ­ a t e d t h r e e e n c lo s u r e s filled w i t h sp iral c lu s te rs o f g ra n a r ie s , plus o n e unfilled. T h e re a l d i a g r a m o f M o k o u l e k s h o w s s e v e r a l u n f i l l e d c i r c l e s — e v i d e n c e t h a t n o t e v e r y t h i n g in t h e a r c h i t e c t u r a l s t r u c t u r e c a n b e a c c o u n t e d fo r by frac tals. N e v ­ e r t h e le s s , a n i m p o r t a n t f e a t u r e is s u g g e s te d by t h e s i m u l a t i o n . I n t h e first i t e r a t i o n we see t h a t t h e large e x t e r n a l a c t i v e li n e is t o t h e left o f t h e c ir c le . B u t s i n c e it is a t a n a n g l e , t h e n e x t i t e r a t i o n finds t h i s a c t i v e lin e a b o v e a n d t o t h e r i g h t. If we fo llo w t h e i t e r a t i o n s , we c a n see t h a t t h e dynam ic constrwction o f t h e c o m p l e x lias a s p ira l p a t t e r n ; t h e r e p l i c a t i o n s w h o r l a b o u t a c e n t r a l l o c a t io n . T h i s sp iral d y n a m i c c a n be m issed w i t h j u s t a s t a t i c view — I c e r ­ t a i n ly d i d n ’t see it b e fo r e I tr ie d t h e s i m u l a t i o n — b u t o u r p a r t i c i p a t i o n . i n t h e v i r ­ tu a l c o n s t r u c t i o n m a k e s t h e spiral q u i t e e v i d e n t . 3 T h e sim ilarity b e t w e e n t h e sm all s p irals o f g ra n a r ie s in s id e t h e e n c l o s u r e s a n d t h i s l a r g e - s c a l e s p ira l s h a p e o f t h e c FIGURE 2 . 4 M o k o u le k ( a ) M o k o u l e k , C a m e r o o n . T h e s m a l l b u i l d i n g s in s i d e t h e s t o n e w a l l a r e g r a n a r i e s . T h e r e c t a n g u l a r b u i l d i n g ( t o p r i g h t ) h o l d s t h e s a c r e d a l t a r , ( b ) A r c h i t e c t u r a l d i a g r a m o f M o k o u l e k . ( c ) F ir s t t h r e e i t e r a t i o n s o f t h e M o k o u l e k s i m u l a t i o n . T h e s e e d s h a p e is c o m p o s e d o f a c i r c l e d r a w n w i t h p a s s i v e lin es ( b la c k ) a n d w it h gray a c t i v e lin es b o t h inside a n d o u tsid e th e c irc le , (d ) F o u rth ite ra tio n o f the M o k o u le k sim u lation , f a a n d b, b y permission f r o m S e i g n o b o s 1 9 8 2 .) Fractals in A fr ic a n settlement architecture c o m p l e x as a w h o l e i n d i c a t e s t h a t t h e f r a c t a l a p p e a r a n c e o f t h e a r c h i t e c t u r e is n o t m e r e ly d u e t o a r a n d o m a c c u m u l a t i o n o f v a rio u s - s iz e d c i r c u l a r fo rm s. T h e i d e a o f c i r c l e s o f i n c r e a s i n g size, s p i r a l i n g f r o m a c e n t r a l p o i n t , h a s b e e n a p p li e d a t tw o d i f f e r e n t sc ales, a n d t h i s s t r u c t u r a l c o h e f e n c e is c o n f i r m e d b y t h e a r c h i ­ tects’ o w n concepts. I n o u r s im u la tio n th e activ e lin e b e c a m e lo c a te d to w a rd th e c e n te r o f th e s p i ra l. T h e M o f o u a ls o t h i n k o f t h e i r a r c h i t e c t u r e as s p i r a l i n g f r o m t h i s c e n t r a l l o c a t i o n , w h i c h h o l d s t h e i r s a c r e d altar. T h e a l t a r is a k i n d o f c o n c e p t u a l “a c t i v e l i n e ” in t h e i r s c h e m a ; it is r e s p o n s ib l e f o r t h e i t e r a t i o n s o f life. S e i g n o b o s ( 1 9 8 2 ) n o te s t h a t th is a re a o f t h e c o m p l e x is t h e site o f b o t h religio us a n d p o litic a l a u t h o r ­ ity; it is t h e l o c a t i o n for r i t u a l s t h a t g e n e r a t e c y c le s o f a g r i c u l t u r a l f e r til it y a n d a n c e s t r a l s u c c e s s io n . T h i s g e o m e t r i c m a p p i n g b e t w e e n t h e s c a l in g c ir c le s o f t h e . a r c h i t e c t u r e a n d t h e s p i r i t u a l c y cle s o f life is r e p r e s e n t e d in t h e i r d i v i n a t i o n ( “f o r t u n e t e l l i n g ” ) r i tu a l , in w h i c h t h e p r i e s t c r e a t e s c o n c e n t r i c c ir c le s o f sto n e s ' a n d p la c e s h i m s e l f a t t h e c e n te r. A s in t h e guti p a i n t i n g in L o g o n e - B ir n i , in w h i c h t h e K o t o k o h a d m o d e l e d t h e i r s c a l i n g r e c t a n g l e s , t h e M o f o u h a v e a ls o c r e a t e d th e ir o w n scalin g sim u latio n . By t h e t i m e 1 a r r iv e d a t M o k o u l e k in 1 9 9 4 t h e c h i e f h a d d ie d , a n d t h e o w n ­ e r s h i p o f t h i s c o m p l e x h a d b e e n p ass ed o n to h i s w id o w s. T h e n e w c h i e f to l d me thac t h e d esig n o f th is a r c h i t e c t u r e , in c l u d i n g t h a t o f h is n e w c o m p l e x , b e g a n w ith a p re c i s e k n o w l e d g e o f t h e a g ri c u lt u ra l y ield . T h i s v o l u m e m e a s u r e w as t h e n c o n ­ v e r t e d t o a n u m b e r o f g r a n a r ie s , a n d t h e s e w e r e a r r a n g e d in spirals. T h e d e sig n is t h u s n o t s i m p l y a m a t t e r o f a d d i n g o n g r a n a r i e s as t h e y a r e n e e d e d ; in fact, it h a s a m u c h m o r e q u a n t i t a t i v e basis t h a n m y c o m p u t e r m o d e l , w h i c h I sim p ly did by e y e b a ll. N o t a ll c i r c u l a r a r c h i t e c t u r e s , in A f r i c a h a v e t h e k i n d o f c e n t r a l i z e d l o c a t i o n t h a t w e sa w in M o k o u l e k . T h e S o n g h a i v i l l a g e o f L a b b e z a n g a in M a l i (fig. 2 . 5 a ) , for e x a m p l e , s h o w s c i r c u l a r sw irls o f c i r c u l a r h o u s e s w i t h o u t a n y s in g le fo c u s . B u t c o m p a r i n g t h i s t o t h e f r a c t a l im a g e o f figure 2 .5 b , w e see t h a t a l a c k o f c e n t r a l fo cu s d o e s n o t m e a n a l a c k o f s e lf-s im ila r ity . It is i m p o r t a n t to r e m e m b e r c h a t w h ile “s y m m e t r y ” in E u c l i d e a n g e o m e t r y m e a n s s i m ila rity w i t h i n o n e s c a l e (e.g ., s i m i l a r i t y b e t w e e n o p p o s i t e s id e s in b i l a t e r a l s y m m e t r y ) , f r a c ta l g e o m e t r y is b a s e d o n s y m m e t r y b e t w e e n d i f f e r e n t sc a le s. E v e n t h e s e d e c e n t r a l ­ ized sw ir ls o f c i r c u l a r b u i l d i n g s s h o w a s c a l i n g s y m m e t r y . . P a u l S to l le r , a n a c c o m p l i s h e d e t h n o g r a p h e r o f t h e S o n g h a i , te lls m e t h a t t h e r e c t a n g u l a r b u i l d i n g s t h a t c a n b e s e e n in figure 2 . 5 a a r e d u e to Is la m ic in flu ­ e n c e , a n d t h a t t h e o r i g i n a l s t r u c t u r e w o u l d h a v e b e e n c o m p l e t e l y c ir c u la r . T h a n k s t o P e t e r B ro a d w e ll, a c o m p u t e r p r o g r a m m e r f r o m S i l i c o n G r a p h i c s In c., ' w e w e r e a b le t o r u n a q u a n t i t a t i v e te s t o f t h e p h o t o t h a t c o n f i r m e d w h a t o u r eyes In tro d u c tio n 32 FIGURE 2 .5 Labbezanga (a) Aerial view of the village of Lahhetanga in Mali, (b) Fractal graphic. ( a , photo by G e o rg G c r s te r ; b , by p erm issio n o f B enoit M a n d e lb ro t.} w e r e t e l l i n g us: t h e S o n g h a i a r c h i t e c t u r e c a n b e c h a r a c t e r i z e d by a f r a c ta l d i m e n ­ s i o n s i m i l a r to t h a t o f t h e c o m p u t e r - g e n e r a t e d f r a c ta l o f figure 2.5b.** T h i s k i n d o f d e n s e c i r c u l a r a r r a n g e m e n t o f c ir c le s , w h i l e o c c u r r i n g in all so r ts o f v a r i a t i o n s , is c o m m o n t h r o u g h o u t i n l a n d w e s t A f r i c a . B o u r d i e r a n d T r i n h ( 1 9 8 5 ) , for e x a m p l e , d e s c r ib e a sim ila r c ircu lar a r c h i t e c t u r e in B u r k i n a Faso. T h e s c a l i n g o f i n d i v i d u a l b u i l d i n g s is b e a u t i f u l l y d i a g r a m m e d in t h e i r c o v e r i l l u s t r a t i o n (fig. 2 . 6 a ) , a p o r t i o n o f o n e o f t h e large b u i l d i n g c o m p l e x e s c r e a t e d by t h e N a n k a n i society. A s for t h e S o n g h a i , fo r e ig n c u l t u r a l i n f lu e n c e s h a v e n o w i n t r o d u c e d r e c t a n g u l a r b u il d in g s as w ell. I n t h e N a n k a n i c o m p l e x t h e o u t e r m o s t e n c l o s u r e ( t h e p e r i m e t e r o f t h e c o m p l e x ) is s o c ia lly c o d e d a s m a l e . A s w e m o v e in , t h e s u c c e s s iv e e n c l o s u r e s b e c o m e m o r e f e m a l e a s s o c i a t e d , d o w n t o t h e c i r ­ c u l a r w o m a n ’s dego (fig. 2 . 6 b ) , t h e c i r c u l a r f i r e p l a c e , a n d f i n a l l y t h e s c a l i n g s t a c k s o f p o t s (fig. 2 .6 c ). U s i n g a t e c h n i q u e q u i t e c lo s e t o t h a t o f t h e K o t o k o w o m e n , t h e w o m e n o f N a n k a n i a ls o w a t e r p r o o f a n d d e c o r a t e t h e s e w alls. T h e r e c u r r e n t i m a g e o f a Fractals m A fr ic a n se ttle m e n t architecture t r i a n g l e in th e s e d e c o r a t i o n s (s e e w alls o f d e g o ) r e p r e s e n t s t h e zalanga, a n e s t e d s t a c k o f c a l a b a s h e s ( c i r c u l a r b o w ls c a r v e d f r o m g o u r d s ) t h a t e a c h w o m a n k e e p s in h e r k i t c h e n (fig. 2 .6 d ). S i n c e th e s e c a l a b a s h e s a r e s t a c k e d fr o m large to sm all, th ey (a n d th e rope th a t h o ld s th e m ) form a tr ia n g le — th u s th e trian g u lar •v d e c o r a t i o n s also r e p r e s e n t s c a l in g c irc le s , j u s t in a m o r e a b s t r a c t way. T h e s m a l l ­ e s t c o n t a i n e r in a w o m a n ’s z a la n g a is t h e /cumpio, w h i c h is a s h r i n e for h e r soul. W h e n s h e dies, th e zalanga, alo n g w i t h h e r p o ts , is sm a sh e d , a n d h e r soul is released t o e t e r n i t y . T h e e t e r n i t y c o n c e p t , ' a s s o c i a t e d w i t h w e l l - b e i n g , is s y m b o li c a ll y FIGURE 2 . 6 N a n k a n i home (a) D r a w i n g o f a N a n k a n i h o m e , ( b ) T h e w o m a n ’s m a i n r o o m (dego) in s i d e t h e N a n k a n i h o m e , ( c ) A s c a l i n g s t a c k o f p o t s in d i e f i r e p l a c e , (d) T h e ?a!an ga . (a, Bonrd ier a n d T rinh 1 5 8 5 ; co urte sy o f the authors; b - d , p/totos fro m B ourdier a n d T rin h 19 8 5 , by permission o f the a u th o rs .) 33 Introduction 34 r e p r e s e n t e d by t h e c o il s o f a s e r p e n t o f i n f i n i t e l e n g t h , s c u l p t e d i n t o t h e w alls o f th e s e hom es. F r o m t h e 2 0 - m e t e r d i a m e t e r o f t h e b u i l d i n g c o m p l e x t o t h e 0 . 2 - r f te t e r k u m p i o — a n d n o t s im p ly a t o n e o r t w o le v e ls in b e t w e e n , b u t w i t h d o z e n s o f selfs i m i l a r s c a l in g s — t h e N a n k a n i f r a c t a l s p a n s t h r e e o r d e r s o f m a g n i t u d e , w h i c h is c o m p a ra b le to th e re s o lu tio n o f m o s t c o m p u te r screens. M o re o v e r, th ese scalin g circles a re far from u n c o n s c i o u s a c c i d e n t : as in sev eral o t h e r a r c h i t e c t u r e s w e h a v e e x a m i n e d , t h e y h a v e m a d e c o n s c i o u s u s e o f t h e s c a l in g in t h e i r s o c i a l s y m b o l ­ ism. In th is case, t h e m o s t p r o m i n e n t s y m b o li s m is t h a t o f b i r t h i n g . W h e n a c h il d is b o r n , fo r e x a m p l e , i t m u s t r e m a i n i n t h e i n n e r m o s t e n c l o s u r e o f t h e w o m e n ’s r d e g o u n t i l it c a n c ra w l o u t by itself. E a c h s u c c e s siv e e n t r a n c e is— s p a t ia l ly as w e ll as so c ia lly — a r i t e o f p assag e, s t a r t i n g w i t h t h e b io l o g i c a l e n t r a n c e o f t h e c h i l d f r o m t h e w o m b . I t le a v e s e a c h o f t h e s e n e s t e d c h a m b e r s as t h e n e x t i t e r a t i o n in life ’s sta g e s is b o r n . T h e z a la n g a m o d e l s t h e e n t i r e s t r u c t u r e i n m i n i a t u r e , a n d its d e s t r u c t i o n in t h e e v e n t o f d e a t h m a p s t h e j o u r n e y i n re v e r s e : f r o m t h e c ir c le s o f t h e la r g e s t c a l a b a s h to t h e t i n y k u m p i o h o l d i n g t h e s o u l — f r o m m a t u r e a d u l t t o t h e e t e r n a l r e a l m o f a n c e s t o r s w h o d w e ll i n " t h e e a r t h ’s w o m b . ” T h e r e is a c o n s c i o u s s c h e m e to t h e s c a l i n g c i r c l e s o f t h e N a n k a n i : it is a r e c u r s i o n w h i c h b o t t o m s - o u t a t in f in ity . B r a n c h in g fr a c ta ls W h i l e A f r i c a n c i r c u l a r b u i l d i n g s a r e ty p i c a ll y a r r a n g e d in c i r c u l a r c l u s t e r s , t h e p a t h s t h a t le a d t h r o u g h t h e s e s e t t l e m e n t s a r e ty p i c a ll y n o t c i r c u la r . L i k e t h e b r o n c h i a l passag es t h a t o x y g e n a t e t h e f o u n d a l v e o l i of t h e lu n g s , t h e r o u t e s t h a t n o u r i s h c i r c u l a r s e t t l e m e n t s o f t e n t a k e a b r a n c h i n g f o r m (e .g ., figure 2 . 7 ) . B u t d esp ite m y u n a v o id a b ly o rg a n ic ist m e ta p h o r, th e s e c a n n o t be sim p ly re d u c e d to u n c o n s c i o u s tr a c e s o f m i n i m u m e ff o rt. F o r o n e t h i n g , c o n s c i o u s d e s i g n c r i t e r i a a r e e v i d e n t in c o m m u n i t i e s in w h i c h t h e r e is a n a r c h i t e c t u r a l t r a n s i t i o n f r o m c i r ­ c u l a r t o r e c t a n g u l a r b u i l d in g s , s i n c e t h e y c a n c h o o s e t o e i t h e r m a i n t a i n o r e ra s e t h e b r a n c h i n g fo rm s. D iscussion c o n c e r n i n g s u c h d ecisio ns are a p p a r e n t in t h e s e t t l e m e n t o f B anyo, C a m e ro o n , w h e re th e tra n s itio n h a s a lo n g h isto ry (H u ra u lt 1975)- 1 fo u n d th a t few c i r c u l a r b u i l d i n g s w e r e le f t, b u t t h o s e t h a t w e r e s t il l i n t a c t s e r v e d as a n e m b o d i m e n t o f c u l t u r a l m e m o r y . T h i s ro le w a s h o n o r e d i n t h e c a s e o f t h e c h i e f ’s c o m p l e x a n d e x p l o i t e d in t h e c a s e o f a b l a c k s m i t h ’s s h o p , w h i c h w a s t h e s i t e o f o c c a s i o n a l t o u r i s t v is its. A f t e r p a s s i n g a p p r o v a l by t h e g o v e r n m e n t o f f ic ia ls a n d t h e s u l t a n , 1 w as g r e e t e d b y t h e o f f i c i a l c i t y s u r v e y o r , w h o — c o n s i d e r i n g t h e fa c t t h a t h is r a is o n d ’e t r e w as E u c l i d e a n i z i n g t h e s t r e e t s — s h o w e d s u r p r is in g Fractals in A fr ic a n se ttle m e n t architecture 35 F I G U R E 2.7 B r a n c h in g p a th s in a S e n e g a le s e s e t t l e m e n t {a) Aerial p h o to o f a traditional s e ttle m e n t in n o rth e a st Senegal. T h e space betw een enclosure walls, serving as roads an d footpaths, creates a b ra n c h in g pattern , (b) A branch in g fractal c a n be created by the background of a scaling set of circular shapes. (a, courtesy Institut Geographique du Senegal.) a p p r e c i a t i o n fo r m y p r o j e c t a n d h e l p e d m e l o c a t e t h e m o s t f r a c ta l a r e a o f t h e c it y (fig. 2 .8 a ) . A t t h e u p p e r left o f t h e p h o t o w e se e a p o r t i o n o f t h e E u c l i d e a n grid t h a t c o v e r s t h e re s t o f t h e city, b u t m o s t o f t h i s a re a is s till fr a c ta l. T h e l o c a ­ t i o n o f t h i s c a r e f u ll y m a i n t a i n e d b r a n c h i n g — f a n n i n g o u t fr o m a la r g e p laza t h a t is b o r d e r e d by t h e p a l a c e o f t h e s u l t a n a n d t h e g r a n d m o s q u e — is n o c o i n c i d e n c e . By m a r k i n g m y p o s i t i o n o n t h e a e r i a l p h o t o as 1 t r a v e l e d t h r o u g h (fig. 2 . 8 b ) , I w as l a t e r a b le t o c r e a t e a m a p by d ig i ta l ly a l t e r i n g t h e p h o t o im ag e (fig. 2 .8 c ) . T h i s p r o v i d e s a s t a r k o u t l i n e — l o o k i n g m u c h lik e t h e v e i n s i n a le a f— o f t h e f r a c ta l s t r u c t u r e o f th i s t r a n s p o r t a t i o n n e t w o r k . I m ay h a v e p lu n g e d t h r o u g h a w all o r t w o in c r e a t i n g th i s m a p , b u t it c e r t a i n l y u n d e r e s t i m a t e s t h e fine b r a n c h i n g o f t h e f o o t p a t h s , as 1 d i d n o t a t t e m p t t o i n c l u d e t h e i r e x t e n s i o n s i n t o p r i v a t e h o u s i n g e n c lo s u r e s . H o w d o e s t h e c r e a t i o n o f th e s e s c a l in g b r a n c h e s i n t e r a c t w i t h t h e k i n d s o f i t e r a t i v e c o n s t r u c t i o n a n d s o c ia l m e a n i n g we h a v e s e e n a s s o c ia te d w ich o t h e r e x a m p l e s o f f r a c t a l a r c h i t e c t u r e ? A g o o d i l l u s t r a t i o n c a n b e f o u n d in t h e Position 1— outside palace FIGURE Position 2 — road below mosque 2.8 B r a n c h i n g p a t h s in B a n y o (a) Aerial phoco of the city of Banyo, Cameroon, (b) Successive views of the branching paths, as marked on the photo above. The clay walls require their own roof, which comes in both thatched and metal versions along the walkway in the last photo, (c) Aerial photo of Banyo with only public paths showing, (a, c o u r t e s y N a t i o n a l I n s t i t u t e o f C a r t o g r a p h y , Cameroon.) Position 3 —narrow walkway FIGURE 2 . 9 S t r e e t s o f Cairo (a) M a p o f s t r e e t s o f C a i r o , 1 8 9 8 . ( b ) F r a c t a l s i m u l a t i o n fo r C a i r o s t r e e t s , ( c ) E n l a r g e d v i e w o f fourth iteration. Introduction 38 b r a n c h i n g stre e ts o f N o r t h A f r i c a n c ities. F ig ure 2 .9 a s h o w s a m a p o f C a i r o , E gypt, i n 1 8 9 8 . T h e m a p w as c r e a t e d b y a n i n s u r a n c e c o m p a n y , a n d I h a v e c o l o r e d t h e s t r e e t s b l a c k t o m a k e t h e s c a l i n g b r a n c h e s m o r e a p p a r e n t . F ig u r e 2 . 9 b s h o w s its c o m p u te r sim u la tio n . D ela v a l ( 1 9 7 4 ) h as d escrib ed th e m o r p h o g e n e s is o f S a h a ­ r a n c i t i e s in t e r m s o f s u c c e s s iv e a d d i t i o n s s i m i l a r t o t h e l i n e r e p l a c e m e n t in t h e f r a c ta l a l g o r i t h m s w e h a v e u se d h e r e . T h e first “s e e d s h a p e " c o n s i s t s o f a m o s q u e c o n n e c t e d by a w id e a v e n u e t o t h e m a r k e t p l a c e , a n d su c c e s s iv e i t e r a t i o n s o f c o n ­ s t r u c t i o n a d d su c c e s s iv e c o n t r a c t i o n s o f t h i s form . S i n c e t h e s e f r a c ta l S a h a r a n s e t t l e m e n t a r c h i t e c t u r e s p r e d a t e Is la m (s e e D e v is s e 1 9 8 3 ) , it w o u ld b e m i s l e a d i n g to s e e t h e m as a n e n t i r e l y M u s l i m i n v e n ­ t i o n ; b u t g i v e n t h e p r e v i o u s o b s e r v a t i o n s a b o u t t h e i n t r o d u c t i o n o f Is la m i c a r c h i te c t u r e as a n i n t e rru p t io n o f c ir c u la r fractals in s u b - S a h a r a n A fric a , it is im p o r­ t a n t t o n o t e t h a t Is la m i c c u l t u r a l in f l u e n c e s h a v e m a d e s t r o n g c o n t r i b u t i o n s to A f r i c a n f r a c ta ls as w ell. H e a v e r ( 1 9 8 7 ) d e s c r ib e s t h e " a r a b e s q u e " a r t i s t i c fo r m in N o r t h A f r i c a n a r c h i t e c t u r e a n d d e s i g n in t e r m s t h a t r e c a l l s e v e r a l f r a c ta l c o n ­ c e p t s ( e .g ., “c y c l i c a l r h y t h m s ” p r o d u c i n g a n “ i n d e f i n i t e l y e x p a n d a b l e ” s t r u c ­ t u r e ) . H e d is c u s s e d t h e s e p a t t e r n s as v is u a l a n a l o g u e s t o c e r t a i n I s l a m i c s o c ia l c o n c e p t s , a n d w e will e x a m i n e his idea s in g re a te r d e ta il in c h a p t e r 12 o f th is b oo k. C o n c lu s io n T h r o u g h o u t t h i s c h a p t e r , w e h a v e s e e n t h a t a w id e v a r i e t y o f A f r i c a n s e t t l e m e n t a r c h i t e c t u r e s c a n be c h a r a c t e r i z e d as fra ctals. T h e i r p h y s i c a l c o n s t r u c t i o n m a k e s u se o f s c a l in g a n d i t e r a t i o n , a n d t h e i r s e lf- s im i la r it y is c le a r ly e v i d e n t fr o m c o m ­ p a r i s o n to fractal-grap hic s im u la tio n s. C h a p t e r 3 wilt s h o w t h a t fracta l a r c h i t e c t u r e is n o t sim p ly a ty pic al c h a r a c t e r i s t i c o f n o n - W e s t e r n s e t t l e m e n t s . T h i s a l o n e d o e s n o t a ll o w us t o c o n c l u d e a n i n d i g e n o u s A f r i c a n k n o w l e d g e o f f r a c t a l g e o m e t r y ; in f a c t , I w ill a rg u e in c h a p t e r 4 t h a t c e r t a i n f r a c ta l p a t t e r n s in A f r i c a n d e c o r a ­ ti v e a r t s are m e re ly t h e res u lt o f a n i n t u i t i v e e s t h e ti c . B u t as w e h a v e a lr e a d y s e e n , t h e fractals in A fr i c a n a rc h i te c t u r e a re m u c h m o r e t h a n t h a t . T h e i r d e sig n is lin k ed t o c o n s c i o u s k n o w l e d g e s y s te m s t h a t s u g g e s t s o m e o f t h e b a s i c c o n c e p t s o f f r a c ­ ta l g e o m e t r y , a n d in la t e r c h a p t e r s w e w ill fin d m o r e e x p l i c i t e x p r e s s i o n s o f t h i s i n d i g e n o u s m a t h e m a t i c s in a s t o n i s h i n g v a r i e t y a n d fo r m . CHAPTER 3 -Fractals-------------------------------------------ire -crosS'Cultural—c o m p a r is o n - T h e f r a c t a l s e t t l e m e n t p a t t e r n s o f ' A f r i c a s t a n d in s h a r p c o n t r a s t t o t h e C a r te - , s i a n g rid s o f E u r o - A m e r i c a n s e t t l e m e n t s . W h y t h e d if f e r e n c e ? O n e e x p l a n a t i o n c o u ld b e t h e d if f e r e n c e in so c ia l s t r u c t u r e . E u r o - A m e r i c a n c u lt u r e s are o rg a n iz e d by w h a t a n t h r o p o l o g i s t s w o u ld c a ll a “s t a t e s o c i e t y . ” T h i s i n c l u d e s n o t j u s t t h e m o d e r n n a t i o n - s t a t e , b u t re f e rs m o r e g e n e r a l l y t o a n y s o c i e t y w i t h a la r g e p o l i t i c a l h i e r a r c h y , l a b o r s p e c i a l i z a t i o n , a n d c o h e s i v e , f o r m a l c o n t r o l s — w h a t is s o m e t i m e s c a ll e d “ t o p - d o w n ” o r g a n i z a t i o n . P r e c o l o n i a l A f r i c a n c u lt u r e s in c l u d e d m a n y s t a t e s o c i e t i e s , as w e ll as a n e n o r m o u s n u m b e r o f s m a ll e r , d e c e n t r a l i z e d so c ia l g ro u p s , w i t h l i t t l e p o l i t i c a l h i e r a r c h y — t h a t is, s o c i e ti e s t h a t a re o rg a n iz e d “b o t t o m - u p ” r a t h e r t h a n “ t o p - d o w n .” 1 B u t if f r a c t a l a r c h i t e c t u r e is s i m p l y t h e a u t o m a t i c re su lt o f a n o n s t a t e social o r g a n iz a tio n , t h e n we sh o u ld see fractal s e t t l e ­ m e n t p a t t e r n s in t h e i n d i g e n o u s s o c ie tie s o f m a n y p a rts o f t h e w orld. I n this c h a p ­ t e r w e w ill e x a m i n e t h e s e t t l e m e n t p a t t e r n s f o u n d in t h e i n d i g e n o u s s o c i e t i e s o f t h e A m e r i c a s a n d t h e S o u t h P a c if ic , b u t o u r s e a r c h w ill t u r n u p v e ry few f r a c ­ tals. R a t h e r t h a n d i v i d i n g t h e w o r l d b e t w e e n a E u c l i d e a n W e s t a n d f r a c t a l n o n - W e s t , w e w ill f i n d t h a t e a c h s o c i e t y m a k e s use o f its p a r t i c u l a r d e s i g n t h e m e s in o rg a n i z in g its b u il t e n v i r o n m e n t . A f r i c a n a r c h i t e c t u r e t e n d s t o b e frac­ tal b e c a u s e t h a t is a p r o m i n e n t d e s i g n t h e m e in A f r i c a n c u l t u r e . I n fact, th is c u l ­ tu r a l s p e c i f i c i t y o f d e s i g n t h e m e s is t r u e n o t o n l y fo r a r c h i t e c t u r e , b u t fo r m a n y introduction 40 o t h e r ty p e s o f m a t e r i a l d e s i g n a n d c u l t u r a l p r a c t i c e s as w e ll. W e w ill b e g i n o u r su rv e y w i t h a b r i e f lo o k a t t h e d e s i g n t h e m e s in N a t i v e A m e r i c a n so c ie tie s, w h i c h i n c l u d e d b o t h h i e r a r c h i c a l s t a t e e m p i r e s as w e l l as s m a l l e r , d e c e n t r a l i z e d tr ib a l cu ltu res. 'N ative A m e r i c a n d e sig n T h e A n c e s t r a l P u e b l o s o c i e t y d w e l l e d in w h a t is n o w t h e s o u t h w e s t e r n U n i t e d S tates aro u n d 1 1 0 0 c .e . A e r i a l p h o t o s o f t h e s e sites (fig. 3 .1 ) a re s o m e o f t h e m o s t f a m o u s e x a m p l e s o f N a t i v e A m e r i c a n s e t t l e m e n t s . B u t as we c a n se e fr o m th i s v a n t a g e p o i n t , t h e a r c h i t e c t u r e is p r i m a r i ly c h a r a c t e r i z e d by a n e n o r m o u s circu la r fo r m c r e a t e d fr o m s m a lle r re c ta n g u la r c o m p o n e n t s — c e r t a i n l y n o t t h e s a m e shapea t t w o d i f f e r e n t scales. T h i s j u x t a p o s i t i o n o f t h e c ir c le a n d t h e q u a d r i l a t e r a l ( r e c ­ tan gle or cross-shaped) form is n o t a c o in c i d e n c e ; th e tw o forms are t h e m o s t im p o r­ t a n t d e s i g n t h e m e s in t h e m a t e r i a l c u l t u r e o f m a n y N a t i v e A m e r i c a n s o c i e ti e s , in c lu d in g b o th N o r th a n d S o u th c o n tin e n ts A s far as a r c h i t e c t u r e is c o n c e r n e d , t h e r e are n o e x a m p l e s o f t h e n o n ! in e a r s c a l i n g w e saw in A f r i c a . T h e o n l y N a t i v e A m e r i c a n a r c h i t e c t u r e s t h a t c o m e c lo s e a r e a few i n s t a n c e s o f l i n e a r c o n c e n t r i c fig ures (fig. 3 . 2 a ) . S h a p e s a p p r o x ­ i m a t i n g c o n c e n t r i c c i r c l e s c a n b e s e e n i n t;he P o v e r t y P o i n t c o m p l e x i n n o r t h - h a FIGURE 3 .I E u c l i d e a n g e o m e t r y in N a t i v e A m e r i c a n a r c h i t e c t u r e (a) Aerial photo of Bandelter, one of the Ancestral Pueblo settlements (starting around 1100 c .e .) in norluwestern New Mexico, (b) Aerial photo of Pueblo Bonito, another Ancestral Pueblo . settlement (starting around 950 C. E. ) . N o te t h a t they are mostly rectangular at the smallest scale and circular at the largest scale. ( a , p h o io by T o m B aiter; b , photo by G e o rg G e r s te r .) F ra c ta l m cross-cultural comparison 41 e r n L o u i s i a n a , f o r e x a m p l e , a n d t h e r e w e r e c o n c e n t r i c c i r c l e s o f t e p e e s in t h e C h e y e n n e c a m p s . T h e s t e p - p y r a m i d s o f M e s o a m e r i c a l o o k li k e c o n c e n t r i c s q u a r e s w h e n v i e w e d fr o m a b o v e . B u t l i n e a r c o n c e n t r i c fig ures a r e n o t f r a c ta ls. F irst, t h e s e a re l i n e a r layers: t h e d i s t a n c e b e t w e e n li n e s is alw a y s t h e s a m e , a n d t h u s t h e n u m b e r o f c o n c e n t r i c c ircles w i t h i n t h e largest circle is fin ite. T h e n o n ­ l i n e a r s c a l i n g o f f r a c t a l s r e q u i r e s a n e v e r - c h a n g i n g d i s t a n c e b e t w e e n lin e s , figure 3.2 L i n e a r c o n c e n tr ic f o r m s in N a tiv e A m e r i c a n a r c h i t e c t u r e (a) N a t i v e A m e r i c a n a r c h i t e c t u r e is t y p i c a l l y b a s e d o n q u a d r i l a t e r a l grid s o r a c o m b i n a t i o n o f circ ula r a n d g ri d fo r m s . T h e o n l y e x a m p l e s o f s c a l i n g s h a p e s a re t h e s e li n e a r c o n c e n t r i c fo rm s. In the P o v e r t y P o i n t c o m p l e x , fo r e x a m p l e , c o n c e n t r i c c i r c l e s w e r e u se d, a n d c o n c e n t r i c sq u a r e s c a n be seen if w e l o o k a t t h e M e x i c a n s t e p p y r a m i d s f r o m a b o v e . T h e s e fo r m s a re b e t t e r c h a r a c t e r i z e d as E u c li d e a n t h a n f r a c t a l fo r t w o re a so n s: ( b ) First , t h e y a re lin e a r . H e r e is a n e x a m p l e o f a n o n l i n e a r c o n c e n t r i c c i r c l e . W h i l e t h e l i n e a r v e r s i o n m u s t h a v e a f in it e n u m b e r o f c i r c l e s , t h is figure c o u l d h a v e a n i n f i n i t e n u m b e r a n d st il l fit in t h e s a m e b o u n d a r y , ( c ) S e c o n d , t h e y o n l y s c a l e w i t h r e s p e c t to o n e p o i n t ( t h e c e n t e r ) . H e r e is a n e x a m p l e o f c i r c l e s w i t h m o r e g l o b a l s c a l i n g s y m m e t r y . /rUroduction w h i c h m e a n s t h e r e c a n b e a n i n f i n i t e n u m b e r in a f i n i t e s p a c e (fig. 3 . 2 b ) . S e c ­ o n d , e v e n n o n lin e a r c o n c e n t r ic c ircles are o n ly self-sim ilar w ith re sp e c t to a sin g le locus ( t h e c e n t e r p o i n t ) , r a t h e r t h a n h a v in g t h e g lo b a l se lf-sitn ilarity o f f r a c t a l s (fig. 3 . 2 c ) . T h e i m p o r t a n c e o f t h e c i r c l e is d e t a i l e d i n a f a m o u s p a s s a g e b y B l a c k E l k ( 1 9 6 1 ) , in w h i c h h e e x p l a i n s t h a t “e v e r y t h i n g a n I n d i a n d o e s is in a c i r c l e , a n d t h a t is b e c a u s e t h e P o w e r o f t h e W o r l d alw a y s w o rk s in c ir c le s , a n d e v e r y t h i n g trie s t o be r o u n d . ” B u t h e g o e s o n to n o t e t h a t h is p e o p l e t h o u g h t o f t h e i r w o r ld as “ t h e circle o f t h e four q u a r t e r s ." A s im i la r c o m b i n a t i o n o f t h e c ir c le a n d q u a d r i ­ la t e r a l fo r m c a n b e s e e n m a n y N a t i v e A m e r i c a n m y t h s a n d a rtifa c ts; it is n o t t h e i r o n l y d e s i g n t h e m e , b u t i t c a n b e f o u n d i n a s u r p r is in g n u m b e r o f d i f f e r e n t s o c i ­ eties. B u rla n d (1 9 6 5 ) , for e x a m p le , s h ow s a c e re m o n ia l r a ttle c o n s is tin g o f a w o o d e n h o o p w i t h a cross i n s i d e fr o m s o u t h e r n A l a s k a , a N a v a j o s a n d p a i n t i n g s h o w i n g f o u r e q u i d i s t a n t sta lk s o f c o r n g r o w in g fro m a c i r c u la r lak e, a n d a P a w n e e bu ffaloh i d e d r u m w i t h f o u r a r r o w s e m a n a t i n g f r o m its c i r c u l a r c e n t e r . N a b o k o v a n d E a s t o n ( 1 9 8 9 ) d e s c r i b e t h e c u l t u r a l s y m b o l i s m o f t h e t e p e e in t e r m s o f its c o m ­ b i n a t i o n o f c i r c u l a r h i d e e x t e r i o r a n d t h e fo u r m a i n s t r u t s o f t h e i n t e r i o r w o o d s u p p o r t s . W a t e r s ( 1 9 6 3 ) p r o v i d e s a n e x t e n s i v e i l l u s t r a t i o n o f t h e c u l t u r a l s ig ­ n if ic a n c e o f c o m b i n i n g t h e c i r c u la r a n d cross fo rm in his c o m m e n t a r y o n t h e H o p i c r e a tio n m yth. T h e fo u rfo ld s y m m e t r y o f t h e q u a d r i l a t e r a l fo rm h a s le a d t o s o m e s o p h i s ­ t i c a t e d c o n c e p t u a l s t r u c t u r e s i n N a t i v e A m e r i c a n k n o w l e d g e s y s te m s . I n N a v a j o s a n d p a i n t i n g , for e x a m p l e , t h e c r u c i f o r m s h a p e r e p r e s e n t s t h e “ fo u r d i r e c t i o n s ” c o n c e p t , s i m i l a r to t h e C a r t e s i a n c o o r d i n a t e s y s te m . W h i l e o r d e r l y a n d c o n s i s ­ t e n t , it is by n o m e a n s s i m p l e (s e e W i t h e r s p o o n a n d P e t e r s o n 1 9 9 5 ) . T h e fo u r N a V a j o d i r e c t i o n s a r e a ls o a s s o c i a t e d w i t h c o r r e s p o n d i n g s u n p o s i t i o n s ( d a w n , day , e v e n i n g , n i g h t ) , y e a r l y s e a s o n s ( s p r i n g , s u m m e r , fa ll, w i n t e r ) , p r i n c i p a l c o lo r s ( w h i t e , b l u e , y e llo w , b l a c k ) , a n d o t h e r q u a d r i l a t e r a l d i v i s i o n s ( b o t a n i c a l c a t e g o r i e s , p a r t i t i o n s o f t h e life c y c le , e t c . ) . T h e s e a r e f u r t h e r b r o k e n i n t o i n t e r ­ s e c t in g b ip o la rities (e.g., t h e e a s t /w e s t s u n p a t h is b r o k e n by t h e n o r t h / s o u t h d i r e c ­ t i o n s ) . C o m b i n e d w i t h c i r c u l a r c u r v e s ( u s u a lly r e p r e s e n t i n g o r g a n i c s h a p e s a n d p ro c e s s e s ), t h e r e s u lti n g s c h e m a a r e Tich c u l t u r a l re s o u rc e s f o r i n d i g e n o u s m a t h e ­ m a t ic s (see M o o r e 1994)- B ut, e x c e p t for m i n o r re p e t it io n s ( l ik e t h e s m a ll c ir c u la r k i v a s in t h e C h a c o c a n y o n s i t e o f fig. 3 . 1 ) t h e r e is n o t h i n g p a r t i c u l a r l y f r a c ta l a b o u t th e s e q u a d rila te ra l designs. M a n y M e s o a m e r i c a n . c i t i e s , s u c h a s t h e M a y a n s ’ T e o t i h u a c a n , t h e A z te c 's T e n o c h t i t l a n , a n d t h e T o l t e c ’s T u l a , e m b e d d e d a w e a l t h o f a s t r o n o m i c a l k n o w l ­ e d g e in t h e i r r e c t a n g u l a r l a y o u t s , a l i g n i n g t h e i r s t r e e t s a n d b u i l d i n g s w i t h h e a v ­ e n l y o b j e c t s a n d e v e n t s ( A v e n i 1 9 8 0 ) . J. T h o m p s o n ( 1 9 7 0 ) a n d K l e i n ( 1 9 8 2 ) Fractals in cross-cultural com parison d e s c r ib e t h e q u a d r i l a t e r a l figure as a n u n d e r l y i n g t h e m e in M e s o a m e r i c a n g e o ­ m e t r i c t h i n k i n g , fr o m s m a l l - s c a l e m a t e r i a ! c o n s t r u c t i o n t e c h n i q u e s s u c h as w e a v i n g , t o t h e h e a v e n l y c o s m o l o g y o f t h e f o u r . s e r p e n t s . R o g e l i o D iaz, o f th e .M a th e m a tic s M u seu m a t,th e U n iv e rs ity -o f Q u e re ta r o , p o in ts o u t th a t th e skin p a t t e r n s o f t h e d i a m o n d b a c k r a t t l e s n a k e w e r e u se d by t h e M a y a n s t o s y m b o liz e th i s c o n c e p t (fig. 3 . 3 a ) . C o m p a r i n g t h e M a y a n s n a k e p a t t e r n w i t h a n A f r i c a n w e a v in g b a s e d o n th e c o b r a s k i n p a t t e r n (fig. 3 . 3 b ) , w e c a n see h o w g e o m e t r i c m o d e l i n g o f s i m i l a r n a t ­ u ra l p h e n o m e n a i n t h e s e tw o c u l t u r e s re s u lts in v e ry d i f f e r e n t r e p r e s e n t a t i o n s . T h e N a t i v e A m e r i c a n e x a m p l e e m p h a s i z e s t h e E u c l i d e a n s y m m e t r y w ithin one size fr a m e ( “size f r a m e ” b e c a u s e t h e t e r m “s c a l e ” is c o n f u s i n g in t h e c o n t e x t o f s n a k e s k i n ) . T h i s M a y a n p a t t e r n is c o m p o s e d o f four s h a p e s o f t h e s a m e size, a fo u r fo ld sy m m e try . B u t t h e A f r i c a n e x a m p l e e m p h a s iz e s f r a c ta l s y m m e t ry , w h i c h is n o t a b o u t s i m i l a r i t y b e t w e e n r i g h t / l e f t o r u p / d o w n , b u t r a t h e r s i m i l a r i t y betw een d iffe r e n t size fr a m e s . T h e A f r i c a n s n a k e p a t t e r n s h o w s d i a m o n d s w i t h i n d i a m o n d s w i t h i n d i a m o n d s . N e i t h e r d e s i g n is n e c e s s a r il y m o r e a c c u r a t e : c o b r a s k i n d o e s i n d e e d e x h i b i t a f r a c ta l p a t t e r n — t h e s n a k e ’s " h o o d , ” its t w i n w h i t e p a t c h e s , a n d t h e i n d i v i d u a l sc a le s t h e m s e l v e s a re all d i a m o n d s h a p e d — a n d yet s n a k e s k i n p a t t e r n s ( t h a n k s t o t h e a r r a n g e m e n t o f t h e s c a le s ) a re also c h a r a c ­ te r i s t i c a l l y in d i a g o n a l ro w s, so t h e y a re a c c u r a t e l y m o d e l e d as E u c l i d e a n s t r u c ­ t u r e s as w e ll. E a c h c u l t u r e c h o o s e s t o e m p h a s i z e t h e c h a r a c t e r i s t i c s t h a t b e s t fit its d e s i g n t h e m e . T h e r e a re a few c a s e s in w h i c h N a t i v e A m e r i c a n s h a v e u s e d s c a l i n g g e o ­ m e t r i e s in a r t i s t i c d e s ig n s . S e v e r a l o f t h e s e w e r e n o t , h o w e v e r , p a r t o f t h e t r a ­ d i t i o n a l r e p e r t o i r e . ^ N a v a j o b l a n k e t s , f o r e x a m p l e , w e r e o r i g i n a l l y q u i t e lin e a r; -it w a s -o n l y c n - e x a m i n i n g P e r s i a n - r u g a ::t-hat N a v a j o w e a v e r s b e g a n t o u se m o r e s c a l i n g s t y le s o f d e s i g n ( a n d e v e n t h e n t h e d e s i g n s w e r e m u c h m o r e E u c l i d e a n t h a n t h e P e r s i a n o r i g in a l s ; see K e n t 1 9 8 5 ) . T h e P u e b l o " s t o r y t e l l e r ” figures h a v e s o m e s c a l i n g p r o p e r t i e s , b u t th e y a r e o f r e c e n t ( 1 9 6 0 s ) o r i g i n . P o t t e r y a n d c a l a ­ b a s h ( c a r v e d g o u r d ) a r t i s a n s in A f r i c a o f t e n c r e a t e s c a l i n g by a l l o w i n g t h e d e s i g n a d a p t i v e l y t o c h a n g e p r o p o r t i o n a c c o r d i n g to t h e t h r e e - d i m e n s i o n a l fo rm o n w h i c h it is i n s c r i b e d (s e e “a d a p t i v e s c a l i n g ” in c h a p t e r 6 ) , b u t th i s w a s q u i t e ra r e i n N a t i v e A m e r i c a n p o t t e r y u n t i l t h e 19 60s. F in a lly , t h e r e a r e t h r e e N a t i v e A m e r i c a n d e s i g n s t h a t a re b o t h i n d i g e n o u s • a n d f r a c ta l . T h e b e s t c a s e is t h e a b s t r a c t fig u ra tiv e a r t o f t h e H a i d a , K w a k i u t l , T l i n g u t , a n d o t h e r s in t h e P acific N o r t h w e s t ( H o l m 1 9 6 5 ). T h e figures, p rim a rily c a r v i n g s , h a v e t h e k i n d o f g lo b a l , n o n l i n e a r s e lf- s im i la r it y n e c e s s a r y t o q ualify as f r a c ta ls a n d c le a r ly e x h i b i t r e c u r s i v e s c a l i n g o f u p t o t h r e e o r fo u r it e r a t i o n s . T h e y a ls o m a k e u se o f a d a p t i v e s c a l i n g , as il l u s t r a t e d by t h e s h r i n k i n g se rie s o f FIGURE 3.3 S n a k e s h i n m o d e ls in N a t i v e A m e r i c a n a n d A f r i c a n c u l t u r e s U ) R o g e lio D iaz o f t h e M a t h e m a t ic s M u se u m a t th e U n iv e r s ity o f Q u e r e t a r o sh o w s h o w the sk in p a tterns o f the d ia m o n d b a c k ra ttlesn ak e w ere used by the M a y a n s to sy m bo lize a c o s m o lo g y based on quad rilateral structure, (b) T h e M a n d ia c k w e avers o f G u in e a - B is s a u h a v e also cre a ted an a b stra c t d esign based on a s n a k e s k in p a tte r n , h u t c h o s e to e m p h a siz e t h e fractal c h a ra c te ristic s . Fractals m cross-cwltural comparison figures o n t h e d i m i n i s h i n g h a n d l e s o f s o u p ladles. R e s e a r c h e r s s i n c e A d a m s (1 9 3 6 ) h a v e p o i n t e d to t h e s im ila rity w ith e arly C h i n e s e art, w h i c h also h a s s o m e b e a u ti f u l e x a m p l e s o f s c a l i n g fo r m , a n d its s t y le o f c u r v a t u r e a n d b i l a t e r a l s y m ­ m e try c o u ld i n d e e d b e c u l t u r a l l y rie d '’to t h e s e N e w W o r l d d e s i g n s t h r o u g h a n a n c i e n t c o m m o n o r i g in . H o w e v e r , 1 d o u b t t h a t is t h e c a s e for t h e s c a l in g c h a r - . a c t e r is tic s . T h e P a c i f i c N o r t h w e s t a r t a p p e a r s t o h a v e d e v e l o p e d its s c a l in g s t r u c tu r e as t h e r e s u lt o f c o m p e t i t i o n b e t w e e n a r t is a n s fo r in c r e a s i n g ly e l a b o r a t e c a rv in g s (F a ris 1 9 8 3 ) . A l t h o u g h s o m e r e s e a r c h e r s h a v e a t t r i b u t e d t h e c o m p e t i ­ tio n pressure to E u ro p e a n trad in g influences, th e d e v e l o p m e n t o f t h e scaling designs was c le a rly a n i n t e r n a l i n v e n t i o n . T h e o t h e r t w o t r a d i t i o n a l N a t i v e A m e r i c a n d e s i g n s d o n o t q ua lify as fr a c ­ tals q u i t e as w e ll. O n e i n v o l v e s t h e s a w - t o o t h p a t t e r n f o u n d in s e v e r a l b a s k e t a n d w e a v i n g d e s ig n s . W h e n t w o s a w - t o o t h row s i n t e r s e c t a t a n a n g le , t h e y c r e ­ a te a t r ia n g l e m a d e fr o m t r i a n g u l a r edges. B u t th e s e ty p ic a lly h a v e o n ly tw o i t e r ­ a t i o n s o f s c a l e , a n d t h e r e is n o i n d i c a t i o n in t h e e t h n o g r a p h i c l i t e r a t u r e t h a t th e y are s e m a n t i c a l l y tie d to ideas o f r e c u r s io n o r s c a l in g (see T h o m a s a n d S lo c k ish 1 9 8 2 , 1 8 ). T h e o t h e r is a n a r r a n g e m e n t o f s p i r a l a r m s o f t e n f o u n d o n c o il e d b a s k e t s . A g a i n , t h i s is s e l f - s i m i l a r o n l y w i t h r e s p e c t to t h e c e n t e r p o i n t , b u t t h e r e a r e s o m e n o n l i n e a r s c a l i n g v e r s i o n s ( t h a t is, d e s i g n s t h a t r a p i d l y g e t s m a ll e r as y o u m o v e f r o m b a s k e t e d g e t o c e n t e r ) . H o w e v e r , t h e s e d e s i g n s g e n e r a l ly a p p e a r t o b e a f u s io n b e t w e e n t h e c i r c u l a r fo r m o f t h e b a s k e t a n d t h e cru cifo rm s h a p e o f th e arm s: a g a in m o r e a c o m b i n a ti o n o f tw o E u c lid e a n shapes th a n a fractal. O n e o f t h e m o s t c o m m o n e x a m p l e s o f t h i s f u s i o n b e t w e e n t h e c ir c le a n d t h e cro ss is t h e “ b if o ld r o t a t i o n ” p a t t e r n in w h i c h t h e a r m s c u r v e in o p p o s i t e d i r e c t i o n s , as s h o w n in fig u re 3 .4 a . F ig u r e 3 . 4 b s h o w s t h e fig u re o f 2 b a t fromM i m b r e s p o t t e r y w i t h a m o r e c o m p l e x v e r s i o n o f t h e b if o ld r o t a t i o n . E u c l i d e a n s y m m e t r y h a s b e e n e m p h a s i z e d in t h i s figure; for e x a m p l e , t h e e a rs a n d m o u t h o f th e b a t h a v e b e e n m a d e to lo o k s i m i la r to in c r e a s e t h e b i l a te r a l sy m m e try , a n d th e belly is d r a w n as a r e c t a n g l e . F ig u re 3 . 4 c s h o w s t h e figure o f a b a t fr o m a n A fric a n d esign ; it is a zigzag s h a p e t h a t e x p a n d s in w i d t h fro m t o p to b o t t o m , r e p ­ r e s e n tin g t h e w i n g o f t h e b a t. H e r e w e see n e g l e c t o f t h e b i l a t e r a l s y m m e t r y o f t h e bat, a n d a n e m p h a s i s o n t h e s c a l in g fo lds o f a s i n g le w in g. A g a i n , t h e N a t i v e A m e r i c a n r e p r e s e n t a t i o n m a k e s use o f its q u a d r i l a t e r a l / c i r c u l a r d esig n t h e m e , ju st as th e A f r i c a n r e p r e s e n t a t i o n o f t h e b a t e m p h a s iz e s s c a l i n g d e s i g n . T h e r e is p l e n t y o f c o m p l e x i t y a n d s o p h i s t i c a t i o n in t h e in d i g e n o u s g e o m ­ etry a n d n u m e r ic - s y s te m s o f t h e A m e r i c a s (s e e A s c h e r 1 9 91 , 8 7 - 9 4 ; C lo s s 1986; Eglash i 9 9 8 b ) , b u t w i t h t h e im p re s s iv e e x c e p t i o n o f t h e Pacific N o r t h w e s t c a r v ­ ings, fra c ta ls a r e a l m o s t e n t i r e l y a b s e n t in N a t i v e A m e r i c a n d e sig n s. A rk a n s a s p o ttery (a) Pim a b a sk e t S o u th w e ste rn pottery m otif T h e c i r c u l a r a n d q u a d r i l a t e r a l f o r m s w e r e o f t e n c o m b i n e d i n N a t i v e A m e r i c a n d e s i g n s as a fo u rfo ld o r b ifold rota tio n . (b) T h i s i m a g e o f a b a t , fr o m a M i t n b r e s p o t t e r y i n S o u t h w e s t e r n N a t i v e A m e r ic a n tra dition , sh o w s an em p ha sis o n c ircu la r and q u a d rila te r a l fo r m . T h e ear a n d t h e m o u t h , fo r e x a m p le , are m ad e t o l o o k s i m i l a r t o e m p h a s i z e b i l a t e r a l s y m m e t r y , a n d t h e b e l l y is d r a w n as a r e c t a n g l e . It a l s o s h o w s t h e w i n g b o n e s a s a b i f o l d ro ta tio n pattern. (c) T h i s A f r ic a n scu lp tu re o f a bat, from th e L eg a so c ie ty o f Zaire, pays l i t t l e a t t e n t i o n t o t h e b i l a t e r a l s y m m e t r y o f t h e b a t ’s b o d y b u t g i v e s a n em p h a sis o n th e s c a lin g sym m etry o f the w in g folds, s h o w n as an e x p a n d i n g z ig z a g p a t t e r n . FIGURE 3 . 4 T h e bifold r o ta ti o n in N a t i v e A m e r i c a n d e s ig n ( a : L e f t , fr o m Miles 19 6 3 . C e n t e r , fr om S o u t h w e s t I n d i a n C r a f t A r t s by C l m a Lee T a n n e r . C o p y r ig h t 1 9 6 8 b y the A r i z o n a B o a rd o f Regents. R e p r i n t e d b y permission 0/ the U n i v e r s i t y o f A r i z o n a Press. Right, c o u r t e s y D o n C - r o w e - b , f r o m Z a s l o w 1 9 7 7 , c o u r t e s y o f the a u t h o r , c , cou rt esy o f D a n i e l D i e b u y c k . ) Fractals in cross-culcuTal comparison D e s i g n s o f A s i a a n d t h e S o u t h P a c ific S e v e r a l o f t h e S o u t h P a cific c u l t u r e s s h a r e a t r a d i t i o n o f d e c o r a t i v e c u r v e d a n d s p ira l fo rm s, w h i c h in c e r t a i n M a o r i v e r s i o n s —- p a r tic u la r l y t h e i r r a f t e r a n d t a t • to o p a t t e r n s — w o u ld c e r t a i n l y c o u n t a s ’f r a c ta l (s e e H a m i l t o n 1 9 7 7 ). T h e s e are s t r o n g l y su g g e s tiv e o f t h e c u r v a t u r e o f w a v e s a n d sw irling, w ater. C l a s s i c J a p a n ­ ese p a i n t i n g s o f w a t e r w a v e s w e r e a ls o p r e s e n t e d as fra c ta l p a t t e r n s in M a n d e l ­ b r o t 's ( 1 9 8 2 ) s e m i n a l t e x t ( p l a t e C 1 6 ) . T h e s e m a y h a v e s o m e h i s t o r i c r e l a t i o n to sc a lin g p a t t e r n s in C h i n e s e a r t (see W a s h b u r n a n d C r o w e 1988, fig. 6 .9 ) , w h i c h are b a s e d o n s w ir lin g fo r m s o f w a t e r a n d c lo u d s , a b s t r a c t e d as s p i r a l s c a l i n g ' s t r u c t u r e s . W h i l e b o t h t h e J a p a n e s e a n d C h i n e s e p a t t e r n s a re e x p l i c i t l y a s s o c i­ a te d w i t h a n effort t o i m i t a t e n a t u r e , t h e s e M a o r i d e sig n s are r e p o r t e d t o b e m o r e a b o u t c u l t u r e — in p a r t ic u l a r , th e y e m p h a s iz e m i rro r -im a g e s y m m e t rie s , w h i c h a re a s s o c i a t e d w i t h t h e i r c u l t u r a l t h e m e s o f c o m p l i m e n t a r i t y in s o c i a l r e l a t i o n s ( H a n s o n 1 9 8 3 ). In a l m o s t all o t h e r in d i g e n o u s e x a m p l e s , h o w e v e r , t h e P a c if ic I s la n d e r p a t ­ t e r n s a r e q u i t e E u c l i d e a n . S e t t l e m e n t l a y o u t , f o r i n s t a n c e , is t y p i c a l l y in o n e o r tw o ro w s o f r e c t a n g u l a r b u i l d i n g s n e a r t h e c o a s ts , w i t h c i r c u l a r a r r a n g e m e n t s o f r e c t a n g l e s a ls o o c c u r r i n g i n l a n d (s e e F r a s e r 1 9 6 8 ) . T h e b u i l d i n g c o n s t r u c ­ t i o n is g e n e r a l l y b a s e d o n a c o m b i n a t i o n o f r e c t a n g u l a r g rid s w i t h t r i a n g u l a r o r c u r v e d a r c h roofs. O c c a s i o n a l l y t h e s e t r i a n g u l a r fa c e s a re d e c o r a t e d w i t h t r i ­ angles, b u t o th e rw is e n o n s c a lin g d esig n s d o m i n a te b o th s tru c tu r a l a n d d e c o ­ r a t i v e patterns.-^ A g a i n , it is i m p o r t a n t to n o t e t h a t t h i s la c k o f fra c ta ls d o e s n o t im ply a lack o f s o p h i s t i c a t i o n in t h e i r m a t h e m a t i c a l t h i n k i n g . F o r e x a m p l e , A s c h e r ( 1 9 9 1 ) h a s a n aly z ed s o m e o f t h e a l g o r i t h m i c p r o p e r tie s o f W a rlp ir i (P acific Is la n d e r) sa n d d r a w in g s . S i m i l a r s t r u c t u r e s a re a ls o f o u n d ' i n " A f r i c a ; w h e r e th e y a re called" lu so n a . Buc w h i l e t h e lu s o n a t e n d t o use s i m i l a r p a t t e r n s a t d i f f e r e n t sc a le s (as w e w ill se e in c h a p t e r 5 ) , t h e W a r l p i r i d r a w i n g s t e n d t o use d i f f e r e n t p a t t e r n s at d i f f e r e n t sc ales. A s c h e r c o n c l u d e s t h a t t h e W a r l p i r i m e t h o d o f c o m b i n i n g d if ­ f e r e n t g r a p h m o v e m e n t s is a n a l o g o u s to a lg e b r a ic c o m b i n a t i o n s , b u t t h e A f r i c a n lu s o n a a re b e s t d e s c r i b e d as fractals. C o m p l i c a t i n g m y c h a r a c t e r i z a t i o n o f t h e S o u t h P acific as d o m i n a t e d by E u c l i d e a n p a t t e r n s is t h e e x t e n s i v e i n f l u e n c e o f I n d i a . I t is p e r h a p s n o c o i n c i ­ d e n c e c h a t th e t r ia n g l e of tr ia n g le s m e n t i o n e d a b o v e is m o s t c o m m o n in I n d o n e ­ sia. In a r c h i t e c t u r e , a f a m o u s e x c e p t i o n t o t h e g e n e r a l l y E u c l i d e a n fo r m is t h a t o f B o rob ud ur, a te m p le o f I n d i a n re lig io u s o rig in in J a v a . A l t h o u g h n o r t h e r n I n d i a t e n d s t o w a r d E u c l i d e a n a r c h i t e c t u r e , e x p l i c i t re c u r s iv e d e s ig n is s e e n in s e v e r a l t e m p l e s in s o u t h e r n I n d i a — t h e K a n d a r y a M a h a d e o in K h a j u r a h o is o n e o f t h e Introduction 48 c l e a r e s t e x a m p l e s — a n d is r e l a t e d to re c u r s iv e c o n c e p t s in re l ig i o u s c o sm o lo g y . T h e s e s a m e a re a s in s o u t h e r n I n d i a a ls o h a v e a v e r s i o n o f t h e lu s o n a d r a w in g s , a n d m a n y o t h e r ex am p les o f fractal design. Interestingly, th e s e e x a m p le s from s o u t h ­ e r n I n d i a a r e t h e p r o d u c t s o f D r a v i d i a n c u l t u r e , w h i c h is s u s p e c t e d t o h a v e sig­ n i f i c a n t h i s t o r i c a l r o o t s in A f r i c a . E u r o p e a n desig n s M o s t t r a d i t i o n a l E u r o p e a n f r a c ta l d e sig n s, lik e t h o s e o f J a p a n a n d C h i n a , a r e d u e t o i m i t a t i o n o f n a t u r e — a t o p i c we w ill t a k e u p i n t h e f o l lo w in g c h a p t e r . T h e r e a r e a t le a s t tw o s t e l l a r e x c e p t i o n s , h o w e v e r , t h a t a r e w o r t h n o t i n g . O n e is t h e s c a l in g i t e r a t i o n s o f t r ia n g l e s in t h e floor tile s o f t h e C h u r c h o f S a n t a M a r i a in C o s t n e d i n R o m e (see p l a t e 5 .7 in W a s h b u r n a n d C r o w e 1 9 8 8 ) . I h a v e n o t b e e n a b le to d e t e r m i n e a n y t h i n g a b o u t t h e i r c u l t u r a l o r i g in s , b u t t h e y a r e c le a rly a r t is t ic i n v e n t i o n r a t h e r t h a n i m i t a t i o n o f s o m e n a t u r a l f o r m . T h e o t h e r c a n b e f o u n d in c e r t a i n v a r i e ti e s o f C e l t i c i n t e r l a c e d esig n s. N o r d e n f a l k ( 1 9 7 7 ) su gg ests t h a t th e s e a re h is to ric a lly r e l a te d to t h e sp iral d e sig n s o f p r e - C h r i s t i a n C e l t i c re li­ g i o n , w h e r e t h e y t r a c e t h e f lo w o f a v it a l life fo r c e . T h e y a r e g e o m e t r i c a l l y classified as a n E u le r i a n p a t h , w h i c h is clo sely a s s o c ia te d w i t h m a t h e m a t i c a l k n o t t h e o r y (cf. J o n e s 1 9 9 0 , 9 9 ) . C o n c l u s io n F ra c t a l s t r u c t u r e is by n o m e a n s u n i v e r s a l in t h e m a t e r i a l p a t t e r n s o f i n d i g e n o u s so c ie tie s . I n N a t i v e A m e r i c a n d e sig n s, o n l y t h e P acific N o r t h w e s t p a t t e r n s s h o w a s t r o n g f r a c ta l c h a r a c t e r i s t i c ; E u c l i d e a n s h a p e s o t h e r w i s e d o m i n a t e t h e a r t a n d a r c h i t e c t u r e . E x c e p t fo r t h e M a o r i s p i r a l d e s i g n s , f r a c t a l g e o m e t r y d o e s n o t a p p e a r to be a n i m p o r t a n t a s p e c t o f in d i g e n o u s S o u t h Pacific p a t t e r n s e ith er. T h a t is n o t t o say t h a t f r a c ta l d e s i g n s a p p e a r n o w h e r e b u t A f r i c a — s o u t h e r n I n d i a is full o f f r a c ta l s , a n d C h i n e s e flu id sw irl d e s i g n s a n d C e l t i c k n o t p a t t e r n s a re a lm o s t c e r t a in l y o f i n d e p e n d e n t o rig in .”* T h e i m p o r t a n t p o i n t h e r e is t h a t t h e frac­ tal d e s i g n s o f A f r i c a s h o u l d n o t b e m i s t a k e n for a u n i v e r s a l o r p a n c u l t u r a l p h e ­ n o m e n o n ; th e y are c u ltu ra lly specific. T h e n e x t c h a p t e r w ill e x a m i n e th e q u e s t i o n o f t h e i r m a t h e m a t i c a l sp ecificity. CHAPTER ---------------— -------------------In te n tio n an d-------------------------------------------------in v e n tio n --------------------------------------------in--------------------------------------------------—design------------------------------ 1----— B efo re w e c a n d is c u ss t h e f r a c t a l s h a p e s in A f r i c a n s e t t l e m e n t a r c h i t e c t u r e s as g e o m e tric k n o w le d g e , we n e e d t o t h i n k carefully a b o u t th e re la tio n b e tw e e n m a te rial d e sig n s a n d m a t h e m a t i c a l u n d e rs ta n d i n g. D e s ig n s a re b e s t s e e n as p o s i ti o n e d o n a r a n g e o r s p e c t r u m o f ^ i n t e n t i o n . A t t h e b o t t o m o f t h e ra n g e a re u n i n t e n ­ t i o n a l p a t t e r n s, c r e a t e d a c c i d e n t a l l y as t h e b y - p r o d u c t o f s o m e o t h e r a c ti v it y . In t h e m i d d l e o f rh e r a n g e a re d e s i g n s t h a t a r e i n t e n t ' o n n ! in tu itiv e , w ith n o rules o r g u id e l in e s t o e x p l a i n its c r e a t i o n . A t t h e u p p e r e n d o f t h e r a n g e , we h a v e t h e i n t e n t i o n a l a p p l i c a t i o n o f e x p l i c i t r u l e s t h a t w e a re a c c u s t o m e d to a s s o c i a t i n g w i t h m a t h e m a t i c s . T h e f o l l o w i n g s e c t i o n s w ill e x a m i n e t h e f r a c ta l d e s i g n s t h a t o c c u r in v a r i o u s p o s i t i o n s a l o n g t h i s i n t e n t i o n a l i t y s p e c t r u m . F ra c ta ls fr o m u n c o n s c io its a c tiv ity A n e x c e l le n t e x a m p le o f u n i n t e n t i o n a l fractals c a n b e fo u n d in t h e w ork o f M ic h a e l Batty a n d Paul L ongley (1 9 8 9 ) , w h o e x a m i n e d t h e sh a p e o f large-scale u r b a n sprawl s u r r o u n d i n g E u r o p e a n a n d A m e r i c a n c i t i e s (fig. 4 . 1 ) . W h i l e t h e b l o c k s o f t h e s e citie s a re ty p ic a lly laid o u t in r e c t a n g u l a r g rids, a t v e ry large s cales— a r o u n d 100 sq u a re m i le s — w e c a n see t h a t t h e p r o c e s s o f p o p u l a t i o n g r o w t h h a s c r e a t e d a n irre g u la r p a t t e r n . T h i s ty p e o f f r a c t a l , a “d if f u s i o n l i m i t e d a g g r e g a t i o n , ” also 50 In tro d u c tio n F I G U R E 4.1 U r b a n s p r a w l in L o n d o n L arg e-sca le urban spraw l g e n e ra lly h as a fra ctal stru ctu re. T h e u rb a n spraw l fra ctals o n ly ex ist at v e ry l a r g e s c a l e s — a b o u t 1 0 0 sq. m ile s — a n d re s u lt fro m th e u n co n sciou s accu m u lation o f u rb an p o p u la tio n d yn am ics. A t le v e ls o f co n s c io u s in te n t ( e .g ., th e grid o f c i t y b l o c k s ) , E u r o p e a n citie s are ty p ic a lly E u c l i d e a n . A r e a is 1 0 x 1 0 k ilom eters. ( .R e p r in t e d w i t h p e r m i s s i o n f r o m B a tty et al. 1 9 8 9 ) o c c u r s in c h e m i c a l s y s te m s w h e n p a r t i c l e s in a s o l u t i o n a r e a t t r a c t e d t o a n e l e c ­ t r o d e . F ra c t a l u r b a n s p r a w l is c le a r ly t h e r e s u l t o f u n c o n s c i o u s s o c ia l d y n a m i c s , n o t c o n s c i o u s d esig n . A t t h e s m a ll e r sc ale s in w h i c h t h e r e is c o n s c i o u s p l a n n i n g , E u r o p e a n a n d A m e r i c a n s e t t l e m e n t a r c h i t e c t u r e s a r e ty p i c a ll y E u c l i d e a n . F r a c t a l s fro m n a tu r e :.m im e s is v e rs u s m o d e lin g I t m i g h t b e t e m p t i n g t o t h i n k t h a t t h e c o r ^ a s t j ^ t w e e n . t h e E u c li .d e jn - d e s ig n s o f E u ro p e a n d t h e fra c ta l d e sig n s o f A f r i c a c a n be e x p l a i n e d by t h e i m p o r t a n t ro le o f t h e n a t u r a l e n v i r o n m e n t in A f r i c a n s o c i e ti e s . B u t t h i s a s s u m p t i o n t u r n s o u t to be w rong; if a n y th in g , th e r e is a t e n d e n c y for in d ig en o u s societies to .fay o r E u clid ­ e a n s h a p e s . P h y s i c is t K h . S . , M a m e d o v o b s e r v e d s u c h a c o n t r a s t in h i s r e f le c t io n s o n h i s y o u t h in a n o m a d i c c u l t u r e : My p aren ts an d c o u n try m en . . . u p to t h e s e c o n d w o r ld w a r h a d b e e n n o m a d s . • •. O u ts i d e ou r n o m a d t e n t s we w ere liv in g in a w o n d e rfu l k in g d o m o f variou s c u rv e d lines a n d forms. S o w h y w e re th e a e s t h e t i c signs n o t form ed from t h e m , h a v i n g in s te a d p re s e rv e d g e o m e t r i c p a t t e r n s . . . ? [Jjn t h e cities where th e straight-line geometry’ was p re d o m in a n t th e a esth etic signs were formed . . . w ith n a tu re playing th e d o m i n a ti n g role. . . . [TJhe n o m a d did n o t n eed the “p o rtra it” o f a n o ak to be carried w ith him elsewhere because h e co uld view all sorts o f oaks every day and every h o u r . . . while for th e tow nsfolk th e ir in c lin a ­ tion to n a tu re was m ore a result of nostalgia. ( M a m e d o v 1986, 5 1 2 - 5 1 3 ) I n t e n t i o n a n d i n v e n t i o n in d e s ig n . C o n t r a r y t o r o m a n t i c p o r t r a i t s o f t h e “n o b l e s a v a g e ” li v i n g as o n e w i t h n a tu r e , m o s t in d i g e n o u s s o c i e t i e s s e e m q u i t e i n t e r e s t e d i n d i f f e r e n t i a t i n g t h e m ­ selves fro m t h e i r s u r r o u q d ii ig s . I t is t h e i n h a b i t a n t s o f larg e s t a t e s o c i e ti e s , s u c h as t h o s e o f m o d e r n J i u r o p e } w h o y e a r n ,fo m i m i c t h e n a t u r a l . W h e n E u r o p e a n de signs a r e fr a c ta l , it is u s u a l ly d u e _t_P.an .effo rt.to m i m i c n a t u r e . A f r i c a n fra c ta ls based o n m i m i c r y o f n a t u r a l fo r m a re r e l a t i v e l y ra r e ; t h e ir i n s p i r a t i o n t e n d s to co m e fr o m t h e r e a l m o f c u l t u r e . H o w s h o u l d w e p l a c e s u c h n a t u r e - b a s e d d e s i g n s in o u r i n t e n t i o n a l i t y s p e c ­ tr um ? T h a t d e p e n d s o n t h e d i f f e r e n c e b e t w e e n m i m e s is.an d.m O -d elin g^> 4 im esjs^ is a r ^ a t t e m p t to m i r r o r t h e im a g e o f a p a r t i c u l a r o b j e c t , a g o al e x p l i c i t l y s t a t e d by P l a t o a n d A r i s t o t l e as t h e e s s e n c e o f a r t , o n e t h a t w a s s u b s e q u e n t l y fo l lo w e d in E urope for m a n y c e n tu r ie s (see A u e r b a c h 1953). A p h o to g r a p h is a goo d e x a m p le o f m im esis. A p h o t o m i g h t c a p t u r e t h e f r a c ta l im a g e o f a t r e e , b u t it w o u ld b e foolish to c o n c l u d e t h a t t h e p h o t o g r a p h e r k n o w s f r a c ta l g e o m e try . If a r t i s a n s a re simply try in g t o c o p y a p a rt ic u l a r n a t u r a l o b je c t, t h e n t h e sc alin g is a n .u .n in te n d e d b y -p r o d u c t. T h e m o s t i m p o r t a n t a t t n b u t e s j : h a t s e p a r a t e m i m e s is f r o m Q n o d e l i n g e r e a b s t r a c t i o n a n d g e n e r a l i z a t i o n .( A b s t r a c ti o n ^ is a n a t t e m p t t o l e a v e o u t m a n y o f th e c o n c r e t e d e t a i l s o f t h e s u b j e c t by c r e a t i n g a s i m p l e r figure w h o s e s t r u c t u r e is s till'ro u g h ly a n a lo g o u s t o t h e o r i g i n a l — o f t e n c a ll e d a “stylized" r e p r e s e n t a t i o n , in t h e a rts C jG en etaliz atio n ^ ir i e a n s s e l e c t i n g a n a n a l o g o u s s t r u c t u r e t h a t is^comm o n to all e x a m p l e s o f t h e s u b j e c t ; w h a t is o f t e n re f e r r e d t o as a n “u n d e r l y i n g ” form o r law .1 F or e x a m p l e , M a n d e l b r o t ( 1 9 8 1 ) p o i n t s t o t h e E u r o p e a n B e a u x A r t s : style as a n a t r e m p t n o t m e r e ly ro i m i t a t e n a t u r e , b u t t o “guess its law s.” H e n o t e s th a t th e in te rio r o f t h e Paris o p e ra h o u s e m a k e s use o f scalin g a rc h e s -w ith in -a rc h e s ;, . a p a t t e r n t h a t g e n e ra liz e s s o m e o f t h e s c a l in g c h a r a c t e r i s t i c s o f n a t u r e , but- is nota co py o f a n y o n e p a r t i c u l a r n a t u r a l o b j e c t . S i n c e t h e u l t i m a t e g e n e r a l i z a t i o n is a m a t h e m a t i c a l m o d e l ,, w h y d i d n ’t d e s i g n p r a c t i c e s s u c h as t h e B e a u x A r t s s t y l e r e s u l t i n a n e a r l y d e v e l o p m e n t o f fra c ta l g e o m e t r y ? For E u r o p e a n s , s u c h l u s h o r n a m e n t a t i o n w as p r e s e n t e d —-an d g e n e r a l ly a c c e p t e d — as e m b o d y i n g t h e opposite o f m a t h e m a t i c s ; i t w as ani e ff o rt to c r e a t e d e sig n s t h a t c o u ld o n ly b e u n d e r s t o o d in i r r a ti o n a l, e m o t i o n a l , o r i n t u ­ itiv e te r m s . E v e n s o m e m o v e m e n t s a g a i n s t t h i s a t t e m p t , s u c h as t h e u s e o f d i s ­ t i n c t l y E u c l i d e a n fo r m s in t h e h i g h m o d e r n a r t s s t y le , s i m p l y r e i n f o r c e d t h e a s s o c i a t i o n b e c a u s e it o n l y o f f e r e d a r e v e r s a l , s u g g e s t i n g t h a t “ m a t h e m a t i c a l ” shapes (cubes, sp h eres, e tc .) c o u ld be e s th e tic a lly a p p re c ia te d . W i t h rare e x c e p t i o n s ( e .g ., T h o m p s o n 1 9 1 7 ) , m i m e s i s o f n a t u r e in p r e - W W 11 E u r o p e a n a rt t r a d i t i o n s m e r e ly f u r t h e r e d t h e a s s u m p t i o n t h a t E u c l i d e a n g e o m e t r y w as t h e o n ly tr u e g e o m e t r y . 2 5* In tro d u c tio n T h e d if f e r e n c e b e tw e e n ^ m im e s js a n d m o d e l i n g p ro v i d e s tw o d i f f e r e n t posit i o n s a l o n g t h e j n t e n t i o n 3 l j C y _ s g e c t r u m . T h e le a s t i n t e n t i o n a l w o u ld b e m e r e ly h o l d i n g a m i r r o r t o n a t u r e —-for e x a m p l e , if s o m e o n e w a s j u s t s h o o t i n g , a c a m ­ e ra o r p a i n t i n g a realistic p ic t u r e o u td o o r s a n d h a p p e n e d t o i n c l u d e a fra c ta l o b j e c t ( c l o u d , tr e e , e t c . ) . T h i s m i m e s is d o e s n o t c o u n t as m a t h e m a t i c a l t h i n k i n g . M o r e | i n t e n t i o n a l is a stylized r e p r e s e n t a t i o n o f n a t u r e . If t h e a r t i s t h a s r e d u c e d t h e n a t - i ural im ag e t o a s t ru c tu r a ll y a n a l o g o u s c o l l e c t i o n o f m o r e s i m p l e e l e m e n t s , s h e h a s \ c re a te d a n ab stract m odel. W h e t h e r o r n o t s u c h a b stra c tio n s m o v e to w a rd m o re J m a t h e m a t i c a l m o d e l s is a m a t t e r o f lo c al p r e f e r e n c e . T h e t w o e x a m p l e s o f A f r i c a n r e p r e s e n t a t i o n s o f n a t u r e w e o b s e r v e d in th e previous c h a p te r c e rta in ly sh o w th a t th e a rtisa n s h a v e g o n e b e y o n d m e r e m im e s is. T h e M a n d i a c k c o b r a p a t t e r n w e saw in fig u re 3 .2 . s h o w s a s t r ic t ly sy ste m a tic scalin g p a tte rn . T h is te x tile d esig n c o n v e y s th e sc a lin g p ro p e rty o f 't h e n a tu r a l c o b ra s k in p a t t e r n — d ia m o n d s a t m a n y sc a le s— in a sty lized or a b s t r a c t way. W e c a n t a k e th is- i d e a a s t e p f u r t h e r b y e x a m i n i n g a n o t h e r B w a m i b a t s c u l p t u r e (fig . 4 . 2 ) . T h i s s p i r a l p a t t e r n is a l s o a s t y l i z e d r e p r e ­ s e n t a t i o n o f t h e n a t u r a l s c a l i n g o f t h e b a t ’s w in g , b u t it.is a d i f f e r e n t g e o m e t r i c d e s i g n t h a n t h e e x p a n d i n g zigzag p a t t e r n w e s a w in fi g u r e 3 .4 c . It is m o r e s ty 1- FIGURE 4-2 Stylized, s c u l p t u r e o f a b a t A nother Legn bat sculpture, but unlike the zigzag design we saw in figure 3.4c, here the scaling of the wing folds is represented by a spiral. (B y p erm ission o f th e M u se u m o f A f r i c a n A r t, N .Y .) /m ention an d invention in design iz e d i n t h e s e n s e o f b e i n g f u r t h e r a b s t r a c t e d f r o m t h e o r i g i n a l n a t u r a l b a t ’s w i n g . T h i s p r o v i d e s f u r t h e r e v i d e n c e t h a t t h e sc u l p t o r s w e r e f o c u s e d o n t h e s c a l i n g p r o p e r t i es— t h e g e n e r a l i z e d u n d e r l y i n g f e a t u r e — a n d n o t p a r t i c u l a r c o n ­ c r e te d etails. f T h e g r e a t e s t d a n g e r o f t h i s b o o k is t h a t r e a d e r s m i g h t m i s i n t e r p r e t its ^ \ m e a n i n g in te rm s o f p rim itiv ism . T h e fa c t t h a t A f r i c a n j r acta ls are_rarely t h e r e s u l t 1 o f i m i t a t i n g n a t u r a l fo r m s Jh e lp s r e m i n d us t h a t t h e y a re n o t d u e t o “p r i m i t i v e s l i v in g c lo s e t o n a t u r e . ” B u t e v e n fo r t h o s e r a r e cases i n w h i c h A f r i c a n f r a c ta ls are re p r e s e n ta tio n s o f n a tu r e , it is c lea rly a s £ ^f'C ^scio \js,abstractiQ n , n o t a m im e tic re f le c tio n . T h e g e o m e t r i c c h i n k i n g t h a t g o e s i n t o chesg^exam ples m a y ja g s i m p le, b u t it is q u i t e i n t e n t i o n a l . T h e fr a c ta l e s t h e t i c J u s t as w e saw h o w d e s i g n s b a s e d o n n a t u r e r a n g e fr o m u n c o n s c i o u s t o i n t e n ­ t i o n a l , a r t i f i c i a l d e s i g n s a ls o v a r y a l o n g a r a n g e o f i n t e n t i o n , w i t h s o m e s i m p l y th e result o f an in tu itiv e in s p ira tio n , a n d o th e r s a m o re self-conscious a p p lic a ­ t i o n o f ru le s o r p r i n c ip l e s . T h e e x a m p l e s o f A f r i c a n f r a c ta l s in fig u re 4 .3 d i d n o t a p p e a r to be r e l a t e d t o a n y t h i n g o t h e r t h a n t h e a r t i s a n ’s e s t h e t i c i n t u i t i o n or s e n s e o f b e auty. A s far as 1 c o u l d d e t e r m i n e fr o m d e s c r i p t i o n s in t h e l i t e r a t u r e a n d m y o w n fie ld w o rk , t h e r e w e r e n o e x p l i c i t ru le s a b o u t h o w t o c o n s t r u c t th e s e d e s i g n s , a n d n o m e a n i n g w as a t t a c h e d to t h e p a r t i c u l a r g e o m e t r i c s t r u c t u r e o f t h e fig u re s o t h e r t h a n l o o k i n g g o o d . I n p a r t i c u l a r , I s p e n t s e v e r a l w e e k s in D a k a r w a n d e r i n g t h e s t r e e ts a s k i n g a b o u t c e r t a i n f r a c ta l fa b ric p a t t e r n s a n d j e w ­ e l r y d e s i g n s , a n d cite i n s i s t e n c e c h a t t h e s e p a t t e r n s w e r e “j u s t fo r l o o k s ” w as so a d a m a n t t h a t , if s o m e o n e f i n a l l y h a d o f f e r e d a n e x p l a n a t i o n , I w o u l d h a v e v i e w e d it w i t h s u s p ic io n . S i n c e s o m e p r o f e s s i o n a l m a t h e m a t i c i a n s r e p o r t t h a t t h e i r id e a s w e r e p u re i n t u i t i o n — a s u d d e n flash o f i n s i g h t , o r “A h a ! ” as M a r t i n G a r d n e r p u t s it— we s h o u l d n ’t d i s c o u n t t h e g e o m e t r i c t h i n k i n g o f a n a r t i s a n w h o r e p o r t s “1 c a n ’t tell y o u h o w I c r e a t e d r h a t , it j u s t c a m e t o m e . " E s t h e t i c p a t t e r n s c l e a r ly q u a li fy as i n t e n t i o n a l d e s ig n s . O n t h e j i t h e r hand,, t h e r e is n ’t m u c h we c a n s a y a b o u t t h e m a t h e m a ti c a l ideas b e h i n d th e s e p a tte rn s ; th e y will h a v e t o r e m a in a m ystery unless s o m e t h i n g m o r e is rev eale d a b o u t t h e i r m e a n i n g o r d i e a r t is a n ’s; c o n s t r u c t i o n te c h niqu£S.sl t is w o r t h n o t i n g , h o w e v e r , t h a t t h e y d o c o n t r i b u t e to t h e f r a c ta l d e sig n t h e m e in A f r i c a . E s t h e t i c p a t t e r n s h e l p i n s p ir e p r a c t i c a l d esig n s, a n d v ic e .y e rsa . S i n c e a n c i e n t tr a d e n e t w o r k s w e r e w e ll e s t a b l i s h e d , t h e d if fu sio n o f e s t h e t i c p a t ­ t e r n s is p r o b a b l y o n e p a r t o f t h e e x p l a n a t i o n for h o w fra c ta ls c a m e t o be so w i d e ­ s p r e a d ac ro s s t h e A f r i c a n c o n t i n e n t . FIGURE 4 .3 E s t h e t i c fr a c ta l s ( a ) M e u r a n t ( q u o t e d in R e i f 1 9 9 6 ) r e p o r t s t h a t t h e M b u t i w o m e n w h o c r e a t e d t h is f r a c t a l d e s i g n , a b a r k - c l o t h p a i n t i n g , t o ld h i m t h e d e s i g n w a s n o t “ t e l l i n g s t o r ie s , " n o r w a s it “ r e p r e s e n t i n g a n y vp a r t i c u i a r o b j e c t . ” ( b ) S c a l i n g p a t t e r n s c a n be fo u n d in m a n y A f r i c a n d e c o r a t iv e designs t h a t a r e r e p o r t e d t o b e “ ju st fo r b e a u t y . ” U p p e r l e f t , S h o o w a R a f f i a c l o t h ; l o w e r l e ft, S e n e g a l e s e t i e d y e ; rig ht, S e n e g a l e s e pendant. ( a , co u rt e sy G e o r g e s M e n i a n l . b : U ppe r le ft , B r it is h M u s e u m ; l o w e r l e f t , f r o m M u s e c R oy al d e l A / r i q u c C e n t r a l , Belgiu m ; right, photo co urte sy I F A N , D a k a r .) Im e n iio n a n d in v e n tio n in design 55 FIGURE 4 . 4 T h e q u in c u n x fra cta l A c u s t o m e r in T o u b a , S e n e g a l , s e l e c t s a f r a c t a l q u i n c u n x p a t t e r n f o r h is l e a t h e r n e c k b a g . T h e q u i n c u n x is h i s t o r i c a l l y i m p o r t a n t b e c a u s e o f its u se b y e a r l y A f r i c a n A m e r i c a n “ m a n o f s c i e n c e " B enjam in Banneker. O f c o u r s e , t h e r e a r e p l e n t y o f A f r i c a n d e s i g n s t h a t a re s t r i c t l y E u c l i d e a n , b u t e v e n t h e s e c a n o c c u r i n “ f r a c t a l i z e d ” v e r s io n s . O n e p a r t i c u l a r l y i n t e r e s t i n g ex a m p l e is t h e quincunx (fig. 4 . 4 ) . T h e b a sic q u i n c u n x is a p a t t e r n o f five sq uares, w i t h o n e a t t h e c e n t e r a n d o n e a t e a c h c o r n e r . T h e d e s i g n is c o m m o n i n S e n e ­ gal, w h e r e it is sa id t o r e p r e s e n t t h e “ li g h t o f A l l a h . ” T h e q u i n c u n x is h i s t o r i ­ cally i m p o r c a n t b e c a u s e t h e im a g e w as r e c o r d e d a s j a e i n g o f re lig io u s s i g n if i c a o c e to t h e early A f r i c a n A m e r i c a n “ m a n o f s c i e n q e ” B e n j a m i n B a n n e k e r . S i n c e e v i ­ d e n c e s h o w s t h a t B a n n e k e r ’s g r a n d f a t h e r ( B a n n a k a ) c a m e f r o m S e n e g a l , t h e q u i n c u n x is a f a s c i n a t i n g p o s s i b i l i t y fo r g e o m e t r y in t h e A f r i c a n d i a s p o r a (s e e E glnsh 19 9 7 c fo r d e t a i l s ) . B e c a u s e o f t h e f r a c t a l e s t h e t i c , t h i s re l ig i o u s s y m b o l h is o f t e n . a r r a n g e d in a r e c u r s i v e p a t t e r n — fiv e s q u a r e s q f five s q u a r e s — a s s h o w n / . in figure 4 .4 in t h e d e s i g n fo r a l e a t h e r n e c k bag. J ‘ F in a lly , t h e r e a r e a ls o e x a m p l e s o f t h e f r a c t a l e s t h e t i c i n c o m m o n h o u s e ­ h o l d f u r n i s h i n g s . E u r o - A m e r i c a n f u r n i t u r e is d i f f e r e n t i a t e d by f o r m a n d f u n c ­ ti o n — sto o ls are s tr u c tu r e d d if fe re n tly fro m c h a irs, w h ic h are s tr u c tu r e d d i f f e r e n t l y f r o m c o u c h e s . B u t in A f r i c a n h o m e s o n e o f t e n se e s d i f f e r e n t sizes o f t h e s a m e s h a p e (fig. 4 .5 ). A s i m i l a r d i f f e r e n c e h a s b e e n n o t e d in c r o s s - c u l tu r a l co m p a riso n s o f h o u sin g . W h e r e a s E u r o - A m e r ic a n s w o u ld n e v e r t h i n k to h a v e a g o v e r n e r ’s m a n s i o n s h a p e d l i k e a p e a s a n t ’s s h a c k ( o r v ic e v e r s a ) , p r e c o l o n i a l A f r i c a n a r c h i t e c t u r e ty p i c a l l y u s e d t h e s a m e fo r m a t d i f f e r e n t sizes (as w e saw for t h e sta tu s d i s t i n c t i o n s in t h e B a -ila s e t t l e m e n t in c h a p t e r 2 ). I t is u n f o r t u n a t e \ f chat t h i s A f r i c a n s t r u c t u r a l c h a r a c t e r i s t i c is ty p i c a l l y d e s c r i b e d in t e r m s o f a ] ! lack— as d i e a b s e n c e o f s h a p e d i s t i n c t i o n s r a t h e r t h a n a s t h e p r e s e n c e o f a se a l- / \ ing d e s i g n t h e m e . Introduction 56 FI GURE 4 .5 T h e fr a c t a l e s t h e t i c i n h o u s e h o l d o b j e c t s A fr ic 3 n s to o ls , chairs, an d ben ch es are often created in a scaling series. (P h o to co u rtesy o f A f r i c a P l a c e , I n c .) C o n c lu s i o n W e n o w h a v e s o m e g u id e lin e s to h e l p d e t e r m i n e w h ic h fracta l designs s h o u ld c o u n t as m a t h e m a t i c s , w h i c h s h o u l d n o t , a n d w h i c h a re i n b e t w e e n . F ig u r e 4.-6 s u m ­ m arizes t h i s s p e c t r u m . F r a c t a ls p r o d u c e d b y ^ u n c o n s c i o u s a c t i v i t y , o r as t h e u n i n ­ t e n t i o n a l b y - p r o d u c t fr o m so m e o t h e r p u rp o s e , c a n n o t b e a t t r i b u t e d t o i n d i g e n o u s c o n c e p t s . B u t s o m e a r t i s t i c a c t i v i t i e s , s u c h as t h e c r e a t i o n o f sty liz e d r e p r e s e n - Llumtentioiial Unconscious activity •urban sprawl Accidental/ractals • “mirror” portrait of nature (moneys; e.g., photography) I n te n tio n a l I n te n tio n a l but implicit and explicit I 1 Construction techniques C o n s c i o u s m e o f n a t u r a l s c a lin g •stylistic abstraction of natural scaling Esthetic design •intuitive fractal design theme f ig u r e 4 .6 F r o m u n c o n s c i o u s a c c i d e n t to e x p lic it d e s i g n Knowledge systems / n t e n t i o n a n d i n v e n t i o n in d e s ig n c a tio n s o f n a t u r e o r p u r e l y e s t h e t i c d e s i g n s , d o s h o w i n t e n t i o n a l a c t i v i t y fo cu s ed o n f r a c ta ls. S u c h e x a m p l e s m a y be r e s t r i c t e d in t e r m s o f g e o m e t r i c t h i n k i n g — tja.e-mt-isans m a y o n l y r e p o r t t h a t t h e d e s i g n s u d d e n l y c a m e t o t h e m in a flash o f in t u it io n - V -b u t th e s e a r e c le a r ly d i s t i n g u i s h e d fr o m th o s e w h i c h a re u n c o n s c i o u s o T "a c c id e n ta l. T h e f o l l o w i n g c h a p t e r s will c o n s i d e r e x a m p l e s t h a t a re n o t o n ly i n t e n t i o n a l , b u t also in c l u d e e n o u g h e x p li c it in f o r m a t i o n a b o u t design te c h n i q u e s a n d k n o w le d g e sy stem s to be easily id en tifiab le as m a t h e m a t i c a l p ra ctice a n d ideas. PA RT -A frican------------------ — “fractal-m athem atics- CHAPTER 5 G eom etric algorithms- T h e w ord (J^g o rith m J) d e riv e s fro m th e n a m e o f a n A r a b m a t h e m a tic ia n , A l-K h w arizm i (c. 7 8 0 - 8 5 0 c .e .), w h o s e b o o k H is a b a l'ja b r w ’ al-m u q a b a lci ( C a l ­ c u l a t i o n by R e s t o r a t i o n a n d R e d u c t i o n ) a l s o g a v e us t h e w o r d “ a l g e b r a . ” A l t h o u g h A l - K h w a r i z m i f o c u s e d o n n u m e r i c p r o c e d u r e s fo r s o l v i n g e q u a t i o n s , t h e m o d e r n t e r m ^ a l g o r i t h m / a p p li e s to a n y for m a lly s p e c ifie d p r o c e d u r e . A p e o m e t r i c a l g o r i t h m g iv es e x p l i c i t i n s t r u c t i o n s for g e n e r a t i n g a part.ifi.u la r s e t o f s p atia l p a t t e r n s . W e h a v e a l r e a d y s e e n h o w i t e r a t i o n s o f s u c h p a t t e r n - g e n e r a t i n g p r o c e d u r e s c a n p r o d u c e f r a c t a l s o n a c o m p u t e r s c r e e n ; i n t h i s c h a p t e r we will e x a m i n e t w o i n d i g e n o u s a l g o r i t h m s t h a t a ls o use i t e r a t i o n t o p r o d u c e s c a l i n g designs: t h e 4 5 -degreg_-angle p o q s tr u e tjo n s o f t h e M a n g b e t u , a n d t h e lu s o n a d r a w ­ ings o f t h e C h o k w e . g e o m e t r y in M c in g b e tu d e s i g n T h e M a n g b e t u o c c u p y t h e D e l e R i v e r a r e a in t h e n o r t h e a s t e r n p a r t o f t h e D e m o c r a t i c R e p u b l i c o f C o n g o (f o rm a lly Z a ir e ) . A r c h a e o l o g i c a l e v i d e n c e s h o w s ir o n s m e l t i n g in t h e are a s i n c e 2 3 0 0 b .c . e ., b u t th e M a n g b e tu , c o m in g from drier lands a ro u n d p re s e n t-d a y U g a n d a , d id n o t arriv e u n til a b o u t 1000 c .e. T hrough b o th c o n flic t a n d c o o p e ra tio n , th e y e x c h a n g e d c u ltu ra l tra d itio n s w ith o th e r 61 62 A fr ic a n fra c ta l m athem atics s o c i e ti e s o f t h e a re a : B a n t u - s p e a k i n g p e o p le s s u c h a s t h e B u d a , B u a a n d L ese, a n d U b a n g i a n - s p e a k i n g p e o p l e s s u c h as t h e A z a n d e , B a n g b a , a n d B ara-m bo. A r o u n d 180 0 a n um b,er o f s m a ll c h ie f d o m s w e re c o n s o l id a t e d i n t o tjfie first M a n g b e t u k i n g ­ d o m . A l t h o u g h it la s te d o n l y t w o g e n e r a t i o n s , a t r a d i t i o n o f c o u r t ly p re s tig e c o n t i n u e d e v e n in sm all villages a n d spread t o m a n y o f th e M a n g b e t u ’s tr a d in g p artn ers. T h i s c o m b i n a t i o n o f c u l t u r a l d iv e r s it y , e x c h a n g e , a n d p r e s t i g e r e s u l t e d in a th r iv in g artistic tra d itio n . / A d e t a i l e d a c c o u n t o f M a n g b e t u h is t o r y a n d t r a d i t i o n s c a n b e f o u n d in African Reflections: A n from Northeastern Zaire. S c h i l d k r o u t a n d K e im ( iggo) beg in - t h e i r a n a ly s is b y s h o w i n g t h a t t h e m o s t f a m o u s a s p e c t o f Man_gbe.Lti-.3rt, t h e ‘‘rjat.u ra lis tic.Jb o k ,’’ w a s a c t u a l l y q u i t e rare in t h e t r a d i t i o n a l M a n g b e t u s o c i e t y o f t h e n i n e t e e n t h c e n t u r y . D u r i n g a r e s e a r c h e x p e d i t i o n t o t h e C o n g o in *914 ( th e o rig in o f th e p h o to s used h e re ) , m a m m alo g ist H e r b e r t L a n g b e c a m e f a s c i n a t e d w i t h life lik e c a r v i n g s o f h u m a n figures, a n d as w o r d s p r e a d t h a t h e w a s p a y ­ in g h i g h p r i c e s fo r t h e m , m o r e o f t h e s e c a r v i n g s w e r e p ro d u c e d . O t h e r co lle c to r s c a m e to b uy th ese piec e s,.an d e v e n t u a l l y t h e e c o n o m i c r e w a r d s for p r o d u c i n g n a t u ­ ra l is t ic M a n g b e t u a r t b e c a m e so s t r o n g t h a t it r e p l a c e d o t h e r s t y le s . S c h ild k ro u t a n d K eim sh o w t h a t o rig in ally th e m o s t i m p o r t a n t e s t h e t i c w as n o t n a tu r a lis m , b u t a b stra c t g e o m e t r i c d esig n . T h e in d i g e n o u s f a s c t n a t i o n w i t h a r t i ­ fic e a n d a b s t r a c t i o n w a s j g n o r e d - by c o l o n i z e r s , a n d t h e i r p r e c o n c e p t i o n s o f A f r i c a n s as nature_-Ioving “ c h i l d r e n o f t f i e f o r e s t” b e c a m e a self-fu lfillin g e x p e c ­ t a t i o n . B u t t h e a r t if a c ts a n d p h o t o g r a p h i c r e c o r d s f ro m t h e 191 4 e x p e d i t i o n p r o v i d e us w i t h e x c e l l e n t e x a m p l e s o f t r a d i t i o n a l M a n g b e t u p a t t e r n s , as w e ll as a n o p p o r ­ t u n i t y to in fe r s o m e o f t h e i r t e c h n i q u e s . F ig u re 5.1 s h o w s t h e d e c o r a t i v e e n d o f a n iv o ry h a t p i n . L ik e t h e a r c h i t e c t u r e a n d e s t h e t i c p a t t e r n s we h a v e s e e n , t h i s is c le a r ly a s c a l i n g d e s i g n , b u t t h e p r e ­ c i s i o n o f t h e p a t t e r n s u g g e s ts t h a t t h e r e m a y b e a m o r e FIGURE 5 . I M a n g e b e t u iv o ry s c u l p t u r e ( T ra n sp a re n cy no . 3 9 3 5 , photograph b y L i n t o n G a rd in e r , co urte sy A m e r i c a n M u s e u m o f R l a t v r a l H is t o ry.) G eom etric algorithms fo rm a l g e o m e t r i c pro cess a t w ork. S im il a r d e s ig n c a n be s e e n a t w ork in th e M a n g b e t u ’s g e o m e t r i c style o f p e rs o n a ! a d o r n m e n t . F ig ure 5 .2 a s h o w s a M a n g b e t u h a i r ­ sty le, p o p u l a r d u r i n g t h e t i m e t h a t th i s c a r v i n g w a s ,p r e a t e d ( a b o u t 1 9 1 4 ) , w h i c h f e a t u r e d a d is k a n g l e d t o t-he v e r t i c a l a t 4 5 d e g r e e s . M e n o f t e n w o re a h a t w i t h t h e t o p f l a t t e n e d , f o r m i n g t h e s a m e a n g l e , as s e e n in figure 5 .2 b . j u s t as a p l a n e c u t s d i a g o n a l l y t h r o u g h t h e t o p o f t h e h e a d s in t h e iv o r y s c u l p t u r e o f figure 5.1, real M a n g b e t u h e a d d r e s s e s also t e r m i n a t e d in a 4 5 ' d e g r e e a n g le . T h i s w a s o n ly o n e p a r t o f a n e l a b o r a t e g e o m e t r i c e s t h e t i c b a s e d o n m u l ­ t ip le s o f t h e 4 5 - d e g r e e a n g le . F igu re 5 .2 b s h o w s a n iv o ry h a t p i n , e n d i n g in a disk p e r p e n d i c u l a r to it, in s e r t e d p e r p e n d i c u l a r t o t h e h a t . T o its r i g h t, a s m a ll ivory a r r o w p i n n e d t o t h e h a t p o i n t s h o riz o n ta lly , t h u s f o r m i n g a n a n g l e o f 135 d e g re e s w ith th e h a tp i n . E a c h p a rt o f th e e n s e m b le w as a lig n e d by a m u ltip le o f th e 4 5 - d e g r e e a n g le . T h i s a d o r n m e n t sty le i n c l u d e d a rtificia l e l o n g a t i o n o f t h e h e a d , w h i c h is c le a r ly v is ib le in t h e p h o t o g r a p h in figure 5 .2 b . E l o n g a t i o n w as a c c o m ­ p l i s h e d b y w r a p p i n g a c l o t h b a n d a r o u n d t h e h e a d o f i n f a n t s ; t h e w o m a n in fig u re 5 . 2 a is w e a v i n g o n e o f t h e s e b a n d s . H e a d e l o n g a t i o n r e s u l t e d in a n a n g l e o f 135 d e g r e e s b e t w e e n t h e b a c k o f t h e h e a d a n d t h e n e c k . FIGURE 5.2 Q e o m e tr i c d e sig n in M a n g b e tu p e r s o n a l a d o r n m e n t (a) M a n g b e t u w o m a n w e a v i n g h e a d b a n d , (b ) M a n g b e t u c h i e f . fa, negative n o . 1 1 1 9 1 9 , photograph by H . L a n g , co urtesy A m e r ic a n M u s e u m o f N a tu r a l H isto ry; b , n e g a t i v e n o . 2 2 4 1 0 5 , p h o t o g r a p h by H . L a n g , c o u r t e s y A m e r i c a n M u s e u m o f N a t u r a l H i s t o r y . ) 63 A fr ic a n fractal m athem atics 64 W h i l e t h e M a n g b e t u g e o m e t r i c c o n c e p t i o n o f t h e b o d y m a y h a v e i n s p ir e d t h e 4 5 'd e g re e -a n g le d esign t h e m e , th o s e desig ns were c erta in ly n o t lim ited to sim ple m i m i c r y o f a n a t o m y . W e c a rt c l e a r l y s e e t h i s in t h e i r m u s ic a l i n s t r u m e n t s . T h e d r u m in figure 5 .3 a, for e x a m p l e , h a s its u p p e r s u r f a c e c u t a t a 4 5 ' d e g r e e a n g l e t o t h e v ertical. T h e s t r i n g e d i n s t r u m e n t s h o w n in figure 5 .3 b h a s a r e s o n a t o r t h a t m e e t s th e v e r t ic a l t u n i n g s t e m a t a 13 5 ' d e g r e e a n g le . E v e n in t h e c a s e o f a n t h r o ­ p o m o r p h i c d esig ns, t h e a r t is a n s e l a b o r a t e d o n t h e h u m a n fo r m in w ays t h a t s h o w b F I G U R E 5 .3 Q e o m e tric d esig n in M a n g b e tit m u s ic a l in s tr u m e n ts (a) Drum, (b) Harp. (a , negative no. t 1 1 8 9 6 , J)/Kitogm|?h b y H - L ang, c o u rtesy A m e ric a n M u s e u m o f N a tu r a l H isto ry ; b , c o u rtesy R i c t b c r g M u se u m Z u ric h , p h o tofpaph by W euscctn a n d K a u f.) G e o m e tric algorithms c re a tiv e — a n d n o t m erely im ita tiv e — a p p lic a tio n s of g e o m e tric a l th in k in g .. F or e x a m p l e , t h e r e is a n a n t h r o p o m o r p h i c d e c o r a t i v e m o t i f a t t h e e n d o f t h e t u n i n g s t e m s h o w n in fig u re 5 . 3 b , b u t t h e s e h u m a n h e a d s a r e n o r s i m p l y m i m ­ i c k i n g h u m a n fo r m . I n fig u re 5 . 2 b w e sa'W t h a t t h e M a n g b e t u h a d a 13 5 - d e g r e e a n g l e b e t w e e n t h e b a c k o f t h e h e a d a n d t h e n e c k . T h e c a r v e d h e a d s in fig u re 5 .3b h a v e a 9 0 -d e g re e a n g le b e tw e e n t h e b ac k o f th e h e a d a n d th e n e c k . S u c h d i s t o r t i o n s i n d i c a t e a c t i v e g e o m e t r i c t h i n k i n g r a t h e r t h a n p a s s iv e r e f l e c t i o n o f n a t u r a l a n a t o m i c a l a n g le s ( w h i c h , r e c a l l i n g t h e a r t ifi c ia l h e a d e l o n g a t i o n , w e re n o t so n a t u r a l to b e g i n w i t h ) . T h e r e are also purely a b stra c t d esig ns t h a t m a k e use o f m u ltip les o f 45 degrees, ,a s w e see in figure 5 .4 . M o d e r n M a n g b e t u r e p o r t t h a t t h e c r e a t i o n o f a d e s i g n re f le c te d t h e a r t i s a n ’s d e s ire t o “m a k e it b e a u t i f u l a n d s h o w t h e i n t e l l i g e n c e o f th e c re a to r” (S c h ild k ro u t a n d K eim 1990, 1 0 0 ). T h i s sug gests a n o t h e r r e a s o n f o r a r t i ­ sa n s to a d h e r e to a n g le s t h a t a r e m t i l r i p l e s o f 45 d e g r e e s: if t h e r e w e r e n o ru le s t o fo llow , t h e n it w o u ld h a v e b e e n d if fic u lt t o c o m p a r e d e sig n s a n d d e m o n s t r a t e o n e ’s i n g e n u i ty . By r e s t r i c t i n g t h e p e r m is s ib l e a n g l e s t o a s m a ll se t, th e y w e r e b e t t e r a b l e t o d i s p l a y t h e i r g e o m e t r i c a c c o m p lis h m e n t s . C o m b in in g th is 45-d eg v ee-an g le c o n ­ stru c tio n te c h n iq u e w ith th e scalin g p r o p ­ e r t ie s o f t h e iv o r y c a r v i n g in fi g u r e 5.1 c a n re v e a l irs u n d e r l y i n g s t r u c t u r e . T h e c a r v i n g lias t h r e e i n t e r e s t i n g g eo uie lii> - f e a U u e s : 1 First, e a c h h e a d is la rg e r t h a n t h e o n e above! ) ir a n d f a c e s i n t h e o p p o s i t e d i r e c t i o n . S e c 1 o n d , e a c h h e a d is f r a m e d b y t w o l i n e s , o n e f o r m e d by t h e j a w a n d o n e f o r m e d b y t h e h a ir ; t h e s e li n e s i n t e r s e c t a t a p p r o x i m a t e l y 9 0 d e g re e s . T h i r d , t h e r e is a n a s y m m e t r y ; t h e le ft s i d e s h o w s a d i s t i n c t a n g l e a b o u t |j2o d e g r e e s fr o m t h e v e r t i c a l . FIGURE 5 .4 M a n g e b e t u iv o ry sc u lp tu r e (Transparency n o . 3 9 2 9 , p h o to g ra p h b y L y r u o n G a rd in e r , courtesy A m e r i c a n M u s e u m o f N a t u r a l H i s t o r y . ) 65 FIGURE 5.5 Q e o m e t r i c a n a l y s i s o f a n ivory s c u l p t u r e FIGURE 5 .6 (g e o m e tric r e l a t i o n s i n t h e M a n g b e t u ite r a tiv e s q u a r e s s t r u c t u r e S i n c e 9[ a n d 02 a r e t h e a l t e r n a t e i n t e r i o r a n g l e s o f a t r a n s v e r s a l i n t e r s e c t i n g t w o p a r a l l e l lin e s , 0 ,-6 ,. A fr ic a n fractal m athem atics 68 A l l o f t h e s e f e a t u r e s c a n b e a c c o u n t e d fo r b y t h e s t r u c t u r e s h o w n in f i g ' u r e 5 .5 . T h i s s e q u e n c e o f s h r i n k i n g s q u a r e s c a n b e c o n s t r u c t e d by a n i t e r a t i v e p rocess, b is e c tin g o n e s q u a re to c re a te t h e - l e n g t h o f th e sid e for t h e n e x t s q u a r e , as i n d i c a t e d i n t h e d i a g r a m . W e w ill n e v e r k n o w fo r c e r t a i n if t h i s ite ra t i v e - s q u a r e s c o n s t r u c t i o n w as t h e c o n c e p t u n d e r l y i n g t h e s c u l p t u r e ’s d e s ig n , bu t. it d o e s m a t c h t h e f e a t u r e s i d e n t i f i e d a b o v e . I n t h e iv o r y s c u l p t u r e , t h e le f t s id e is a b o u t 2 0 d e g r e e s fr o m t h e v e r t i c a l . I n t h e i t e r a t i v e - s q u a r e s s t r u c t u r e , t h e le ft s i d e is a b o u t 18 d e g r e e s f r o m t h e v e r t i c a l , as s h o w n i n fig u re 5 .6 . H e r e w e s e e s / t h a t t h e c o n s t r u c t i o n a l g o r i t h m c a n b e c o n t i n u e d i n d e f i n i t e l y , a n d t h e r e s u ltI in g s t r u c t u r e c a n b e a p p li e d to a w id e v a r i e ty o f m a t h t e a c h i n g a p p l i c a t i o n s , fro m / \s im p le p ro c e d u ra l c o n s tr u c tio n to tr ig o n o m e try (E g iash 1998a). Lusona T h e C h o k w e p e o p le o f A n g o ja /r a d a tm d itiq n .q f c r e n tin g p a tte r m ^ li n e s c a l l e d j ' l u s o n a ” in t h e _ s a n d . G e r d e s ( 1 9 9 1 ) n o t e s t h a t t h e l u s o n a s a n d d raw ings sh o w th e c o n s tr a in ts necessary to d efine w h a t m a th e m a tic ia n s c a ll an “E u l e r i a n p a t h ” : t h e s ty lu s n e v e r le a v e s t h e s u r f a c e a n d n o l i n e is r e t r a c e d . T h e lu s o n a also t e n d to use t h e s a m e p a t t e r n a t d i f f e r e n t scales, t h a t is, s u c c e s siv e i t e r ­ a t i o n s o f a s in g le g e o m e t r i c a l g o r i t h m . F ig u r e 5.7 s h o w s t h e first t h r e e i t e r a t i o n s o f o n e o f t h e d o z e n s o f l u s o n a t h a t w e re r e c o r d e d by m i s s i o n a r i e s d u r i n g t h e n i n e ­ t e e n t h c e n t u r y , w h e n t h e lu s o n a t r a d i t i o n w a s s tiii i n t a c t . A s in t h e c a s e o f t h e M a n g b e c u 4 5 - d e g r e e c o n s t r u c t i o n s , t h e r e s t r i c t io n t o a n E u le ria n p a th pro v id es t h e C h o k w e w ith a m e a n s to c o m p a re designs w ith in a single fram ew ork, a n d to sh o w h o w in c re a sin g c o m p l e x it y c a n b e a c h ie v e d w i t h i n t h e s e c o n s t r a i n t s o f s p a c e a n d lo g ic. B u t u n l i k e t ’h e ' c o m p e t i r i v e ba sis fo i co m -p a r i s o n t h a t t h e M a n g b e t u d e s c r ib e , th e C h o k w e m a d e use o f th e s e figures t o c r e ­ a t e g ro u p id e n tity . T h e re p o r ts i n d i c a t e t h a t t h e l u s o n a w e re u se d in a n a g e -g r a d e i n i t i a t i o n s y s te m ; r i tu a l s t h a t a l l o w e d e a c h m e m b e r t o a c h i e v e t h e s t a t u s o f r e a c h i n g t h e n e x t , m o r e s e n i o r le v e l o f i d e n t it y . By u s i n g m o r e c o m p l e x l u s o n a , t h e i t e r a t i o n s o f s o c ia l k n o w l e d g e p a s s e d o n in t h e i n i t i a t i o n b e c o m e v is u a liz e d by t h e g e o m e t r i c i t e r a t i o n s . In c h a p t e r 8 w e w ill s e e o t h e r e x a m p l e s o f i t e r a t i v e scalin g p a tt e r n s in in i ti a ti o n rituals. T h i s tr a d it io n o f g ro u p id e n tity t h r o u g h k n o w l­ e d g e o f t h e lu s o n a w a s a ls o d e p l o y e d b y t h e C h o k w e as a w a y t o d e f l a te t h e e g o o f o v e r c o n f i d e n t E u r o p e a n visito rs, w h o f o u n d t h e m s e l v e s u n a b l e t o r e p l ic a t e t h e lu s o n a o f m a n y c h i l d r e n . C o n c lu sio n T h e s e t w o e x a m p l e s , t h e M a n g b e t u iv o ry c a r v i n g a n d t h e lu s o n a d r a w in g s , h e l p us see t h a t A f r i c a n fractals are n o t ju s t t h e resu lt o f s p o n t a n e o u s i n t u i t i o n ; in so m e G eom etric algorithms cases th e y are c re a te d u n d e r r u le -b o u n d t e c h n i q u e s e q u iv a le n t to W e s te rn m a t h e m a t i c s . A n d t h e i r c u l t u r a l s i g n i f i c a n c e m a k e s it c l e a r t h a t a ll m a t h e ­ m a t i c a l a c t i v i t y — n o m a t t e r in w h i c h s o c i e t y it i s j b u n d — is p r o d u c e d t h r o u g h a n i n t e r a c t i o n b e t w e e n t h e f r e e d o m o f lo c a l h u m a n i n v e n t i o n a n d t h e u n i v e r ­ sal c o n s t r a i n t s w e d i s c o v e r in s p a c e a n d lo gic. ‘Myombo"— trees of the ancestors. F I G U R E 5 .7 L usona (a) These figures, “lusona," were traditionally drawn in sand by the Chokwe people of Angola. Successive iterations of the same algorithm were sometimes used to produce similar patterns of increasing size, (b) The first and third iterations of another lusona algorithm carved into a wooden box lid. (a, based on drawings in Gerdes 1995.) 69 A fr ic a n fra c ta l m a th em a tics R e c a ll t h a t in b o t h e x a m p l e s t h e role o f “c o n s t r a i n t ” w as c ru c ia l t o t h e d e v e l ­ o p m e n t o f t h e i r sc a lin g g eo m etry . F o r t h e M a n g b e t u ’s d e s i g n it w as t h e c o n s t r a i n t s o f s t r a i g h t - e d g e c o n s t r u c t i o n w i t h a n g l e s a t m u l t i p l e s o f 4 5 degrees'.--For t h e C h o k w e ’s lu s o n a it w as t h e c o n s t r a i n t s o f a n E u l e r i a n p a t h . B u t in e a c h c a s e t h e c h o i c e o f p a r t i c u l a r o b j e c t i v e c o n s t r a i n t s — d e c i d i n g w h i c h o f t h e i n f i n i t e laws o f s p a c e a n d log ic w e a r e c o n c e r n e d w i t h — w a s e s t a b l i s h e d b y a n d fo r t h e s o c ia l re la tio n s o f th e co m m u n ity . In th e case o f th e M a n g b e tu it w as a rtis tic c o m p e ­ t i t i o n , a n d in t h e c a s e o f t h e C h o k w e it w as a g e - g r a d e i d e n t i t y . I n o t h e r w o rd s, t h e i n v e n t i o n a n d d is c o v e ry c o m p o n e n t s o f m a t h e m a t i c s a r e i n e x t r i c a b l y l i n k e d t h r o u g h s o c i a l e x p r e s s io n . P h i l o s o p h i c p e r s p e c t i v e s o n t h e r e l a t i o n o f c u l t u r e a n d m a t h e m a t i c s will b e f u r t h e r d is c u s s e d in p a r t 11, b u t t o d o so w e n e e d a f u l le r p o r t r a i t o f A f r i c a n f r a c ta l ge om e try . T h e n e x t c h a p t e r w ill e x a m i n e A f r i c a n c o n c e p t i o n s o f t h e m o s t f u n d a m e n t a l c h a r a c t e r i s t i c o f fr a c ta ls : n o n l i n e a r s c a l in g . CHAPTER Scaling- 6 W e h a v e a lr e a d y s e e n m a n y e x a m p l e s o f s c a l i n g in A f r i c a n d esig ns. I n t h e s e t t l e ­ m e n t a r c h i te c t u r e o f c h a p t e r 2, for e x a m p l e , t h e c o m p u t e r sim u la tio n s c learly sh ow t h a t w e c a n t h i n k a b o u t t h e s e p a t t e r n s in t e r m s o f f r a c ta l g e o m e try . H o w d o t h e A f r i c a n a r t i s a n s t h i n k a b o u t s c a l in g ? Is it j u s t i n t u i t i o n , o r d o t h e y u se e x p l i c i t m a t h e m a t i c a l p r a c t i c e s in t h i n k i n g a b o u t s i m i l a r i t y a t d i f f e r e n t sizes? By e x a m ­ in in g v a r i e t i e s o f d e s i g n s w i t h d i f f e r e n t s c a l i n g p r o p e r t i e s , a n d c o m p a r i n g th e s e w ith t h e a r t i s a n s ’ d is c u s s io n s o f t h e p a t t e r n s , w e c a n g a i n s o m e i n s i g h t i n t o s c a l ­ ing as a m a t h e m a t i c a l c o n c e p t in A f r i c a n c u l t u r e s . P ow er lo u ' s c a lin g in iv in d s c r e e n s fro m th e S a h e l T h ^ S a h e l ^ s a b r o a d b a n d o f a r i d l a n d b e t w e e n t h e S a h a r a D e s e r t a n d t h e rest o f s u b - S a h a r a n A f r i c a . S i n c e t h e r e a r e few t r e e s a n d a g r e a t d e a l o f m i l l e t c u l ­ ti v a ti o n , it is n o t s u r p r i s i n g t h a t a r t i s a n s u se m i l l e t s t a lk s to w e a v e f e n c e s , walls, an d o t h e r c o n s t r u c t i o n s . B u t t h e c o n s i s t e n t use o f a n o n l i n e a r s c a l in g p a t t e r n in these stra w s c r e e n s (fig. 6 . 1 a ) isja b j t o d d . R a t h e r t h a n u n if o r m l e n g t h s , t h e row s o f m ille t s t r a w g e t s h o r t e r a n d s h o r t e r as t h e y go up. I n t h e U n i t e d S t a t e s w e are used to t h e im a g e o f “ t h e w h i t e p i c k e t f e n c e ” as a s y m b o l o f u n c h a n g i n g , l i n e a r r e p e t it io n , y e t h e r e t h e f e n c e s a re d i s t i n c t l y n o n l i n e a r . W h i l e 1 w as in M a l i o n The straw windscreen in Niger. FI GURE 6. 1 A n A f r i c a n w in d screen (a) The diagonallengths of these rows from bottom to top-. L = 16 12 8 6 5 , 5 3 3 2 2 This pattern is quantitatively determined by the African artisans. Here we see how the bundles of straw are first laid in long diagonal rows, then a row at the opposite angle is interlaced in back of it. T he length of each diagonal tow— how high up you go before doing the interlace step—is determined by counting a certain number o f diagonals to be crossed. In the first layer (c) we go over eight, then six, then four, then three. Each bundle is about 2 inches across the diagonal, which is why the lengths go as dovible the number of crossings. The odd numbered lengths are created by splitting the bundles in two. Why do the lengths repeat in pairs as we go toward the top? There is a discrete approximation to the continuous nonlinear scale that the African artisans follow. ( a , p h o t o by p e r m i s s i o n o f G a r d i 1 9 7 3 . ) (f ig u re c o n t i n u e s ) Scaling 73 t h e o u t s k i r t s o f t h e c a p i t a l c it y o f B a m a k o , I h a d t h e o p p o r t u n i t y to i n t e r v i e w s o m e o f t h e a r t i s a n s w h o c r e a t e t h e s e s c r e e n s a n d w as p r o v i d e d w i t h a s t r i k i n g e x a m p le o f in d ig en o u s a p p lic a tio n o f th e scalin g c o n c e p t. T h e a r t i s a n s b e g a n by e x p l a i t n n g th a .t.in .'T e rd le . a r e a s ” s u c h as t h e forests o f t h e s o u t h , t h e s c r e e n s a re n o t m a d e w i t h s c a l i n g ro w s b u t r a t h e r w i t h ro w s o f lo n g , u n i f o r m l e n g t h . T h i s is b e c a u s e t h e l o n g ro w s use less s t r a w a n d ta k e less t i m e to m a k e . B u t h e r e in t h e S a h e l , t h e y s a id , w e h a v e s t r o n g w in d s a n d d u st. T h e s h o r t e s t ro w s a re t h e o n e s t h a t k e e p o u t d u s t t h e b e s t, b e c a u s e th e y a re th e t i g h t e s t w e a v e . B u t t h e y a ls o t a k e m o r e m a t e r i a l s a n d effort. “W e k n o w t h a t . t h e w i n d b lo w s s t r o n g e r as y o u g o u p fr o m t h e g r o u n d , so we m a k e t h e w i n d s c r e e n t o m a t c h — t h a t w ay w e o n l y u se t h e s t r a w n e e d e d a t e a c h l e v e l . ” T h e r e a s o n in g t h e a r r i s a n s r e p o r t e d is e q u i v a l e n t t o w h a t a n e n g i n e e r w o u ld c a ll a “c o s t - b e n e f i t ” a n a l ysis; d e v e l o p i n g t h e m a x i m u m in f u n c t i o n ( k e e p ­ in g o u t d u s t ) fo r a m i n i m u m o f c o s t ( e ff o rt a n d m a t e r i a l s ) . M y p r i m a r y i n t e r e s t h e r e is in s h o w i n g t h a t t h e sc a l i n g c o n c e p t in A f r i c a c a n b e m u c h m o r e s o p h ist i c a t e d t h a n j u s t a n o b s e r v a t i o n , “th e s a m e t h i n g in d i f f e r e n t sizes." T h e c r e a t i o n A ssum ing d e c re a se in wind penetration is reciprocal of length: a = I (wind engineers: a = 1/3 ) • - 0 .4 . -0 .6 . < -0 .8 - - 1 .0 . ^ ♦ ?- » ♦ ♦ ♦ ♦ L o g(H ) FIGURE 6 .1 (continued) (d) T h e r e l a t i o n b e t w e e n w i n d s p e e d a n d v e r t i c a l h e i g h t as s h o w n in t h e W in d E n g in e e rin g H a n d b o o k , ( e ) T h e A f r i c a n w i n d s c r e e n m a k e r s say t h a t t h e y h a v e s c a l e d t h e r o w s o f s tr a w to m a t c h t h e c h a n g e o f w i n d s p e e d w i t h h e i g h t . I f w e a s s u m e , j u s t f o r s i m p l i c i t y , t h a t t h e d e c r e a s e in w i n d p e n e t r a t i o n is t h e r e c i p r o c a l o f t h e l e n g t h , t h e n w e c a n g e t t h e A f r i c a n e s t i m a t e f o r a. b y m easu rin g th e s lo p e o f ro w le n g t h v ersu s h e ig h t o n a lo g - lo g gra p h . T h is g iv e s a = e n g i n e e r s u s e Ct = 16 — 1, w h e r e a s the n o t b a d for a b a llp a r k e s t im a t e . N o t e t h a t t h e g r a p h is i n a v e r y s t r a i g h t l i n e , e x c e p t w h e r e t h e d i s c r e t e n a t u r e o f t h e s c r e e n (the s c r e e n m a k e r s m u s t c o u n t in w h o l e n u m b e r u n it s d u e t o t h e s tr a w b u n d le s ) fo rce s a n a p p r o x i ­ m ation by re p e a tin g th e s a m e le n g th tw ic e . A fr ic a n fra c ta l m athem atics 74 o f t h e w i n d s c r e e n as a n o p t i m a l d e s i g n r e q u i r e d m a t c h i n g t h e s c a l i n g v a r i a t i o n o f w i n d s p e e d v e r s u s h e i g h t t o a s c a l i n g v a r i a t i o n in l e n g t h s o f straw . By t r a n s ­ ferring th is c o n c e p t b e tw e e n tw o c o m p le te ly d iffe re n t d o m a in s , th e - a r tis a n s h a v e d e m o n s t r a t e d t h a t th e y u n d e r s t a n d sc a lin g in t h e a b s t r a c t ; i n d e e d , t h e d e sig n e s s e n t i a l l y p l o t s t h e r e l a t i o n o f w i n d s p e e d to h e i g h t o n a s t r a w g r a p h . A l t h o u g h 1 w as c o n c e r n e d o n l y w i t h t h e o v e r a l l r e l a t i o n o f s c a l i n g a n d r e a s o n i n g , I m e a s u r e d t h e ro w s j u s t t o s e e h o w c l o s e t h e y c a m e t o w h a t a W e s t ­ e r n e n g i n e e r w o u ld d e v e l o p f o r a n o p t i m a l m a t c h w i t h w i n d s p e e d . If t h e s t r a w s c r e e n h a d l i n e a r s c a l in g , t h e n e a c h ro w w o u ld d e c r e a s e in l e n g t h by t h e s a m e a m o u n t (e.g., 12 in c h e s , 10 i n c h e s , 8 in c h e s , e t c . ) . B u t t h e ro w s d e c r e a s e less a n d less w i t h h e i g h t ; it t u r n s o u t t h a t t h e s c r e e n d e s i g n s h o w s a c lo s e fit t o w h a t is c a l l e d a " p o w e r l a w " — t h a t is, it s c a l e s a c c o r d i n g t o a n e x p o n e n t (fig . 6 . 1 c ) . F ig u re 6 . 1 b , r e p r i n t e d fr o m t h e W in d E ng in eerin g H a n d b o o k , s h o w s t h e e q u a t i o n o f w i n d s p e e d w i t h h e i g h t m o s t c o m m o n l y u s e d by e n g i n e e r s — a ls o a p o w e r law. S o t h e S a h e l w i n d s c r e e n is n o t o n l y a p r a c t i c a l a p p l i c a t i o n o f t h e a b s t r a c t s c a l ­ in g c o n c e p t , it is also a fa irly a c c u r a t e o n e . O f c o u r s e , o n e m i g h t o b j e c t t h a t t h e in d ig e n o u s e n g in e e rs d id n o t a c tu a lly se t up th e alg eb ra a n d p e rfo rm th e o p ti­ m i z in g c a l c u l a t i o n . B u t I a s k e d t h r e e A m e r i c a n m a t h e m a t i c i a n s h o w t h e y w o u ld s e t u p th e s e e q u a ti o n s to d e t e r m i n e t h e o p t i m a l d esig n , a n d all t h r e e said t h e s a m e th i n g : “ 1 w o u l d n ’t so lv e it a n a l y t i c a l l y , I ’d j u s t g r a p h t h e e q u a t i o n s o n t h e c o m ­ p u t e r a n d see w h e r e t h e f u n c t i o n s p e a k e d . " W h e t h e r w e m a k e o u r g r a p h s o n a c o m p u t e r s c r e e n o r a stra w s c r e e n d o e s n ’t m a tte r, as lo n g as w e g e t t h e ri g h t answ er. S tr e tc h in g s p a c e in k e n t e c lo th If s o m e o n e in A m e r i c a w e re a sk ed to t h i n k o f a n A fr i c a n t e x t ile, k en te c lo t h w o u ld b e t h e m o s t likely im ag e. Its c o m b i n a t i o n o f stro jig c o lo r s ,.h o ld d e s i g n s , a n d ja s so c i a t i o n s w i t h a n c i e r ^ kingdoms.of-.We.s.t-Africa h a s m a d e it a f a v o r i te fo r im p o rts. B u t m o s t o f t h e i m p o r t e d k e n t e c l o t h is c r e a t e d by a u t o m a t e d m a c h i n e , a n d w h i l e 1 w o u l d fi e rc e l y d e f e n d it a s “a u t h e n t i c , ” t h e n e e d for p a t t e r n r e p e t i t i o n in a u to m a tio n has e lim in a te d a w o n d erfu l scalin g tr a n sfo rm a tio n th a t c a n be seen in t h e o l d e r p a t t e r n s c r e a t e d o n h a n d lo o m s (fig. 6 .2 a ) . T h e s c a l i n g - c h a n g e is n o t ju s t s m a ll a n d large v e r s i o n s o f t h e s a m e t h i n g ; r a t h e r , it is as if t h e d e s i g n w as d ra w n o n a ru b b e r sh e e t, w h ic h was h a lf s tr e tc h e d a n d h a lf c o n tr a c te d . In G h a n a I tr a v e le d to th e villag e o f B o n w i r e , w h e r e h a n d - l o o m w e a v in g is still p r a c ­ ticed , a n d asked th e artisan s th e r e w h y th is scalin g tr a n s f o r m a tio n was c re a te d . ^ | T h e w e a v e r s r e p l i e d t h a t t h e y t h i n k o f t h e c o m p r e s s e d v e r s i o n as t h e o rig in a l p a t t e r n , a n d said th e y call it “s p r e a d i n g " w h e n t h e y c r e a t e t h e s t r e t c h e d ver- \ s i o n . T h e r e a s o n th e y g a v e for t h e s p r e a d i n g p a t t e r n c a n b e s t b e u n d e r s t o o d w i t h I I \ FIGURE 6. 2 K e n te c lo th (a) In this traditional kente cloth design, stretched and compressed versions of the same pattern appear. The weavers call this “spreading” the pattern, (b) Why are weavers spreading the pattern? They say that our eyes give "heavy looks” to the face, and only “light looks” to the rest of the body. This is what neurobiologists call “saccadic” eye movements. Unlike “tracking” eye movements, which are continuous, saccadic movements are discrete and tend to leap about. Since kente cloth was traditionally worn as a toga over the shoulder, the part near the face was given a compressed pattern, and the part along the body a stretched pattern, to match the scaling of the saccadic eye movements, (c) T h e compression of space is used in mathematics to model scaling patterns, like chat of the saccadic eye movements. Mathematicians call this a “contractive affine transformation." 76 A fric a n , fra c ta l ?natliernatics t h e f o llo w in g e x p e r i m e n t . H o l d y o u r finger i n f r o n t o f y o u r face, a n d w i t h o u t m o v ­ in g y o u r h e a d , t r a c k t h e fin g er w i t h y o u r eyes as yo u m o v e it s lo w ly a c ro s s t h e v is u a l field. N o w try t h e s a m e t h i n g a g a in , s m o o t h l y t r a c k i n g t h e v is u a l field, b u t w i t h o u t t h e fin g er t o g u id e y o u r eyes. Y o u ’ll find t h a t it c a n ’t b e d o n e ! Y o u r eye m o v e s i n v o l u n t a r i l y in l i t t l e j u m p s , c a l l e d “sa c c a d ic ” m o v e m e n t s . W h e n a p e r ­ s o n c o m e s i n t o y o u r v is u a l field, t h o s e s a m e s a c c a d i c m o v e m e n t s d e n s e l y c o v e r t h e face, a n d t h e n m a k e a few g la n c e s a t t h e ' b o d y (fig. 6 . 2 b ) . T h e w e a v e rs in B o n w ire r e p o r t e d t h e s a m e idea: “W h e n y o u se e a p e r s o n y o u g iv e h e a v y lo o k s t o t h e face , a n d l i g h t lo o k s t o t h e b o d y . ” T h e y e x p l a i n e d t h a t t h e p u r p o s e o f t h e s c a l - \ in g c h a n g e is t o m a t c h t h i s v is u a l s c a l in g : t h e c o m p r e s s e d p a r t o f t h e p a t t e r n is \ t h e c l o t h w o r n o v e r t h e s h o u l d e r , a n d t h e s t r e t c h e d p a r t is w o r n d o w n t h e I l e n g t h o f t h e body. T h e m a t h e m a t i c a l t e r m for t h i s o p e r a t i o n is “c o n t r a c t i v e affin e t r a n s f o r ­ m a t i o n ” (fig. 6 .2 c ), w h i c h c a n be used for c r e a t i n g j r a c t a l s t h r o u g h a m e t h o d c a l led “ i t e r a t e d f u n c t i o n s y s te m s ” ( s e e W a h l 1 9 9 5 , 1 5 6 - 1 5 7 ) . I n k e n t e c l o t h t h e r e is n o i t e r a t i o n — t h e o p e r a t i o n is d o n e o n l y o n c e — b u t i t d o e s s h o w a c t i v e t h i n k ­ in g a b o u t a s c a l in g t r a n s f o r m a t i o n . A s in t h e case o f t h e w i n d s c r e e n , t h e w eav ers are t a k i n g a r a t h e r a b s t r a c t o b s e r v a t i o n a b o u t a t i m e - v a r y i n g q u a n t i t y a n d m a p ­ pin g this m o d el in to a m a te ria l design. L o g a r i t h m i c s p ira ls I n c h a p t e r 3 (fig. 3 . 2 ) w e e x a m i n e d t h e c o n t r a s t b e t w e e n n o n l i n e a r c o n c e n t r i c c ir c le s a n d l i n e a r c o n c e n t r i c c irc le s . I n t h e s a m e way, n o n l i n e a r s p ir a ls a re easy to u n d e r s t a n d if w e * c o n tia * t t h e m w i t h lin e a i^ s p ira ls (fig. 6 . 3 a ) . T h e l i n e a r s p i ­ ral, a lso c a lle d a n A r c h e m e d e a n sp iral in h o n o r o f t h e G r e e k m a t h e m a t i c i a n w h o fa v o re d it, is i n j h e s h a p e o f a c o il e d r o p e o r w a t c h sp r in g . E a c h r e v o l u t i o n b rin g s y o u o u t by t h e s a m e d i s t a n c e ( j u s t as e a c h la y e r in t h e l i n e a r c o n c e n t r i c c i r c l e w a s t h e s a m e t h i c k n e s s ) . F o r t h a t re a s o n , a l i n e a r s p i r a l o f a fi n it e d i a m e t e r c a n h a v e o n ly a fin ite n u m b e r o f tu r n s . A n o n l i n e a r sp iral o f fi n it e d i a m e t e r c a n h a v e a n infinite n u m b e r o f turns, b eca u se e v e n th o u g h t h e r e is less a n d less sp ace r e m a i n ­ in g as o n e g o e s t o w a r d t h e c e n t e r , t h e d i s t a n c e b e t w e e n e a c h r e v o l u t i o n c a n g e t sm aller a n d s m a lle r,. A g o o d e x a m p l e o f t h i s n o n l i n e a r s c a l i n g c a n b e s e e n in t h e l o g a r i t h m i c sp ira l (fig. 6 . 3 b ) . L o g a r i t h m i c sp ira ls a r e ty p i c a l s t r u c t u r e s in t w o d i f f e r e n t c a t ­ e g o rie s o f natu ra l^ p h e n o m e n a . O n t h e o n e J m n d , t h e y a r e f o u n d in a s t o n i s h i n g » v a r i e t i e s o f o r gaja.Lc--gr.o_w.th. T h e o d o r e C o o k ’s T h e C u r v e o f L ife ( 1 9 1 4 ) , fo r \ J e x a m p le , s h o w s d oz ens o f lo g a r ith m ic spirals from ev e ry b r a n c h o f t h e e v o l u t io n a r y tree : s n a i l a n d n a u t i l u s sh e lls; t h e h o r n s o f ra m s a n d a n t e l o p e ; a lg a e , p i n e c o n e s , / Scaling 77 FIGURE 6 .3 S pirals ( a ) I n t h e l i n e a r s p i r a l o f A r c h i m e d e s , t h e r e is a c o n s ta n t d ista n c e b e tw e e n e a c h rev o lu tio n . O n ly a f i n i t e n u m b e r o f t u r n s c a n fit i n t h i s finite sp a c e , (b) In t h e lo g a r ith m ic spiral, th e re is a n i n c r e a s i n g d i s t a n c e b e t w e e n e a c h r e v o l u t i o n . A n i n f i n i t e n u m b e r o f t u r n s c a n fit i n t h i s f i n i t e sp a c e . j a n d su n flo w e rs; a n d e v e n a n a t o m i c a l p a r t s o f t h e h u m a n e a r a n d h e a r t . M a n y I r e s e a r c h e r s h a v e s p e c u l a t e d o n w h y t h i s is so; t h e i r a n s w e r is ty p ic a lly t h a t l i v ­ ing sy s tem s n e e d to k e e p t h e sa m e p r o p o r t i o n s as th e y grow, .and so a scaling.cu.rve a ll o w s t h e s a m e f o r m to be m a i n t a i n e d . I p r e f e r t o t h i n k o f it as r e c u r s i o n : if we \ l o o k a t t h e c h a m b e r e d n a u t i l u s , fo r e x a m p l e , w e c a n t h i n k o f each, n e w c h a m ­ b e r as t h e n e x t i t e r a t i o n t h r o u g h t h e s a m e s c a l i n g a l g o r i t h m . O n t h e o t h e r h a n d , l o g a r i t h m i c sp ira ls a re a ls o f o u n d in flu id t u r b u l e n c e . W e b e c o m e a w a re o f th is w h e n we w a t c h a h u r r i c a n e fr o m sp ac e, o r sim p ly a d m i re th e swirls o f w a t e r a l o n g a r i v e r b a n k . E x p l a n a t i o n s for th e s e fluid c u rv e s aye m u c h less s p e c u l a t i v e , s i n c e w e can _w rite e q u a t i o n s fo r t u r b u l e n c e a n d s h o w t h e m p r o ­ d u c i n g l o g a r i t h m i c s p irals in c o m p u t e r s i m u l a t i o n s (as w e w ill see in c h a p t e r 7). B u t t h e E u r o - A m e r i c a n t r a d i t i o n is n o t t h e o n ly o n e i n t e r e s t e d in sim u la c ra . T h e a r t is t s o f w h a t is n o w G h a n a — p a r t i c u l a r l y t h o s e o f t h e A k a n s o c i e ty — lo n g ag o / a b s t r a c t e d t h e l o g a r i t h m i c sp iral for p re c i s e l y t h e s e tw o c a te g o r ie s . T h e i r sym -\ b o ls for t h e life fo rce (fig. 6 .4 a ) a re c le a rly r e l a te d to t h e " c u r v e s o f life,” a n d icons \ for T a n u , t h e r i v e r g o d (fig. 6 . 4 b ) , s h o w t h e l o g a r i t h m i c sw irls o f t u r b u l e n c e . A fr ic a n fra c ta l m athem atics 78 a b FIGURE 6 .4 L o g a r i t h m i c spirals (a) S e v e r a l G h a n a i a n i c o n i c figure s, s u c h as t h i s g o l d w e i g h t , l i n k a s p i r i t u a l f o r c e w i t h t h e s t r u c t u r e o f l i v i n g s y s t e m s t h r o u g h l o g a r i t h m i c s p i r a l s . T h i s e x a m p l e is p a r t i c u l a r l y s t r i k i n g s i n c e it s h o w s h o w s p i r a l s c a n be c o m b i n e d w i t h b i l a t e r a l s y m m e t r y t o c r e a t e o t h e r s e l f - s i m i l a r s h a p e s ( t h e l a r g e d i a m o n d s h a p e c r e a t e d b y t h e m e e t i n g o f t h e la r g e s p i r a l a r m s is r e p e a t e d o n e i t h e r sid e b y t h e s m a l l d i a m o n d a t t h e m e e t i n g o f t h e s m a l l s p i r a l a r m s ) , ( b ) T h i s figure, a g a i n b a s e d o n l o g a r ith m ic spirals, a p p e a r s o n t h e t e m p le s o f T o n u , t h e r iv e r g o d , a n d lin k s th is s p ir itu a l fo rc e to t h e g e o m e t r i c s t r u c t u r e o f flu id t u r b u l e n c e . ( a , pho to courtesy D o r a n Ross.) ^ A g a i n , we n e e d to a v o i d t h e a s s u m p t i o n t h a t t h e G h a n a i a n lo g s p i r a ls a r e ^ ' s i m p l y m i m e t i c “r e f l e c t i o n s ” o f n a t u r e , a n d e x a m i n e h o w ..they, a r e used. .a n d < < ' d e s i g n e d . T h e A k a n a n d o t h e r s o c i e t i e s o f G h a n a c r e a t e d a c o l l e c t i o n o f specific icons t h a t several resea rch ers h a v e c o m p a r e d to a w r it te n language . B u t r a t h e r t h a n c o m p o s e d o f t h e v a s t n u m b e r o f s y m b o ls we c a ll “w o rd s ,” t h e G h a n a i a n s y m b o li c v o c a b u l a r y is m u c h s m a lle r, a n d e a c h s y m b o l refers n o t t o a s in g le w o r d b u t a n e n t i r e s o c ia l, re l ig i o u s ,.o r p h i l o s o p h i c a l c o n c e p t . M o r e o v e r , in m a n y c a s e s t h e s t r u c t u r e o f t h e s y m b o l is n o t a r b i t r a r y (as G r e g o r y B a t e s o n said, “T h e r e is n o t h ­ in g ‘s e v e n i s h ’ a b o u t t h e n u m e r a l 7 "), b u t r a t h e r is s h a p e d so t h a t e a c h ic o n 's g e o ­ m e t r i c s t r u c t u r e r e c a lls t h e c o n c e p t it r e p r e s e n t s . In o c h e r w o rd s , t h e y a r e n o t o n ly a b s tra c tio n s in t h e sense o f b e i n g stylized, b u t also g e n e ra liz a tio n s in t h e sense o f t h e d e s i g n e r s ’ i n t e n t t o find a n u n d e r l y i n g s t r u c t u r e t h a t all e x a m p l e s h a v e in c o m m o n . F o r th is r e a s o n w e c a n a c c u r a t e l y d e s c r ib e t h e G h a n a i a n log sp iral .icons as g e o m e t r i c m o d e l s f o r .th e p h e n o m e n a o f o r g a n i c g r o w t h a n d fluid t u r b u l e n c e . S o m e a s p e c ts o f t h e s e d e s i g n s illu s tr a r e a c o n s c i o u s r e f l e c t i o n o n t h e i r g e o ­ m e tric pro pertie s. Figure 6 .4 a, for e x a m p l e , n o t o n ly displays t h e log s p ira l’s Euclid- Scaling 79 ear* s y m m e t r y — for we c a n see h o w c l o c k w i s e a n d c o u n t e r c l o c k w i s e spirals c o m ­ p a r e — b u t a ls o e x p e r i m e n t s w i t h o t h e r k i n d s o f s c a l i n g s y m m e t r y : n o t e t h a t t h e large d i a m o n d s h a p e c r e a t e d by t h e m e e t i n g o f t h e la r g e s p ira l a r m s is r e p e a t e d ’**>v o n b o t h s id e s by t h e s m a ll d i a m o n d a t t h e ^ m e e t i n g o f t h e s m a ll s p ira l a rm s . C a n th is s c a l in g b e c o n t i n u e d in f u r t h e r i t e r a t i o n s ? 1 w ill l e a v e t h a t q u e s t i o n as a n e x e r c i s e fo r t h e read ers. ' T h e r e are h in t s th a t th e p r e c o lo n ia l G h a n a i a n desig n ers w ere h e a d e d t o w a r d a q u a n t i t a t i v e a p p r o a c h i n t h e i r lo g s p i r a l d e s i g n s . F i g u r e 6 . 5 a s h o w s t h e s c u l p t u r e o f a w a t e r b u f f a l o in w h i c h t h e y h a v e i n s c r i b e d u n i f o r m d i s c r e t e s te p s . 1 d o n ’t t h i n k t h i s w as m o t i v a t e d b y n u m e r i c m e a s u r e s , b u t r a t h e r t h e r e v e r se . By c u t t i n g t h e s e s te p s w e c a n c l e a r l y g a u g e t h e n o n l i n e a r n a t u r e o f t h e spiral— th e w ay steps o f a c o n s t a n t i n c r e m e n t sh o w a n in c re a sin g a m o u n t of c u r v e g e n e r a t e d — a n d t h i s p r a c t i c e c o u l d h a v e le d t o q u a n t i t a t i v e m e a s u r e s . A n o t h e r m o v e in t h a t d i r e c t i o n w o u l d g e n e r a l i z e s u c h d i s c r e t i z e d l o g a r i t h m i c f ig u r e 6.5 L o g a rith m ic sc a lin g in Q h a n a ia n d e sig n s (a) L o g a r i t h m i c s c a l i n g c a n b e d e m o n s t r a t e d in a t h r e e - d i m e n s i o n a l c u r v e b y s h o w i n g how d isc re te step s o f th e sam e vertical in c re m e n t lead to rap id ly in c re a sin g area. (b) O v e r h e a d v i e w o f p y r a m i d - s h a p e d goldw eight. (c ) L o g a r ith m ic p lo t of goldw eight tria n g le le n g th s. (a, p h o i o f r o m the M e t r o p o l i t a n M u s e u m o f A r t . b, photo co urte sy G e o r g e A rt h u r, M a rsh a ll ‘ University.,) no e -a ■ £ step o f pyram id FIGURE 6 . 6 A d a p t i v e s c a l i n g - w i t h tr ia n g le s (a) A n t e l o p e h e a d d r e s s c r e a t e d b y t h e K r u m b a o f B u r k i n a F a s o , ( b ) M a s k s o l d in A c c r a , G h a n a , ' b a s e d o n d e s i g n u se d in t h e S n k a r a - B m m o u r e l i g i o u s d a n c e s - ( c ) R e p r e s e n t a t i o n o f t h e w a t e r s p i r i t c r e a t e d b y t h e B a g a o f G u i n e a , ( d ) S c u l p t u r e fr o m t h e C o n g o , ( e ) A K i k u y u w o o d e n s h i e l d . T h e w o o d h a s a n o n l i n e a r c u r v e t o w a r d t h e c e n t e r , a n d t h e t r i a n g l e s a re s c a l e d t o m a t c h . (a, courtesy Musee M u s e u m o f A r t . c, <!e I ' H o m m e . c , M e t r o p o l i t a n M u s e u m o f A r t ; />liofo b y E l io t E l i s n / o n . d , D e t r o i t British M u s e u m ; fr o m Z aslavsky 1 9 7 3 .) Scaling s c a l i n g to fo r m s o t h e r t h a n s p ira ls , a n d t h a t d i d i n d e e d o c c u r , as w e c a n s e e in figure 6 .5b , o n e o f t h e A k a n g o ld w e ig h ts . A p l o t o f t h e l e n g t h o f th e s e tr ia n g l e s (fig. 6 . 5 c ) i n d i c a t e s t h a t r e a s o n a b l e a c c u r a c y w as a c h i e v e d in t h i s i n d i g e n o u s lo g arith m ic_ scajin g p ra c tic e . ' A d a p tiv e sc a lin g S o far this c h a p t e r h a s focused o n q u e s tio n s o f intentional.ity, precision, a n d m a t h e ­ m a t i c a l r e a s o n i n g in A f r i c a n s c a l i n g d e s i g n s . A d a p t i v e s c a l i n g h a s l i t t l e m a t h e ­ m a t ic a l s o p h i s ti c a ti o n , b u t it to o is a n i m p o r t a n t p a r t o f t h e A f r i c a n fra c ta l de sign \ t h e m e . By a d a p t i n g t h e s c a l e o f a p a t t e r n t o fit v a r i o u s fo rm s, a n u m b e r o i { e s t h e t i c a n d p r a c t i c a l effects c a n b e a c h i e v e d . T h e s e e x a m p l e s fall i n t o tw o c a t-^ egories. In c o n f o r m a l m a p p in g , t h e p a t t e r n sim p ly fits alo n g t h e c o n t o u r s o f a c o n ­ c r e t e , p r e e x i s t i n g s t r u c t u r e . I n g l o b a l m a p p i n g , t h e p a t t e r n is d i s t o r t e d by c o m p r e s s i o n o r e x p a n s i o n — as w e sa w h a p p e n a l o n g o n e d i m e n s i o n in k e n t e c l o t h — a j x o r d i n g t q a m o r e u n i v e r s a l, a b s t r a c t t r a n s f o r m a t i o n . F ig u re 6 .6 s h o w s s e v e r a l e x a m p l e s o f c o n f o r m a l m a p p i n g o n t r ia n g l e s . .My s e a r c h o f t h e f a c ia l m a r k i n g s o f a n t e l o p e o f t h e w e s t e r n S u d a n d id n o t t u r n u p a n y t h i n g lik e t h e s c a l i n g p a t t e r n o f fig u r e 6 .6 a ; t h e s e t r i a n g l e s a r e d e c o r a t i v e a d d itio n s , sized to fit in to t h e s h a p e o f t h e s c u l p tu re . O t h e r e x a m p le s (fig. 6 . 6 b - e ) s h o w a s e r ie s o f t r i a n g l e s c o n f o r m i n g t o t h e s c a l i n g c o n t o u r s o f a m a s k , a s i n ­ u o u s c u r v e , a c a r v e d h u m a n fig u re , a n d a s h i e l d . F i g u r e 6 . 7 a s h o w s c o n f o r m a l m a p p i n g in t h e h a i r s t y l e A m e r i c a n s c a ll “c o r n - r o w i n g ”; its s i m u l a t i o n is s h o w n in fig u re 6 . 7 b . T h e Y o ru b a n a m e f o r t h i s s t y l e is ipako e le d e , w h i c h m e a n s t h e n a p e o f t h e n e c k ofTi_boar— b e c a u s e t h e b o a r ’s b ristle s s h o w a s i m i l a r n o n l i n e a r s c a l in g . F ig u r e 6 . 7 c s h o w s a h a i r s t y l e t h a t c o m b i n e s c o n f o r m a l m a p p i n g w i t h i t e r a t i o n . A d a p t i v e s c a l i n g o f c i r c l e s c a n b e s e e n in t h e S e n e g a l e s e t e x t i l e in fig u re 6 . yd. A p r a c t i c a l a p p l i c a t i o n o f c o n f o r m a l m a p p i n g a p p e a r s in figure 6 . ye, a n a e r i a l p h o t o o f t h e N k o n g - m o n d o q u a r t e r i n t h e c i t y o f E d e a in s o u t h e r n C a m e r o o m j v h e r e w e s e e a s c a l i n g s e r ie s o f h o u s e s . A s e x p l a i n e d t o m e by o n e m e m b e r o f t h e n e ig h b o r h o o d , Mr. S o ss o , t h e h o u s e s w ere c o n s t r u c t e d a lo n g a n a r ­ ro w in g rid g e , a n d t h e s c a l i n g w as s i m p l y c o n f o r m i n g to t h e n a t u r a l la n d s c a p e . H o w e v e r , t h e o l d e s t i n h a b i t a n t o f t h i s B assa n e i g h b o r h o o d , M r. B e l l m b o c k , to ld m e t h a t t h e p a t t e r n w as c r e a t e d b e c a u s e p e o p l e w a n t e d neighbors.o(_a..sim ­ ilar e c o n o m i c class n e x t d o o r , so t h a t t h e r a n g e in h o u s e size re f le c te d a n e c o ­ n o m i c g r a d i e n t , fr o m p o o r e s p t p _ w e a l t h i e s t . M r. B e l l m b o c k liv e d in t h e s m a ll e s t] h o u s e , a n d Mr. S o s s o in t h e la r g e s t, s o I w o u ld n o t d i s c o u n t t h e p o s s ib ility t h a t / t h e r e was a n e c o n o m i c s c a l i n g as w ell. 81 FIGURE 6 . 7 A d a p tiv e s c a lin g b a s e d on v a r io u s s h a p e s (a, b ) A Y o r u b a h a ir sty le , l p a k o E le d e , a d a p t s th e s c a lin g o f th e bra id s to t h e n o n l i n e a r c o n t o u r s o f t h e h e a d , (c) T h i s h a ir s t y le b e g in s b y b r a id in g a sm a ll h o r s e s h o e sh a p e in t h e 't o p c e n t e r , a n d th e n t r a c in g th e c o n t o u r in in c r e a s in g p e r im e te r s — a c o m b i n a t i o n o f a d a p t iv e s c a l i n g a n d ite r a t io n . ( d ) F i t t i n g c i r c le s b e t w e e n i n t e r s e c t i n g c u r v e s c r e a t e s a s c a l i n g s e r ie s in t h is t e x t i l e d e s i g n fro m G u i n e a , ( e ) A n a e r ia l p h o t o o f t h e N k o n g - m o n d o q u a r t e r in t h e c i t y o f E d e a in s o u t h e r n C a m e r o o n , w h e re w e see a sc a lin g series o f houses, f a , fr o m Sagay 1 9 8 3 . c , fr o m Sagay 1 9 8 3 . d , p h o to c o u rt e s y I F A N , D a k a r .) Scaling + infinity - in f in it y FIGURE 6 . 8 M a p p in g f r o m th e p la n e to a s p h e r ic a l s u r fa c e (a) M apping bars of infinite length from the plane to a sphere, ( l i ) A Y o i n h a h a i r s t y l e , Koioba (“hurker”) ......................... ...... (b,/nmi Sagay j 983.) It is p o s s ib le t o m i s r e a d th e s e e x a m p l e s o f c o n f o r m a l m a p p i n g as b e i n g th e p r o d u c t o f a r t i s a n s w h o a r e s t r o n g l y g u i d e d by c o n c r e t e f o r m s r a t h e r t h a n a b s tr a c t th o u g h t . B u t a d a p tiv e sc a lin g c a n a ls o b e s e e n in m o re a b s tra c t e x a m p le s : g lo b a l t r a n s f o r m a t i o n s in w h i c h s p a c e it s e l f is d is t o r t e d . T h i s is a c o m ­ m o n o p e ra tio n in W e s te rn geom etry, th e m o st fre q u e n t e x a m p le b e in g a m a p ­ p i n g b e t w e e n t h e p l a n e a n d a s p h e r e (fig. 6 . 8 a ) . F ig u r e 6 . 8 b s h o w s a h a i r s t y l e t h a t a p p e a r s t o h a v e a p l a n a r d e s i g n m a p p e d o n t o a s p h e r i c a l su rfac e. F ig u re 6 .9 p ro v id e s a n e v e n m o r e a b s t r a c t illu s tratio n , th e in ve rse o f t h e p re v io u s m a p p i n g — n o w g o in g fr o m s p h e r i c a l to r e c t a n g u l a r — a n d u tiliz in g t h r e e d i m e n s i o n s in s te a d 83 A fr ic a n fra c ta l m athem atics 84 o f t w o . I n t h i s C h o k w e s c u l p t u r e , t h e e n t i r e h u m a n fig u re is d i s t o r t e d as if its sp h erical v o lu m e h a d b e e n m a p p e d to a c u b ic v o lu m e; th e re s u ltin g n o n lin e a r s c a l i n g is d r a m a t i c a l l y i l l u s t r a t e d by t h e d i s c r e t e s t e p s in t h e h e a d d r e s s . A r t h i s ­ to r ia n W illia m Fagg (1 9 5 5 ) m a d e a sim ila r su g g e stio n a b o u t o t h e r A fric a n d e s ig n s , w h i c h h e c o m p a r e d to t h e d r a w i n g s o f n a t u r a l g r o w t h by b i o l o g i s t D ’A r c y T h o m p s o n : “I b e l i e v e t h a t t h e m o r p h o l o g y o f A f r i c a n s c u l p t u r e m a y b e usefully s t u d ie d . . . by r e f e r e n c e to m a t h e m a t i c s . . . . F o r e x a m p l e in c e r t a i n m asks • , FIGURE 6 .9 M a p p in g fr o m a s p h e r ic a l v o lu m e to a r e c ta n g u la r , v o lu m e (a ) B astin (1 9 9 2 , 6 8 ) sh o w s th a t th is C h o k w e cro w n , t h e C i p e n y a - M u t w e , is m a d e u p o f l i n e a r b a n d s i n r e a l life. T h e n o n l i n e a r s c a l i n g w e s e e in t h i s s c u l p t u r e c a n b e e x p l a i n e d as t h e i n v e r s e o f t h e t r a n s f o r m a t i o n w e saw i n figure 6 .8 a . R a t h e r t h a n s h r i n k i n g a s w e m o v e fro m th e c e n te r to th e m arg in s, th e in v erse m a p p in g causes e x p a n s i o n f r o m c e n t e r t o m a r g i n s . T h i s is n o t o n l y t h e in v e rs e o f t h e p re v io u s m a p p in g , b u t also o p e ra te s o n th re e -d im e n s io n a l v o lu m e ra th e r th a n surface. S im ilar t r a n s f o r m a t i o n s a r e u s e d in n e u r o s c i e n c e t o m o d e ! t h e ways th a t ta ctile receptors are m a p p e d from bo d y to ^ b r a i n , s i n c e t h e r e is a t n u c h g r e a t e r d e n s i t y o f s e n s o r y n e u r o n s a t t h e e x t r e m e s , ( b ) T h e r e a s o n f o r t h i s t r a n s f o r m a t i o n is t o i n v o k e t h e i m p r e s s i o n o f p o w e r a n d s t a b ilit y ( C h a n d a 1 9 9 3 ). T h e m e a n in g h a s n o t h in g in p a r t ic u la r to d o w it h g e o m e t r ic m a p p i n g , o t h e r t h a n a c h i e v i n g t h e d e s ir e d e f fe c t , b u t it is i n t e r e s t i n g t o n o t e t h a t t h e t r a n s f o r ­ m a t i o n is u n i f o r m l y a p p l i e d t o n l! e x t e r n a l a r e a s , e v e n t o t h e e x t e n t o f d e f o r m i n g t h e f o r e h e a d . ( a , c o u r t e s y J a c q u e s K e r c h a c h e a n d M u s e u m o f M a n k i n d , L o n d o n , b , c o u r t e s y M u s e u m o f the P h i l a d e l p h i a C ivic C e n te r .) Scaling f o r t h e G e l d e s o c i e t y t h e n a t u r a l . . . p h y s i o g n o m y is ‘b l o w n u p , ’ so t o s p e a k , in a w ay w h ic h c o u ld b e p lo tte d o n a set o f flarin g e x p o n e n ti a l c o o rd in a te s .” ( 1 9 1 7 . 4 3 )- C o n c lu sio n T h e e x a m p l e s o f s c a l i n g d e s i g n s in t h i s c h a p t e r v a ry g r e a t l y in p u r p o s e , p a t t e r n , a n d m e t h o d . W h i l e it is n o t difficult to i n v e n t e x p l a n a t i o n s b a sed o n u n c o n s c i o u s s o c ia l fo r c e s— fo r e x a m p l e , t h e f l e x ib ility in c o n f o r m i n g d e s i g n s to m a t e r i a l s u r ­ fa c e s as e x p r e s s i o n s o f so c ia l f l e x ib ility — I d o n o t t h i n k t h a t a n y s u c h e x p l a n a ­ t i o n c a n a c c o u n t fo r t h i s d iv e rsity . F r o m o p t i m i z a t i o n e n g i n e e r i n g , t o m o d e l i n g o r g a n i c life, t o m a p p i n g b e t w e e n d if fe re n t s p a tia l s tru c tu r e s , A f r i c a n a rtis a n s h a v e d e v e l o p e d a w id e r a n g e o f t o o l s , t e c h n i q u e s , a n d d e s i g n p r a c t i c e s b a s e d o n t h e c o n s c i o u s a p p l i c a t i o n o f s c a l in g g e o m e t r y . I n t h e n e x t c h a p t e r , w e w ill see t h a t A f r i c a n n u m e r i c s y s te m s a ls o s h a r e m a n y f r a c t a l c h a r a c t e r i s t i c s . 85 C H A P T E R N um eric -systems- S o far vve h a v e fo cu s ed o n g e o m e t r i c s t r u c t u r e s r a t h e r t h a n n u m e r i c sy stem s. T h e o n l y e x c e p t i o n w as in t h e w i n d s c r e e n , w h e r e t h e n o n l i n e a r s c a l i n g w a s c r e a t e d by c o u n t i n g a s p e c ific s e q u e n c e o f d i a g o n a l s t r a w ro w s . B u t t h e r e a r e m a n y o t h e r i n s t a n c e s in w h i c h t h e A f r i c a n a p p r o a c h t o f r a c t a l g e o m e t r y m a k e s use o f num bers* ' ...................... ........................... N o n lin e a r a d d itiv e se rie s in A f r i c a T h e c o u n t i n g n u m b e r s ( 1 ,2 ,3 • • •) c a n b e t h o u g h t o f as a k i n d o f i t e r a t i o n , b u t o n l y in t h e m o s t tr iv ia l w a y .1 I t is t r u e t h a t w e c o u l d p r o d u c e t h e c o u n t i n g n u m ­ b e r s fr o m a r e c u r s i v e lo o p , t h a t is, a f u n c t i o n in w h i c h t h e o u t p u t a t o n e sta g e b e c o m e s t h e i n p u t for t h e n e x t : X n+( = X n + 1 . B u t t h i s is a s t r i c t l y l i n e a r series, i n c r e a s i n g by t h e s a m e a m o u n t e a c h t i m e — t h e n u m e r i c j e q u i v a l e n t o f w h a t we sa w in t h e li n e a r c o n c e n t r i c c ir c le a n d l i n e a r s p i r a l.'- A d d i tio n e a n , h o w e v e r , prod u c e n o n l i n e a r se r ie s ,2__and t h e r e a r e a t l e a s t tw o e x a m p l e s o f n o n l i n e a r a d d i t i v e se ries in A f r i c a n c u l t u r e s . T h e t r i a n g u l a r n u m b e r s ( 1 , 3 , 6 , 1 0 , 1 5 . . .) a r e u se d in a g a m e c a l l e d " t a r u m b e t a " in e a s t A f r i c a (Z a s la v s k y j 9 7 3 , 1 1 1). F ig u r e 7.1 s h o w s h o w th e s e n u m b e r s a re d e r i v e d from t h e s h a p e o f t r i a n g l e s o f in c r e a s i n g size, a n d h o w t h e n u m e r i c series c a n b e c r e a t e d by a r e c u r s i v e lo o p . A s in t h e c a s e o f c er- N u m e ric systems 87 num ber o f s to n e s: num ber o f it e r a t io n s : A g a m e c a l l e d ‘' t a r u m b e t a " i n E a s t A f r i c a m a k e s u s e o f t h e t r i a n g u l a r n u m b e r s , s t a r t i n g w i t h 3 ( 3 , 6 , 1 0 , 15 In th is g a m e , o n e p la y e r calls o u t a c o u n t as h e re m o v e s s to n e s co n se c u tiv e ly , l e ft t o r i g h t a n d b o t t o m t o t o p , w h i l e t h e o t h e r p l a y e r , w i t h h i s b a c k t u r n e d , m u s t s i g n a l w h e n e v e r t h e f ir st s t o n e i n a r o w h a s b e e n r e m o v e d . T h e s t o n e s i n e a c h t r i a n g u l a r a r r a y c a n b e b u i l t u p i n a n i t e r a t i v e f a s h i o n , t h a t is, t h e n e x t tria n g le c a n be c r e a t e d by a d d in g a n o t h e r la y e r to a n y s id e o f t h e p r e v io u s tria n g le . T h e n u m b e r to b e a d d e d i n e a c h a d d i t i o n a l l a y e r is s i m p l y t h e n u m b e r o f i t e r a t i o n s . F o r e a c h i t e r a t i o n i, a n d t o t a l n u m b er o f sto n es N , we have: Ni*i = N , + 1 1 = +.1 3 = + 2 “ '6 = + 3 10 = + 4 15 = 10 + 5 (sta rtin g w ith N 0 = 0 ) (a triv ia l array, n o t u s e d in t h e g a m e ) I n o t h e r w o r d s , t h e n e x t n u m b e r w i l l b e g iv e n b y t h e la s t n u m b e r p lu s t h e it e r a t io n c o u n t : FIGURE 7.1 T h e tr ia n g u la r n u m b e r s in a n E a s t A fr ic a n g a m e t a i n f o r m a l a g e - g r a d e i n i t i a t i o n p r a c t i c e s (s e e c h a p t e r s 5 a n d 8 ) , che sim p le ve rsion s a re u sed by s m a l l e r c h i l d r e n , a n d t h e h i g h e r i t e r a t i o n s a re p i c k e d u p w ith increasing age. W h i l e th e r e is n o in d i c a ti o n o f a form al r e l a ti o n s h ip in this in stance, t h e r e is s till a n u n d e r l y i n g p a r a l l e l b e t w e e n t h e .ite ra tiv e c o n c e p t o f a g i n g c o m ­ m o n t o m a n y A f r i c a n c u h u r e s — e a c h i n d i v i d u a l p a s s in g t h r o u g h m u l t i p l e t u r n s o f t h e “h f e c ycle.*.’.— a n d t h e k e r a t i v e n a t u r e o f t h e t r i a n g u l a r n u m b e r series. A n o t h e r n o n l i n e a r a d d it iv e series was fo u n d in a r c h a e o l o g ic a l e v id e n c e from N o r t h A fric a . B ada w y ( 1 9 6 5 ) n o t e d w h a t a p p e a rs t o .b e use o f t h e F ib o n a c c i series in t h e la y o u t o f t h e t e m p l e s o f a n c i e n t E gypt. U s i n g a s l ig h t ly d if fe re n t a p p r o a c h , T h e Fibonacci series .(I, 1 , 2 , 3 , 5 , 8 , 13 . . . ) was found by Badawy (1965) in his study o f the layout o f th e temples of Egypt. H is analysis was quite complex, but it ismot difficult to create a simple visualization. Here we see th e series in the successive c ham be rs of the tem ple of Karnak. T h e Fibonacci series is p roduced by adding the previous n u m b e r to the curren t n u m b e r to get the n ex t number, starting with 1 + 1 = 2 . For each iteration i, th e n u m b er N in th e series is given by: G ray rectangles added fo r m easurem ent 9 N i +1 = N i t h a t is, N n e x t = H cu rten t + ^previous 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 = 8 3 + 5 5 + 8 = 13 next N previous current FIGURE 7 . 2 T h e F ib o n a c c i series in a n c ie n t E g y p t N um eric systems I f o u n d a v is u a lly d i s t i n c t e x a m p l e o f t h i s se r ie s in t h e s u c c e s s iv e c h a m b e r s o f t h e t e m p l e o f K a r n a k , as s h o w n in figure 7 .2 a. F ig u re 7 .2 b s h o w s h o w th e s e n u m ­ b e r s c a n b e g e n e r a t e d u s i n g a r e c u r s i v e lo o p . T h i s f o r m a l s c a l in g p l a n m a y h a v e • b e e n d e r i v e d f r o m t h e n o n n u m e r i c v e r s i o n s o f s c a l i n g a r c h i t e c t u r e w e see t h r o u g h o u t A f r i c a . A n a n c i e n t s e t o f b a l a n c e w e ig h ts , a p p a r e n t l y u se d in E gypt, S y ria, a n d P a l e s t i n e c irca 1200 b .c .e ., also a p p e a r to e m p lo y a F ib o n a c c i s e q u e n c e ( P e t r u s o 1 9 8 5 ) . T h i s is a p a r t i c u l a r l y i n t e r e s t i n g use, s i n c e o n e o f t h e s t r i k i n g m a t h e m a t i c a l p r o p e r t i e s o f t h e se r ie s is t h a t o n e c a n c r e a t e a n y p o s i t i v e in t e g e r t h r o u g h a d d i t i o n o f s e l e c t e d m e m b e r s — a p r o p e r t y t h a t m a k e s it id e al for a p p l i ­ c a t i o n to b a l a n c e m e a s u r e m e n t s ( H o g g a t t 1 9 6 9 , 7 6 ) . T h e r e is n o e v i d e n c e t h a t a n c i e n t G r e e k m a t h e m a t i c i a n s k n e w o f t h e F i b o n a c c i series. T h e r e w as u se o f t h e F i b o n a c c i se rie s in M i n o a n d e s i g n , b u t P re zio si ( 1 9 6 8 ) c i t e s e v i d e n c e i n d i ­ c a t i n g t h a t it c o u l d h a v e b e e n b r o u g h t f r o m E g y p t by M i n o a n a r c h i t e c t u r a l w orkers e m p lo y e d a t K ahun. D o u b lin g se ries in A f r i c a S o m e a c c o u n t s r e p o r t t h a t A f r i c a n s u se a “ p r i m i t i v e ” n u m b e r s y s te m in w h i c h th e y c o u n t b y m u l t i p l e s o f tw o . It is tr u e t h a t m a n y c ase s o f A f r i c a n a r i t h m e t i c a re b a s e d o n m u l t i p l e s o f tw o , b u t as w e w ill s e e , b a s e - 2 s y s te m s a re n o t c r u d e a r t if a c ts fr o m a f o r g o t t e n past. T h e y h a v e s u r p r is in g m a t h e m a t i c a l s ig n if i c a n c e , n o t o n ly in r e l a t i o n t o A f r i c a n fractals, b u t to t h e W e s t e r n h is to ry o f m a t h e m a t i c s a n d c o m p u t i n g as well. T h e p re s e n c e o f d o u b lin g as a c u ltu ral t h e m e occurs in m a n y different A fric a n so cieties a n d in m a n y d ifferen t social d o m a i n s , c o n n e c t i n g th e sa c re d n e ss o f tw ins, ' s p i r i i 'd o u b i e s , a n d d o u b l e v is io n w i t h m a t e r i a l o b j e c t s , s u c h a s t h e b l a c k s m i t h 's tw in b e llo w s a n d t h e d o u b l e i r o n h o e g i v e n in b r i d e w e a l t h (fig. 7 .3 ). F ig ure 7 .4 a s h o w s t h e I s h a n g o b o n e , w h i c h is a r o u n d 8 , 0 0 0 y e a r s o ld a n d a p p e a r s t o s h o w a d o u b l i n g s e q u e n c e . j D o u b l i n g is f u n d a m e n t a l t o m a n y o f th e c o u n t i n g s y s te m s o f A f r i c a in m o d e r n t i m e s as well. It is c o m m o n , fo r e x a m p l e , t o h a v e t h e w o rd for . .* a n ' e v e n n u m b e r 2 N m e a n “N p lu s N ” (e .g ., t h e n u m b e r 8 in t h e S h a m b a a l a n ­ guage o f T a n z a n i a is “n e n a n e , ” literally “fo u r a n d f o u r ”). A s im ilar d o u b li n g takes p lac e fo r t h e p re c i s e l y a r t i c u l a t e d s y s te m o f n u m b e r h a n d g e s tu re s (fig. 7 . 4 b ) , for ex am ple, Ilfour” re p r e s e n te d by t w o gro ups o f tw o fingers, a n d “e i g h t ’’ by tw o g roups o f four. P e t i t t o ( 1 9 8 2 ) f o u n d t h a t d o u b l i n g w as u s e d in m u l t i p l i c a t i o n a n d d iv i s i o n t e c h n i q u e s in W e s t A f r i c a (fig. 7 . 4 c ) . G i l l i n g s ( 1 9 7 2 ) d e t a i l s t h e p e r ­ s i s t e n t use o f p o w e r s o f t w o in a n c i e n t E g y p t i a n m a t h e m a t i c s as w e l l , a n d Z aslav sky ( 1 9 7 3 ) s h o w s a r c h a e o l o g i c a l e v i d e n c e s u g g e s tin g t h a t a n c i e n t E g y p t ’s use o f b a s e - 2 c a l c u l a t i o n s d e r i v e d f r o m t h e use o f b a s e - 2 in s u b - S a h a r a n A f r i c a . A frican fractal m athematics 90 D o u b l i n g practices; w e r e a ls o u se d by A f r i c a n d e s c e n d a n t s in cine A m e r i ­ cas. B e n j a m i n B a n n e k e r , for e x a m p l e , m a d e u n u s u a l u se o f d o u h l i n g in b is c a lc u l a t t o n s , w h i c h m a y h a v e d e r i v e d fr o m t h e t e a c h i n g s o f h i s A fric a iV T a th e r a n d g r a n d f a t h e r ( E g la s h 1 9 9 7 c ) . G a t e s ( 1 9 8 8 ) e x a m i n e d t h e c u l t u r a l s i g n i f i c a n c e o f d ou b.ling in W e s t A f r i c a n re l ig i o n s s u c h as v o d u n a n d its t r a n s f e r t o “v o o d o o ” in t h e A m e r i c a s . In t h e r e l i g i o n o f S h a n g o , for e x a m p l e , t h e v o d u n g o d o f t h u n ­ d e r a n d l i g h t n i n g is r e p r e s e n t e d by a d o u b l e - b l a d e d a x e (fig. 7 . 5 a ) , u s e d by S h a n g o d e v o te e s in t h e n e w w o rld as well (R. T h o m p s o n 198,3). F igure 7.51-) show s figure 7 .3 D o u b lin g in A fr ic a n so c ia l p r a c tic e s ( a ) T h i s f i g u r e is u s e d b y w o m e n i n G h a n a t o e n c o u r a g e t h e b i r t h o f t w i n s , ( b ) A d o u b l e i r o n h o e is s o m e t i m e s u s e d a s p a r t o f th e b rid e p rice c ere m o n y , (c) T h e d o u b le b ello w s o f th e b la c k s m ith , (d) D o u b le v is io n ; a c o m m o n th e m e .in sev eral A fric a n sp iritu a l p ra c tic es, o fte n im p ly in g th a t o n e c a n see b o t h t h e m a te r ia l w o rld a n d t h e s p irit w orld. ( b , M a r c a n d E v e l y n B e r n h e i m f r o m R a / d io G i u / l i o n c u e ; c o u r t e s y o f U g a n d a N a t io n a l M u se u m , c , Jihoio c o u r t e s y I F A N , D a k a r . d , fro n B e rjo n n e a u a n d Som tery 1 9 8 7 .) N um eric systems 9i (a) T h e Ishango bone, estimated to be over 8, 000 years old, shows wh at appears to be use of doubling: 3 + 3 = 6, 4 + 4 = 8, 10 = 5 + 5 . 10 (b) Even numbers are typically represented by doubli ng in t h e precisely articulated system of Afri can h a n d gestures. (c) D o u b l i n g w a s t r a d i t i o n a l l y u s e d b y t a i l o r s in W e s t A f r i c a w h e n d o i n g la r g e m e n t a l m u l t i p l i c a t i o n s ; it is e s s e n t i a l l y b a s e d o n w h a t w e w o u l d c a l l f a c t o r i n g . For e x a m p l e , 3 x 2 7 3 ( “ 3 t a k e n 2 7 3 t i m e s ” ) w o u l d b e c a l c u l a t e d b y s u c c e s s i v e l y d o u b l in g 3 ( 6 , 1 2, 2 4 . . . ) w h i l e k e e p i n g c r a c k o f t h e c o u n t e r p a r t in p o w e r s o f t w o ( 2 , 4 , 8 . . . ). W h e n th e n e x t p ow er o f tw o w o u ld o v e r s h o o t 2 7 3 , h e t h e n h as to m em orize th e n u m b e r re a ch e d so fa r t h r o u g h d o u b l i n g s o f 3 ( 7 6 8 ) . w h i l e s u b t r a c t i n g t h e p o w e r o f t w o t h a t w a s r e a c h e d ( 2 7 3 - 2 5 6 = 1 7 ) . T h e n h e s t a r t s a g a i n , d o u b l i n g 3, a n d k e e p i n g t r a c k o f t h e p o w e r s o f tw o. W h e n t h e n e x t p o w e r o f t w o w o u l d o v e r s h o o t 1 7 , h e a g a i n m e m o r i z e s t h e n u m b e r re a ch e d t h r o u g h d o u b l i n g s o f 3 ( 4 8 ) a n d s u b t r a c t s t h e p o w e r o f t w o ( 1 7 — 1 6 — 1 ) . S i n c e o n e is left o v e r , h e j u s t n e e d s t o a d d a n a d d i t i o n a l . 3 . T h e a n s w e r is t h e n g i v e n b y . t h e s u m o f t h e u n d e r l in e d t e rm s : 7 6 8 + 4 8 + 3 = 8 1 9 . D e s p it e t h e c o m p l e x i t y o f t h e m e t h o d , t h e t a i l o r s w e r e q u i t e fast a t p e r f o r m i n g t h e s e s i l e n t m e n ia l o p e r a t i o n s . FI GURE 7.4 D o u b l i n g in A f r i c a n a r i t h m e t i c f t i i n t l b, f r o m F r o m Z a s l a v s k y 1 9 7 3 .) t he use o f a d o u b l i n g s e q u e n c e in t h e ser uccure o f a S h a n g o t e m p l e a n d in r e l i ­ gi ous c e r e m o n i e s ( r i t u a l c h o r e o g r a p h y a l i g n i n g t w o p r i es t s , f o u r c h i l d r e n , eight legs). A c u r a t o r a t t h e M u s e e E t h n o g r a p h i q u e in P o r t o N o v o , B e n i n , w h o speci ali zed in S h a n g o e x p l a i n e d t o m e t h a t t h e s e d o u b l i n g s t r u c t u r e s w e r e u s e d I9.ecause.the g o d o f l i g h t n i n g r e q u i r e d a p o r t r a i t o f t h e f o r k e d s t r u c t u r e oFaTigft tni ng bol t . T h e m o d e l is p a r t i c u l a r l y i n t e r e s t i n g in t h a t t h e l e n g t h s o f e a c h i t e r ­ ation are s h o r t e n e d , so t h a t o n e c o u ld h a v e in fin ite d o u b li n g s in a finite (a) S h an g o , th e god o f li gh tnin g, is p a rt of the vodun religion o f B enin an d was one of t h e im p o rta n t c o m p o n en ts in th^, c reatio n of the voodoo religion in th e N e w World. Here we see th e double-bladed " t h u n d e r ax e ,” w ith a n o th e r double blade w ith in each side. a b (b) S h ang o temple arid initiation. H e re we see th e doubling sequence carried o u t further, using the bilateral symmetry of t h e h u m a n body itself in tbe last iteration. T h is is used to symbolize the bifurcating p attern of the lig h tn in g bolt. FIGURE 7 .5 D o itb lin g in th e re lig io n o f S h a n g o fa, courtesy / F AN, Dakar, b: hath center j>ho(os, courtesy / F A N , Dakar; lower right, courtesy Dave C ro w ley, xoxuw.storinguy.cnm.) N u m e r ic system s sp a c e — a tr u e fra c ta l. T h e self-sim ilar s tru c tu r e o f lig h tn in g h a s b e e n a favorite e x a m p le fo r fr a c ta l g e o m e try te x ts (see M a n d e lb r o t 1977 ). T h e d o u b lin g s e q u e n c e u s e d t o m o d e l t h e f r a c t a l s t r u c t u r e o U i g h t n i n g in S h a n g o w o u ld n o t . g iv e a n a c c u r a t e v a l u e fo r t h e e m p i r i c a l ' f r a c t a l d i m e n s i o n — re a l l i g h t n i n g t e n d s to b r a n c h m u c h m o r e t h a n d o u b l i n g a llo w s for— b u t i t ’s e n o u g h to k n o w t h a t t h e • v o d u n r e p r e s e n t a t i o n offers a t e s t a b l e ^ i u a n t i .t a t i v e . m o d e l . T h e m o s t m a t h e m a t i c a l ly s ig n i f i c a n t a s p e c t o f d o u b l i n g in A f r i c a n relig io n o c c u r s in t h e d i v i n a t i o n ( “f o r t u n e t e l l i n g ’^) t e c h n i q u e s o f v o d u n a n d its r e l i­ gious re la tiv e s (E g la s h 19 9 7 b ). T h e f a m o u ^ j f a d j v i n a t i o n sy stem (fig. 7.6) is b ased o n to ssin g pairs o f flat sh e lls o r see d s s p l it in tw o. E a c h l a n d s o p e n - s i d e o r c losed side (lik e “h e a d s o r t a i ls ” in a c o i n to s s ). T h e y a re c o n n e c t e d by a d o u b l e d c h a i n to m a k e f o u r p airs. E a c h g r o u p o f fo u r p a ir s g iv e s o n e o f t h e 16 d i v i n a t i o n s y m ­ b ols, w h i c h te ll t h e f u t u r e o f t h e d i v i n e r ’s c l i e n t . T h e Ifa s y s t e m is w h a t a m a t h e m a t i c i a n w o u ld c a ll “s t o c h a s t i c , ” t h a t is, i t o p e r a t e s by p u r e c h a n c e . B u t a clo sely r e l a t e d d i v i n a t i o n s y s te m , C e d e n a, h a s a n o n s t o c h a s t i c e l e m e n t— it is c lo s e r t o w h a t m a t h e m a t i c i a n s c a ll “d e t e r m i n i s t i c c h a o s . ” . M y i n t r o d u c t i o n t o c e d e n a , o r s a n d d i v i n a t i o n , t o o k p l a c e in D a k a r, S e n e ­ gal, w h e r e t h e lo c a l Is la m ic c u l t u r e c r e d i t s t h e B a m a n a (a ls o k n o w n as B a m b a r a ) w i t h a p o t e n t p a g a n m y s ti c is m . A l m o s t all d i v i n e r s h a d s o m e k i n d o f p h y s ic a l d e f o r m i t y — “ t h e p r i c e p a id for t h e i r p o w e r . ” -5 O n e d i v i n e r s e e m e d q u i t e w ill in g to t e a c h m e a b o u t t h e s y s te m , s u g g e s tin g t h a t it “w o u ld b e ju s t lik e s c h o o l . ” T h e first few se s s io n s w e n t s m o o t h l y , w i t h t h e d i v i n e r s h o w i n g m e a s y m b o l i c c o d e in w h i c h e a c h s y m b o l, r e p r e s e n t e d by a s e t o f f o u r v e r t i c a l d a s h e d li n e s d r a w n in t h e s a n d , s t o o d for s o m e a r c h e t y p i c a l c o n c e p t ( t r a v e l , d esire, h e a l t h , e t c . ) w ith w h i c h h e a s s e m b l e d n a r r a t i v e s a b o u t t h e f u t u r e . B u t w h e n I fin ally a s k e d h o w • he d eriv ed th e sym bols in p a r t i c u l a r , t h e m e a n i n g o f s o m e o f t h e p a t t e r n s d r a w n p r i o r t o t h e s y m b o l w r i t i n g — t h e y all l a u g h e d a t m e a n d s h o o k t h e i r h e a d s . “T h a t ’s t h e s e c r e t ! ” M y o ff e rs o f i n c r e a s i n g l y h i g h p a y m e n t s w e r e m e t w ith d is in t e r e s t . Finally, 1 tr ie d to e x p l a i n t h e so c ia l sig n ific a n c e o f c ro s s -c u ltu ra l m a t h e m a t i c s . I h a p p e n e d to h a v e a c o p y o f L i n d a G a r c i a ’s F ra cta l E x p lo r e r w ith m e a n d b e g a n by s h o w i n g a g r a p h o f t h e C a n t o r s e t, e x p l a i n i n g its re c u r s iv e c o n ­ s t r u c t i o n . T h e h e a d d iv i n e r , w i t h a n e x p r e s s i o n o f e x c i t e m e n t , s u d d e n l y s t o p p e d m e , s n a p p e d t h e b o o k s h u t , a n d s a i d “s h o w - h i m w h a t h e w a n t s ! ” A s it t u r n s o u t , t h e re c u r s iv e c o n s t r u c t i o n o.fJihe.,.Can.tQ.r_set w as j u s t t h e rig h t t h i n g to sh o w , b e c a u s e t h e B a m a n a d i v i n a t i o n is a ls o b a s e d o n r e c u r s io n Xs (fig- l - l ) - T h e d i v i n a t i o n b e g in s w i t h f o u r h o r i z o n t a l d a s h e d lin es, d r a w n rapidly, / so th a t th e r e is so m e r a n d o m v a r i a ti o n in t h e n u m b e r o f dashes in e a c h . T h e dashes \ are t h e n c o n n e c t e d in p airs, s u c h t h a t e a c h o f t h e fo u r lin e s is left w i t h e i t h e r \ o n e s i n g le d a s h ( i n t h e case o f a n o d d n u m b e r ) o r n o d a s h e s (a ll p a irs, t h e case b c FIGURE 7 .6 B i n a r y codes in d i v i n a t i o n (a) This Nigerian priest is telling the future by Ifa divination, in which pairs of flat shells or seeds split in two are tossed with each landing open-side or closed-side. They are connected hy a doubled chain to make four pairs, giving a total of 16 divination symbols. In this version of lfa (used in the Abigba region of Nigeria) they use two doubled chains and consider the cast more accurate if there is a correlation between the two sets, (b) Here we see a chain using split seeds. Each half lands eititer “closed" (meaning we see the rounded outside) or "open" (meaning we see tire interior). By using open to represent 0 (double lines), and closed to represent 1 (single line), we can see how the divination symbol is obtained, (c) The divination chain is interpreted as pairs summing to odd (one stroke) or even (two strokes). (a, j ) h o t o b y E . M. McClelland, courtesy R o y a l A n t h r o p o l o g i c a l I n s t i t u t e ) N u n ie r ic sy ste m s f o i a n e v e n n u m b e r ) . T h e n a r r a t i v e s y m b o l is t h e n c o n s t r u c t e d as a c o l u m n o f four I v e r t i c a l m a r k s , w i t h d o u b l e v e r t i c a l li n e s r e p r e s e n t i n g a n e v e n n u m b e r o f d a s h e s i a n d s i n g l e li n e s R e p r e s e n t in g a n o d d n u m b e r . A t t h i s p o i n t t h e sy s te m is s im ila r ^ t o t h e f a m o u s I f m d i v i n a t i o n : t h e r e a re t w o p o s s ib le m a r k s in fo u r p o s i t i o n s , so 16 p o s s ib le s y m b o ls . U n l i k e lfa, h o w e v e r , t h e r a n d o m s y m b o l p r o d u c t i o n is r e p e a t e d fo u r ti m e s r a t h e r t h a n tw o . T h e d i f f e r e n c e is q u i t e s i g n ifi c a n t. E a c h o f t h e lfa s y m b o l p a irs a re i n t e r p r e t e d as o n e o f 2 5 6 p o s s ib le O d u , o r verses. T h e lfa d i v i n e r m u s t m e m o r i z e t h e O d u ; h e n c e , f o u r s y m b o ls w o u ld be t o o c u m b e r ­ s o m e ( 6 5 , 5 3 6 p o s s ib le v e rs e s ). B u t t h e B a m a n a d i v i n a t i o n d o e s n o t r e q u i r e a n y verse m e m o riz a tio n ; as w e will see, its use o f re c u r s io n allows for verse self-assembly. A s in t h e a d d i t i v e s e q u e n c e s w e e x a m i n e d , t h e d i v i n a t i o n c o d e is g e n e r ­ a t e d b y a n i t e r a t iv e lo o p in w h i c h t h e o u t p u t o f t h e o p e r a t i o n is used as t h e i n p u t for t h e n e x t stage. I n t h i s c a se , t h e o p e r a t i o n is a d d i t i o n m o d u l o 2 ( “m o d 2” for shortTT w h i c h sim p ly g iv e s t h e r e m a i n d e r a f t e r d i v i s i o n by tw o. T h i s is t h e s a m e e v e n / o d d d i s t i n c t i o n u se d in t h e p a r i t y b it o p e r a t i o n t h a t c h e c k s for e rro rs o n c o n t e m p o r a r y c o m p u t e r sy s te m s . T h e r e is n o t h i n g p a r t i c u l a r l y c o m p l e x a b o u t m o d 2; in f a c t , 1 w as q u i t e d i s a p p o i n t e d a t f i rs t b e c a u s e its r e a p p l i c a t i o n d e s t r o y e d t h e p o t e n t i a l f o r a b i n a r y p l a c e h o l d e r r e p r e s e n t a t i o n in t h e B a m a n a d i v i n a t i o n . R a t h e r t h a n i n t e r p r e t e a c h p o s i t i o n in che c o l u m n as h a v i n g s o m e m e a n i n g (as w o u ld o u r b i n a r y n u m b e r 1 0 1 1, w h i c h m e a n s o n e 1, o n e 2, zero 4s, a n d o n e 8 ) , t h e d i v i n e r s r e a p p l i e d m o d 2. t o e a c h ro w o f t h e first tw o .s y m b o l s a n d t o e a c h ro w o f t h e last t w o . T h e r e s u l t s w e r e t h e n a s s e m b le d i n t o tw o n e w s y m b o ls , a n d m o d 2 w as a p p l i e d a g a i n t o g e n e r a t e a t h i r d s y m b o l. A n o t h e r fo u r s y m b o ls w e r e c r e a t e d by r e a d i n g t h e ro w s o f t h e o r i g i n a l f o u r 'a s c o l u m n s , a n d m o d 2 w a s a g a i n r e c u r s i v e l y a p p l i e d t o g e n e r a t e a n o t h e r t h r e e s y m b o ls . . T h e use o f a n i t e r a t i v e l o o p , p a s s i n g o u t p u t s o f a n o p e r a t i o n b a c k as in p u t s fo r t h e n e x t s ta g e , w as a s h o c k t o m e ; 1 w a s a t le a s t as t a k e n a b a c k by t h e s a n d s y m b o ls as t h e d i v i n e r s h a d b e e n b y t h e C a n t o r sec. It w o u ld b e n a i v e to c l a i m t h a t th i s w as s o m e h o w a l e a p o u t s i d e o f o u r c u l t u r a l b a r r ie r s a n d p o w e r d i f f e r e n c e s — in f a c t , t h a t ’s ju s t t h e s o r t o f p r e t e n s i o n t h a t t h e last t w o d e c a d e s o f r e f l e x i v e a n t h r o p o l o g y h a s b e e n d e d i c a t e d a g a i n s t — b u t it w o u ld a ls o be e t h n o c e n t r i c to ru le o u t th o s e a s p e c t s t h a t w o u ld b e a t t r i b u t e d to m a t h e m a t i c a l c o l l a b o r a t i o n e l s e w h e r e in che w o r l d ; t h e m titiia]_ _d eh gh t in t w o r e c u r s i o n f a n a t i c s d i s c o v e r i n g e a c h o t h e r . A n d t h e a p p e a r a n c e o f t h e sy m b o JsJ U k L o iit in tw o g r o u p s o f " s e v e n — t h e R o s i c r u c i a n ’s m y s t i c n u m b e r — a d d e d s o m e n u m e r olo g ica T icT n g o r T t h e c a k e . T h e f o l l o w i n g d ay 1 f o u n d t h a t t h e p r e s e n t a t i o n h a d n o t b e e n c o m p l e t e : a n a d d it io n a l tw o sym bo ls w ere left o u t. T h e s e w e re also g e n e ra t e d by m o d 2 re c u r ­ s i o n u s i n g t h e t w o b o t t o m s y m b o l s t o c r e a t e a f i f t e e n t h , a n d u s i n g t h a t last A fr ic a n fra c ta l m athem atics s y m b o l w i t h t h e first s y m b o l t o c r e a t e a s i x t e e n t h ( b r i n g i n g t h e t o t a l d e p t h o f r e c u r s i o n t o five i t e r a t i o n s ) . T h e f i f t e e n t h s y m b o l is c a l l e d “t h i s w o r l d , " a n d t h e s i x t e e n t h is “t h e n e x t w o r l d , ” so t h e r e w as g o o d r e a s o n t o s e p a r a t e t h e m fro m t h e o t h e r s . T h e final p a r t o f t h e s y s te m — c r e a t i n g a n a r r a t i v e f r o m t h e s y m b o ls — w a s s till u n c l e a r , b u t I w as a s s u re d t h a t it c o u l d b e l e a r n e d if I c a r e f u l l y fo l lo w e d t h e i r i n s t r u c t i o n s . 1 w as t o g i v e s e v e n c o i n s t o s e v e n le p e rs, p l a c e a k o l a n u t o n b 1 — I 1 < — 1 < — ^ — 1 1 1 1 1 1 I I 1 — ^ 1 I 1 I I — ^ 1 1 1 1 1 — > 1 1 1 1 — > 1 1 I 1 1 1 1 c 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 (e ) A f te r th is , th e o rig in al four a re read sid ew ay s to c re a te four 1 1 1 1 d 1 > 1 ----- > 1 1 1 1 m o re sy m b o ls, a n d th e e n tire N.. 1 - - 1 1 p r o c e s s is r e p e a t e d , p r o d u c i n g 1 --- 1 ----- ^ 1 a n o th e r g ro u p o f sev en . In th e fin a l s t e p , t h e first a n d la s t f r o m e a c h g ro u p d f s e v e n a re p a ire d off to g e n e r a t e t h e final tw o s y m b o ls. FIGURE 7 . 7 (a) F o u r sets o f ra n d o m d a sh e s B a m a n a s a n d d iv in a tio n a r e d r a w n , (b) E a c h o f t h e d a s h e s is p a i r e d , a n d t h e o d d / e v e n r e s u l t s a r e r e c o r d e d , ( c ) T h e p r o c e s s is r e p e a t e d f o u r t i m e s , r e s u l t i n g i n f o u r s y m b o l s . E a c h r o w o f t h e firs t tw o s y m b o ls a n d t h e last t w o s y m b o ls a re p a ire d o ff to g e n e r a t e t w o n e w s y m b o ls , (d ) T h e tw o n e w ly g e n e r a t e d s y m b o ls, n o w p l a c e d b e lo w t h e o r ig i n a l four, a re a g a i n p a i r e d o ff t o g e n e r a t e a s e v e n th sy m b o l. N u m e ric system s a p ile o f s a n d n e x t t o m y b e d a t n i g h t , a n d in t h e m o r n i n g b r i n g a w h i t e c o c k , 1^ w h i c h w o u ld h a v e t o b e s a crificed t o c o m p e n s a t e fo r t h e h a r m f u l e n e rg y re l e a s e d I in t h e t e l l i n g o f t h e s e c r e t. I f o l lo w e d all t h e in s t r u c t i o n s , a n d t h e n e x t m o r n - i in g b o u g h t a large w h i t e c o c k a t t h e m a r k e t . T h e y h e l d t h e c h i c k e n o v e r t h e d iv - ' i n a t i o n s a n d , a n d I w as t o l d t o e a t t h e b i t t e r k o l a n u t as th e y m a r k e d d i v i n a t i o n s y m b o ls o n its f e e t w i t h a n i n k p e n . A l i t t l e s a n d w as r h r o w n in its m o u t h , a n d i t h e n 1 w as to ld t o h o l d it d o w n as p ra y e r s w e r e c h a n t e d . T h e r e w as n o a c t i o n o n j t h e p a r t o f t h e d i v i n e r ; t h e c h i c k e n s i m p l y d i e d in m y h a n d s . W h i l e still a b i t s h a k e n by t h e c h i c k e n ’s d e m i s e (as w ell as e x p e r i e n c i n g a re s p e c ta b le buzz fr o m t h e k o la n u t ) , I was to l d t h e r e m a in i n g mystery. E a c h sy m ­ b o l h a s a “ h o u s e ” in w h i c h it b e lo n g s — fo r e x a m p l e , t h e p o s i ti o n o f t h e s i x t e e n t h s y m b o l is “ t n e n e x t w o r l d " — b u t in a n y g i v e n d i v i n a t i o n m o s t s y m b o ls w ill n o t b e l o c a t e d in t h e i r o w n h o u s e . T h u s t h e s i x t e e n t h s y m b o l g e n e r a t e d m i g h t be “d e sire ,” so we w o u ld h a v e d e s ire in t h e h o u s e o f t h e n e x t w orld , a n d so o n . O b v i ­ o usly t h i s still l e a v e s r o o m f o r c r e a t i v e n a r r a t i o n o n t h e p a r t o f t h e d i v i n e r , b u t t h e b e a u t y o f t h e s y s te m is t h a t n o v e rs e s n e e d to b e m e m o r i z e d o r b o o k s c o n ­ s u l te d ; t h e sy s te m c r e a t e s its o w n c o m p l e x variety. T h e m o s t e l e g a n t p a r t o f t h e m e t h o d is t h a t it r e q u i r e s o n l y f o u r r a n d o m d r a w i n g s ; a f t e r t h a t t h e e n t i r e s y m b o l i c a r r a y is q u i c k l y s e l f - g e n e r a t ed- S e lf g e n e r a t e d v a r i ety is i m p o r t a n t i n j n o c l e r n c o m p u t i n g , w h e r e it is c a l l e d “p s e u d o ­ r a n d o m n u m b e r g e n e r a t i o n ” (fig. 7 . 8 ) . T h e s e a l g o r i t h m s t a k e l i t t l e m e m o r y , b u t c a n g e n e r a t e v e r y l o n g li s t s o f w h a t a p p e a r t o b e r a n d o m n u m b e r s , a l t h o u g h t h e list w ill e v e n t u a l l y s t a r t o v e r a g a i n ( t h i s l e n g t h is c a l l e d t h e “p e r i o d ” o f t h e a l g o r i t h m ) . A l t h o u g h t h e B a m a n a o n l y r e q u i r e a n a d d i t i o n a l 12 s y m b o l s to b e g e n e r a t e d in t h i s f a s h i o n , a m a x i m u m - l e n g t h p s e u d o r a n d o m n u m b e r g e n e r a t o r u s i n g t h e i r i n i t i a l fo u r s y m b o l s w ill p r o d u c e 6 5 . 5 3 5 s y m b o ls b e f o r e it b e g i n s t o r e p e a t . A s i m i l a r s y s te m f o r s e l f - g e n e r a t e d v a r i e t y w as d e v e l o p e d as a m o d e l for t h e “c h a o s ” o f n o n l i n e a r d y n a m i c s by M a r s t o n M o r s e ( 1 8 9 2 - 1 9 7 7 ) . B e fo re th e 1970s, m a t h e m a t i c i a n s h a d a s s u m e d t h a t , b e s i d e s a few e s o t e r i c e x c e p t i o n s (che a l g o r i t h m s for p r o d u c i n g i r r a t i o n a l n u m b e r s s u c h as V 2), t h e o u t p u c o f a n e q u a ­ t i o n w o u l d e v e n t u a l l y s t a r t r e p e a t i n g . T h a t a s s u m p t i o n w as p a r t ly b a s e d o n E u r o p e a n c u l t u r a l id e a s a b o u t fr e e w il U - c o m p l e x b e h a v i o r c o u l d n o t b e t h e result o f p r e d e t e r m i n e d sy ste m s (see P o r t e r 1 9 8 6 ). It was n o t u n t i l r h d ^ i Q h o s ^ o s j t h a t m a t h e m a t i c i a n s realized t h a t e v e n sim ple, c o m m o n eq u a tio n s d escrib in g 'th in g s lik e p o p u l a t i o n g r o w t h o r.flu id flow c o u l d r e s u l t in w h d t t h e y c a l l e d “d e t e r m i n ­ istic c h a o s ”— a n o u t p u t t h a t n e v e r re p e a ts, g iv i n g th e a p p e a r a n c e o f r a n d o m n u m ­ bers fr o m a n o n r a n d o m ( d e t e r m i n i s t i c ) e q u a t i o n . M o r s e d e v e l o p e d t h e m i n i m a l case for s u c h b e h a v io r . ^ A fr ic a n fra c ta l mathematics 98 I 1 1I 0111 001 1 0001 1000 0 100 0010 1001 0110 1 100 01 I 0 1011 0101 1010 1 101 1110 FIGURE 7 .8 P seu d o ra n d o m n u m b e r genera tio n f r o m s h ift r e g is t e r c ir c u its (a) If we think of the two-strokes as zero and single stroke as one, the Bamana divination, system is almost identical to the process of pseudorandom number generation used by digital circuits called “shift registers.” Here the circuit cakes mod 2 of die last two bits in the register and places the result in the first position. T h e other bits are shifted to the right, with the last discarded. This four-bit shift register will only produce 15 binary words before the cycle starts over, but the period of the cycle increases with more bits ( 2 n - 1). For .the entire 16 bits (four symbols of four bits each) that begin the Bamana divination, 65,535 binary words can be produced before repeating the cycle. (b ) Electrical circuit representation of a four-bit shift register combined with exclusive-or to perform the mod 2 operation. W hile school­ teachers are making increasing use of African culture, in the mathematics classroom, few have explored the potential applications to technology education. T h e c o n s t r u c t i o n o f t h e M o r s e se q u e n c e b e g i n s b y c o u n t i n g f r o m z e r o j n b in ary n o ta tio n : 0 0 0 , oot , 01 o , 0 1 1 . . . . I t t h e n t a k e s t h e s u m o f t h e d ig i ts in e a c h n u m b e r - -o + o + o = o , n . + o + 1 = i , e t c . — a n d f i n a l l y m o d 2 o f e a c h s u m . T h e r e s u l t is a s e q u e n c e w i t h m a n y r e c u r s i v e p r o p e r t i e s , ^ b u t o f e n d l e s s v a r i e t y . M o r s e d id t h e s a m e “ m i s r e a d i n g ” o f t h e b i n a r y n u m b e r a s d i d t h e \ ' B a m a n a — a lth o u g h h e did n o t h a v e a n a n th r o p o lo g is t sc o w lin g a t h im for •! i g n o r i n g p l a c e v a l u e — a n d h e d i d it for t h e s a m e r e a s o n : c o m b i n e d w i t h t h e j • m o d 2 o p e r a t i o n , i t m a x i m i z e s v a r i e ty . I n m y r e a d i n g o f d i v i n a t i o n l i t e r a t u r e 1 e v e n t u a l l y c a m e a c ro s s t h e d u p l i ­ c a t e o f t h e B a m a n a t e c h n i q u e 5 , 0 0 0 m i le s t o t h e e a s t in M a l a g a s y sik id y ( S u s s m a n a n d S u s s m a n 1 9 7 7 ) , w h i c h i n s p i r e d a s tu d y o f t h e h i s t o r y o f its d if fu s io n . T h e s tr o n g jj r n i la v i ty o f b o t h s y r n b o lic t e c h n i q u e a n d s e m a n t i c c a t e g o r i e s { a . w h a t E u r o p e a n s t e r m e d “g e o m a n c y ” w a s first n o t e d by F l a c o u r t { r 6 6 r ), b u t it w as n o t u n t i l T r a u t m a n n ( 1 9 3 9 ) t h a t a s e r io u s c l a i m w a s m a d e for a c o m m o n s o u r c e fo r th is A r a b ic , E u r o p e a n , W e s t A f r i c a n , a n d East A f r i c a n d i v i n a t i o n t e c h n i q u e . T h e c o m m o n a l i t y w as c o n f i r m e d in a d e t a i l e d f o r m a l a n a ly s is by J a u l i n ( 7 9 6 6 ) . B u t w h e t e d id it o r i g i n a t e ? j N u m e ric system s ( S k i n n e r ( i q 8 o ) provides a w e ll- d o c u m e n t e d his tory o f t h e diffusion e v id e n c e , i from cPie'first specific w r it te n r e c o r d — a n i n t h - c e n t u r y Jew ish c o m m e n ta r y by A r a n ' b e n J o s e p h — to its m o d e r n use in A l e i s t e r Crq.yvley’s L ib e r 7 7 7 . T h e o l d e s t A r a - [ bic d o c u m e n t s ( t h o s e o f a z - Z a n t i in t h e t h i r t e e n t h c e n t u r y ) c la i m t h e o r ig in o f g e o m a n c y (i!m aLrcim l, “t h e s c i e n c e o f s a n d ” ) t h r o u g h t h e E gy p tian go d Idris ( H e r - ; m e s T ris m e g i s tu s ) ; w h ile we n e e d n o t t a k e t h a t as a n y t h i n g m o r e t h a n a c l a i m ! to a n t i q u i t y , a N i l o t i c i n f l u e n c e is n o t u n r e a s o n a b l e . B u d g e ( 1 9 6 1 ) a t t e m p t s t o j c o n n e c t t h e use o f s a n d in a n c i e n t E g y p t i a n r i tu a l s to A f r i c a n g e o m a n c y , b u t it 1 is h a r d to see th is as u n i q u e . M a t h e m a t i c a l l y , h o w e v e r , g e o m a n c y is s trik in g ly out./ o f p l a c e in n o n - A f r i c a n sy s te m s . L ik e o t h e r li n g u is ti c c o d e s , n u m b e r b a s e s t e n d j o h a v e a n j x t r . e m e l y lo n g h is to r i c a l p e r s is t e n c e . E v e n u n d e r P l a t o n i c r a t i o n a l i s m , t h e a n c i e n t G r e e k s h e l d 10 t o b e t h e m o s t s a c r e d o f all n u m b e r s ; t h e K a b b a l a h ’s A y in S o f e m a n a t e s by 10 S e firo t, a n d t h e C h r i s t i a n W e s t c o u n t s o n its “H i n d u - A r a b i c ” d e c i m a l n o t a ­ t i o n . I n ^ ' f r i c gjl, o n t h e o t h e r h a n d , b a se -2 c a l c u l a t i o n w as u b i q u i t o u s, e v e n for m u l t i p l i c a t i o n a n d d iv is io n . A n d it is h e r e t h a t we find t h e c u lt u ra l c o n n o t a t i o n s o f d o u b l i n g t h a t g r o u n d t h e d i v i n a t i o n p r a c t i c e in its re lig io u s sig n i f i c a n c e. T h e im p lic atio n s o f this trajecto ry — fro m s u b - S a h a r a n A frica to N o r t h A frica to E u r o p e — a re q u i t e s i g n if i c a n t fo r t h e h i s t o r y o f m a t h e m a t i c s . F o l l o w i n g t h e i n t r o d u c t i o n o f g e o m a n c y to E u ro p e by H u g o o f S a n ta l la in tw e lf th - c e n tu r y S p a in , it w as t a k e n u p w i t h g r e a t i n t e r e s t b y t h e p r e - s c i e n c e m y s tic s o f t h o s e t i m e s — a lc h e m is ts , h e r m e t i c i s t s , a n d R o s i c r u c i a n s (fig. 7 .9 ). B u t th e s e E u r o p e a n g eom a n c e r s — R a y m o n d L u ll , R o b e r t F l u d d , d e P e r u c h i o , H e n r y d e P isis, a n d o th e r s — p e rs iste n tly re p la c e d t h e d e t e r m i n i s t i c asp e c ts o f t h e system w i t h c h a n c e . By m o u n t i n g t h e 16 figures o n a w h e e l a n d s p i n n i n g it, t h e y m a i n t a i n e d t h e i r s o c i e t y ’s e x c l u s i o n o f a n y c o n n e c t i o n s b e t w e e n d e t e r m i n i s m a n d u n p r e d i c t a b i l ­ ity. T h e A fric a n s , o n t h e o t h e r h a n d , se e m t o h a v e e m p h a s iz e d s u c h c o n n e c t i o n s . In c h a p t e r 10 we will e x p l o r e o n e s o u r c e of t h i s d if f e r e n c e : t h e A f r i c a n c o n c e p t o f a “t r i c k s t e r " g o d , o n e w h o is b o t h d e t e r m i n i s t i c a n d .u.nor_e d i c t a b l e • O n a v id eo reco rd in g 1 m a d e o f th e B a m a n a d iv in a tio n , I n o tic e d th a t th e p r a c t i t i o n e r s h a d u se d a s h o r t c u t m e t h o d in s o m e d e m o n s t r a t i o n s ( t h i s m a y h a v e b e e n a p a r t in g gift, as th e v id e o was s h o t o n my last day). A s th e y first ta u g h t m e, w h e n th e y c o u n t off t h e p airs o f r a n d o m d a s h e s , t h e y li n k t h e m by d r a w i n g sh o r t cu rves. T h e s h o r t c u t methodI t h e n links t h q s e c u rv e s witly.larger cu rv es , a n d ^th o s e b e lo w w i t h e v e n la r g e r c u r v e s . T h i s u p s i d e - d o w n C a n t o r s e t s h o w s t h a t th e y are n o t s i m p l y a p p l y i n g m o d 2 a g a i n a n d a g a i n in a m i n d l e s s f a s h i o n . T h e s e lf-s im ila r p h y s i c a l s t r u c t u r e o f t h e s h o r t c u t m e t h o d v iv i d ly il lu s t r a t e s a r e c u r ­ sive p r o c e s s ^ a n d as a n g m .tra d itio n a l i n v e n t i q n ( t h e r e is n o r e c o r d o f its use e l s e ­ w h e re ) it sh o w s a c t i v e m a t h e m a t i c a l p r a c t ic e . O t h e r A f r i c a n d i v i n a t i o n p ra c tic e s A fr ic a n fra c ta l m athem atics IOO l i i d in f i i*i ;)c c o lc m '{ iH h n n i YDi f i c f m .p iitn r o ucT .m m po oo xjo oo . /• ' J^ ? Y ) b i t R b m a - c o i t r a t . Of) Hjb nimcrficiipetfifi? 0 .9n hMrQ'Jr Afrufantrtbftfc ° ° A nQ° c£«wii gunXrnprc urtra n 'J ^ r ^ ^ r n b c o v t t m - (ttJtnbicurnimrptti® nr. u Q Ctrvwtca opm tdurbu oo QJ> <WGMubiaa*afcufk m ru a h ta tu o 0 cuuaiw rncabtrfaK u 0 Oft- V - n q {c>orcnramficrr filn . n o m fltiu n rrp tiitw lurt 0im 0 n An o ficwr r a r nHumbert* y w i n r v bu tf u 00 or V&Vw n tittu ffta ftiu . 00 oo ro am uhcgMctfum n u r r r c B iim M 00 ? y to « b w w A tttfrte u c o ty S ir tf- lu r te m a f tf v d imuftfl mnutesriBU f tu V r w u : ' S FIGURE q utram fK tum 0 5? h i r m i f & ftraocrafi 00 00 o u r uroU m m nftnis 0 0 ftp.tT#jBj,ia?j{*ta,._ ^ Oa -O 00 0 0 w jj (V • w tcw iito iflttitin icr* * cutiufottm ttftnt O j*» < V o .................... 7.9 Q eom ancy African divination was taken up under the name “geomancy” by European mystics. This chart drawn for King Richard u in 1391. ( F r o m S k in n e r 1 9 8 0 .) c a n b e lin k e d to r e c u r s io n as w ell; for e x a m p l e D e v i s c h ( 1 9 9 1 ) d e s c r ib e s t h e Yaka d i v i n e r s ’ “s e l f - g e n e r a t i v e ” i n i t i a t i o n a n d u t e r i n e s y m b o l i s m . Before lea v in g d i v i n a t i o n , t h e r e is o n e m o r e i m p o r t a n t c o n n e c t i o n t o m a t h e ­ m a t i c a l h is to ry . W h i l e R a y m o n d L u ll, l i k e o t h e r E u r o p e a n a l c h e m i s t s , c r e a t e d w h e e l s w i t h s i x t e e n d i v i n a t i o n fig ures, h i s p r i m a r y i n t e r e s t w a s in t h e c o m b i ­ n a t o r i a l p o s s ib il it ie s o f f e r e d by b a s e - 2 d i v i s i o n s . L u l l ’s w o r k w as c l o s e ly e x a m ­ i n e d by G e r m a n m a t h e m a t i c i a n G o t t f r i e d L e i b n i z , w h o s e D iss erta tio d e arte c o m b in a to r ia , p u b l i s h e d in 1 6 6 6 w h e n h e w a s t w e n t y , a c k n o w l e d g e s L u l l ’s w o r k as a p c e c u rs o r. F u r t h e r e x p l o r a t i o n le d L e ib n i z t o i n t r o d u c e a b a s e - 2 c o u n t i n g s y s te m , c r e a t i n g w h a t we n o w call, t h e b i n a r y c o d e . W h i l e t h e r e w e r e m a n y o t h e r N u m e r ic sy ste m s i n f l u e n c e s i n t h e li v e s o f L u ll a n d L e i b n i z , it is n o t f a r - f e t c h e d t o see a h i s t o r ­ ic a l p a t h fo r b a s e - 2 c a l c u l a t i o n t h a t b e g i n s w i t h A f r i c a n d i v i n a t i o n , r u n s th r o u g h t h e g e o m a n c y o f E u r o p e a n a lc h e m is ts , a n d is finally tr a n s la t e d i n t o binary c a l c u l a t i o n , w h e r e i t is -n o w a p p l i e d inl,e v e r y d i g i t a l c i r c u i t f r o m a l a r m c l o c k s to su p erco m p u te rs. I n a 1 9 9 5 i n t e r v i e w i n W ir e d m a g a z i n e , t e c h n o - p o p m u s i c i a n B r i a n E n o c la i m e d t h a t t h e p r o b l e m w i t h c o m p u t e r s is t h a t “th e y d o n ’t h a v e e n o u g h A f r i c a n in t h e m . ” E n o w as, n o d o u b t , t r y i n g t o b e c o m p l i m e n t a r y , s a y i n g t h a t t h e r e is s o m e i n t u i t i v e q u a l i t y t h a t is a v a l u a b l e a t t r i b u t e o f A f r i c a n c u lt u r e . B u t in d o i n g so h e o b s c u r e d t h e c u l t u r a l o r i g i n s o f d i g i t a l c o m p u t i n g a n d d id a n i n j u s t i c e to th e very c o n c e p t h e was try in g to c o n v e y . D iscrete se lf-o rg a n iz a tio n in O w a r i F ig u r e 7 .1 0 a s h o w s a b o a r d g a m e t h a t is p l a y e d t h r o u g h o u t A f r i c a in m a n y d if ­ f e r e n t v e r s io n s v a r i o u s ly t e r m e d a y o , b a o , g iu th i, lela, m a n ca la , omi v eso , ow a ri, lei, a n d songo ( a m o n g m a n y o t h e r n a m e s ) . B o a r d s t h a t w e re c u t i n t o s t o n e s , s o m e o f e x t r e m e a n ti q u it y , h a v e b e e n f o u n d f r o m Z i m b a b w e t o E t h i o p i a (see Z aslavsky 1 97 3, fig. 1 1 -6 ) . T h e g a m e is p l a y e d b y s c o o p i n g p e b b l e o r s e e d c o u n t e r s fro m o n e cup, a n d placin g o n e o f th o s e c o u n te r s in to e a c h cup, s ta rtin g jv itJ x th e cup t o t h e r i g h t o f t h e s c o o p . T h e g o a l is t o H a v e t h e last c o u n t e r l a n d in a c u p t h a t h a s o n l y o n e o r t w o c o u n t e r s a l r e a d y in it, w h i c h a ll o w s t h e p l a y e r t o c a p t u r e t h e s e c o u n t e r s . I n t h e G h a n a i a n g a m e o f o w a r i , p la y e r s a r e k n o w n fo r u ti li z in g a se r ie s o f m o v e s t h e y c a ll a “ m a r c h i n g g r o u p . ” T h e y n o t e t h a t if t h e n u m b e r o f c o u n t e r s in a se r ie s o f c u p s e a c h d e c r e a s e s b y o n e (e .g ., 4 - 3 - 2 - ! ) , t h e e n t i r e p a t t e r n c a n - b e r e p l i c a t e d w k h a rig h c -s h ift by s c o o p i n g fr o m t h e largest c u p , a n d t h a t if th e p a t t e r n is left u n i n t e r r u p t e d it c a n p ro p a g a te in th is way as far as n e e d e d for a w i n n i n g m o v e (fig. 7 . 1 0 b ) . A s s i m p l e as it s e e m s , t h i s c o n c e p t o f a selfr e p l i c a t i n g p a t t e r n is a t t h e h e a r t o f s o m e s o p h i s t i c a t e d m a t h e m a t i c a l c o n c e p t s . J o h n v o n N e u m a n n , w h o p l a y e d a p i v o t a l r o l e in t h e d e v e l o p m e n t o f t h e m o d e r n d i g i t a l c o m p u t e r , w a s a ls o a f o u n d e r o f t h e m a t h e m a t i c a l t h e o r y o f s e l f - o r g a n i z i n g s y s te m s . I n i t i a l l y , v o n N e u m a n n ’s t h e o r y was t o b e b a s e d o n se lf-rep ro d u c in g ph y sical ro b o ts. W h y w ork on a th e o ry o f s e lf-re p ro d u c in g m a c h i n e s ? 1 b e l i e v e t h e a n s w e r c a n b e f o u n d in v o n N e u m a n n ’s s o c i a l o u t ­ look. H e i m s ’s ( 1 9 8 4 ) b i o g r a p h y e m p h a s iz e s h o w t h e d is o r d e r o f v o n N e u m a n n ’s p r e c a r io u s y o u t h as a H u n g a r i a n Je w w a s r e f l e c t e d in h is a d u l t e ffo rts t o i m p o s e a s t r i c t m a t h e m a t i c a l o r d e r o n v a r i o u s a s p e c t s o f t h e w o rld . I n v o n N e u m a n n ’s a p p l i c a t i o n o f g a m e t h e o r y to s o c i a l s c i e n c e , fo r e x a m p l e , H e i m s w r it e s t h a t his “H o b b e s i a n ” a s s u m p t i o n s w e r e “c o n d i t i o n e d b y t h e h a r s h p o l i t i c a l r e a l i t i e s o f 102 A fr ic a n fra c ta l mathematics FIGURE 7 - 1 0 O w ari (;i) T h e owari board has i 2 cups, plus one cup on each side for captured counters. This board is hinged in the center, with a beautifully carved cover (see fig. 7.14). (b) Scoop from the first cup, and plant one counter in each succeeding cup. (c) The Marching Group is replicated with a right-shift. Repeated application will allow it to propagate around the board. .his H u n g a r i a n e x i s t e n c e . ” H is e n t h u s i a s m for t h e use o f n u c l e a r w e a p o n s a g a in s t t h e S o v i e t U n i o n is a ls o a t t r i b u t e d t o th i s e x p e r i e n c e . D u r i n g t h e H i x o n S y m p o s i u m ( v o n N e u m a n n 195 1) h e w a s a s k e d if c o m ­ p u t i n g m a c h i n e s c o u l d b e b u i l t s u c h t h a t th e y c o u l d r e p a i r t h e m s e l v e s if “d a rn a g e d in a i r r a i d s ,” a n d h e r e p l i e d t h a t “ t h e r e is n o d o u b t t h a t o n e c a n d e s i g n m a c h i n e s w h i c h , u n d e r s u i t a b l e c i r c u m s t a n c e s , w ill r e p a i r t h e m s e l v e s . " H is w o r k o n n u c l e a r r a d i a t i o n t o l e r a n c e for t h e A t o m i c E n e r g y C o m m i s s i o n in 1 9 5 4 - 1 9 5 5 i n c l u d e d b i o l o g i c a l e ff e c ts as w e ll as m a c h i n e o p e r a t i o n . T u t t i n g t h e s e fa c ts to g e t h e r , 1 c a n n o t e s c a p e t h e c r e e p y c o n c l u s i o n t h a t v o n N e u m a n n ’s i n t e r e s t in s e l f - r e p r o d u c i n g a u t o m a t a o r i g i n a t e d i n f a n t a s i e s a b o u t h a v i n g a m o r e p e r f e c t m e c h a n i c a l p r o g e n y s u r v i v e t h e n u c l e a r p u r g i n g o f o r g a n i c life o n th is p lan et. M o d e l s for p h y s i c a l r o b o t s t u r n e d o u t to b e t o o c o m p l e x , a n d a t t h e s u g ­ g e s t i o n o f h is c o l l e a g u e S t a n i s l a w U l a m , v o n N e u m a n n s e t t l e d for a g r a p h i c a b ­ s t r a c ti o n : “c e llu la r a u t o m a t a .” as t h e y c a m e to b e c a lle d . In th is m o d e l (fig. 7.11 a), e a c h s q u a r e in a grid is s a i d t o b e e i t h e r a l i v e o r d e a d ( t h a t is, in o n e o f t w o p o s ­ sible s t a te s ) . T h e i t e r a t i v e rules for c h a n g i n g t h e s t a t e o f a n y o n e s q u a r e are based ■ In the cellular automaton called “the game of life,’’ each cell in the grid is in one of two states: live or dead. Here we see a live ceil in the center, surrounded by dead h cells in its eight nearest neighbors. The state of each cell in the next iteration is B [ determined by a set of rules. In "classic” life (the rules first proposed by John Horton Conway), a dead cell becomes a live celf'Tf it has three live nearest neighbors, and a cell dies unless it has two or three live neighbors. BBBBBBBB B f l f lf l B B B f l ■■■■■■■■ ■■■ ■ ■■ ■ ■■■ ■■■ ■■ BBBBB BBBBBBBB BBBBBBBB — >■ ■■■■■■■■ ■■■■■■■■ BBBBBBBB. ■■■■■■■■ ■■■■■■■■ BBBBBBBB ■■■ um n ■■ ■ ■■■ BB ■ ■ ■ — > ■ ■ B B B B B — >■ ■ ■ ■■■ ■■■■ ■■ BBB ■■■■■■■■ B B B B B B B fl BBBBBBBB BBBBBBBB B B f lB f l f l B f l. BBBBBBBB BBBBBBBB ■■■ B ■ ■ — =► ■■■ BBB BBBB BBS ■ B B B B B B B fl _ B f lflflfll BBBBBI BBBBBI ■■■ i ■■ ■■ ■ ■ BB BBB I BBBBBI This initial condition produces a fixed pattern after four iterations. The patterns occurring before it settles down to stability are called the “transient.” BB■ BBBBBBBBB ■BB BBBBB ■■ ■ BBBB ■BBBBBBBB ■ B B B BB BBBBB B ■ ■ B B B BB ■■B BBBB ■■■■■■■■■ BBBBBBBB BBBBBBBB BBBB BBB BBBB BBB BBBB BBB BBBBBBBB BBBBBBBB BBBBBBBBB BBB BBBBB BBB BBBBB B BBBBB fl BBBBBB a aa This stable pattern flips back and forch between these two states. This is called a “period-2” pattern. ■ BBB BBB BBB BBB BBBBBBBBB BBBBBBBBB BBBB BBB B B B BBBB. fl B B B B B B BB BBB ' B —>- B B B B B B B B B ' BB BB B BBB BBBBBBBBB BBBBBBBBB BBB BBB BBB BBB B BB fl BBBBBB B ^ BBBBB BBB BBBBB BBB BBBBBBBBB A period-4.pattern. Periods of any length can be produced, as we saw in the previous examples of pseudorandom number generation. Deterministic chaos, in which the pattern never repeats (i.e., a pcrioJ-infinity pattern, like the Morse sequence), is also possible. Ite ratio n 49 Iteration 133 I t e r a t io n 182. A constam-growth pattern, shown in high resolution, looks similar to the cross-section of an internal organ. The rules: a dead cell becomes a live cell if it has three live nearest neighbors, and a cell dies only if it has seven or eight live neighbors. FIGURE 7. 1 I C ellu la r a u to m a ta A fr ic a n fracial mathematics 104 o n t h e e i g h t n e a r e s t n e i g h b o r s (e.g ., if t h r e e o r m o r e n e a r e s t - n e i g h b o r s a r e full, t h e c e l l b e c o m e s full in t h e n e x t i t e r a t i o n ) . A t first, r e s e a r c h e r s e a r n e d o u t o n t h e s e c e l l u l a r a u t o m a t a e x p e r i m e n t s o n c h e c k e r e d t a b l e c l o t h s w i t h p o k e r c h ip s a n d d o z e n s o f h u m a n h e lp e r s (M ay er-K ress, pers. c o m m . ) , b u t by 1 9 7 0 it h a d b e e n d e v e l o p e d i n t o a sim p le c o m p u t e r p r o g r a m ( C o n w a y ’s “ g a m e o f life” ), w h i c h was d e s c r i b e d by; M a r t i n G a r d n e r i n h is f a m o u s “M a t i-------c a l G a m-e s ” c o l u m n in v- a t h e m--/ S cien tific A m erica n . T h e “g a m e o f life" story was a n in s t a n t h i t , a n d c o m p u t e r screens a ll o v e r t h e w o r ld b e g a n to - p u l s a t e w i t h a b izarre a r r a y o f p a t t e r n s (fig. 7.1 i b ) . A s t h e s e a c ti v it ie s d re w in c r e a s i n g p ro f e ssio n a l a t t e n t i o n , a w id e r a n g e o f m a t h e ­ m a t i c a l l y o r i e n t e d s c i e n t i s t s b e g a n to re alize t h a t t h e s p o n t a n e o u s e m e r g e n c e o f s e l f - s u s t a i n i n g p a t t e r n s c r e a t e d in. c e r t a i n , c ejlul.ar a . m o m a t a - ^ e r e e x c e l l e n t m o d e l s for t h e k i n d s o f s e lf-o r g a n i zin g p a t t e r n s t h a t . l m d b e e n _sj?„elu.siyeJ.n s t u d ­ ies o f fluid flow a n d b io l o g i c a l g r o w t h . S i n c e s c a l i n g s t r u c t u r e s a re o n e o f t h e h a l l m a r k s o f b o t h flu id t u r b u l e n c e a n d b io l o g i c a l g r o w t h , t h e o c c u r r e n c e o f f r a c ta l p a t t e r n s i n c e l l u l a r a u t o m a t a a ttra c te d a g reat deal o f in terest. But a m o re sim p le sc a lin g stru c tu re , th e log­ a r i t h m i c s p ira l (fig. 7 -1 2 ), h a s g a r n e r e d m u c h o f t h e a t t e n t i o n . E v e n b a c k i n t h e i 9 ^ o s m a t h e m a t i c i a n A l a n T ux lng , w h o s e t h e o r y - o f c o m p u t a t i o n p r o v i d e d v o n N e u m a n n w i t h ' t h e i n s p i r a t i o n f o r t h e first d i g i t a l c o m p u t e r , b e g a n h i s r e s e a r c h o n “ b io l o g i .c a l x n o r p h o g e n e s i s ” w i t h a n a n a ly s is o f l o g a r i t h m i c s p ira ls in g r o w t h p a t t e r n s . ( M a r k y s ( 1 9 9 1 ) n o t e s t h a t t h e a p p l i c a O o n a j e a s fox,cN^ m o d e l s o f s p ira l w a v e s i n c l u d e n e r v e a x o n s , t h e r e t i n a , t h e s u r f a c e o f fe r tiliz e d eggs, t h e c e r e b r a l c o r t e x , h e a r t tissu e, a n d a g g r e g a t i n g s l i m e m o l d s . I n t h e t e x t for c a l a b , t h e first c o m p r e h e n s i v e s o f t w a r e fo r e x p e r i m e n t i n g w i t h c e l l u l a r a u t o m a t a , m a t h e m a t i c i a n R u d y R u c k e r ( 1 9 8 9 , 1 6 8 ) refers t o s y s te m s t h a t p r o ­ d u c e p a ir e d log spirals as “Z h a b o t i n s k y C A s , " a f t e r t h e c h e m i s t w h o first o b s e r v e d s u c h s e lf-o r g a n iz i n g p a t t e r n s in a r t ifi c ia l m e d i a : “W h e n y o u l o o k a t Z h a b o t i n ­ sky C A s , y o u a r e s e e i n g v e r y s t r i k i n g t h r e e d i m e n s i o n a l s t r u c t u r e s ; t h i n g s like p a i r e d v o r t e x s h e e t s in t h e su rfa c e o f a r i v e r b e l o w a d a m , t h e s c r o ll p a i r s t r e t c h ­ i n g all t h e w a y d o w n t o t h e r i v e r b o t t o m . . . . In t h r e e d i m e n s i o n s , a Z h a b o t i n ­ sky r e a c t i o n w o u ld b e like tw o p a ir e d n a u t i l u s s h ells, fa c in g e a c h o t h e r w i t h t h e i r lips b l e n d i n g . T h e s u cce s siv e layers o f s u c h a g r o w i n g p a t t e r n w o u ld b u il d u p v ery li k e a f e t u s ! ” / F ig u r e 7 .1 3 s h o w s h o w t h e o w a r i m a r c h i n g - g r o u p s y s te m c a n b e u s e d as a I o n e -d im e n s io n a l cellu lar a u to m a to n to d e m o n s tr a te m a n y o f th e d y n a m ic phe- ( n o m e n a p r o d u c e d o n t w o - d i m e n s i o n a l s y s te m s .- ’ E a r l i e r w e n o t e d t h a t t h e A k a n a n d o t h e r G h a n a i a n s o c i e t i e s h a d a r e m a r k a b l e p r e c o l o n i a l u s e o f lo g a ­ r i t h m i c sp ira ls in i c o n i c r e p r e s e n t a t i o n s fo r l i v i n g s y s te m s . T h e G h a n a i a n f o u r ­ fo ld s p i r a l (fig. 6 . 4 a ) a n d t h e f o u r - a r m e d c o m p u t e r g r a p h i c in fig u re 7 . 1 2 b a r e (a) Paired spirals emerge from a three-state cellular automation. Black cells are live, white cells are dead, aiu! gray cells are in a refractory or f‘ghosc" state. The rules: Any dead nearest neighbors of a live cell become live in the next iteration, and any live cell goes into the ghost state in the next iteration. The refractory layer acts as a memory, providing the directed growth (i.e., the breaking of symmetry) needed to create a spiral pattern. (b) This four-armed logarithmic spiral from Markus (1991) was produced by a six-state cellular automaton in which a sequence of ghost states corresponds to increasingly dark shades of gray. The system makes use of a very highresolution grid as well as some random noise to prevent the tendency for the patterns to follow the grid shape (as in the square contours of the spiral above). Compare with che Ghanaian fourfold spiral in figure 6.4a. • Bivalve shell. (From H a e c k e l 1904.) Mushroom cut in half. North African sheep. (From Cook 1914.) (c) Paired logarithmic spirals often occur in natural growth forms. (J) Recursive line replacement, as we saw for other fractal generations, can also produce such paired spirals. FIGURE 7 . 1 2 S p ira ls in c e llu la r a u to m a ta W e c a n v i e w t h e o w a r i b o a r d as a o n e - d i m e n s i o n a l c e l l u l a r a u t o m a t o n . O n e d i m e n s i o n is n o t n e c e s s a r i l y a d i s a d v a n t a g e ; in f a c t , m o s t o f t h e p r o f e s s i o n a l m a t h e m a t ic s o n c e llu la r a u t o m a t a (see W o lfr a m 19 8 4 , 1 9 8 6 ) h a v e b e e n d o n e on o n e - d i m e n s i o n a l v e r s i o n s , b e c a u s e it is e a s i e r t o k e e p t r a c k o f t h e r e s u l t s . T h f e y c a n sh ow a ll th e d y n a m ic s o f tw o dim en sion s. T h e p a ttern s n o t e d by tra d itio n a l o w a r i players offer a g re a t d ea l o f in s ig h t in to s e l f - o r g a n i z i n g b e h a v i o r . T h e i r o b s e r v a t i o n o f a c la s s o f s e l f - p r o p a g a t i n g p a t t e r n s , t h e ‘' m a r c h i n g g r o u p , " p r o v i d e s a n e x c e l l e n t s t a r t i n g p o i n t . 3 4 2 1 —> 5 3 2 —> 4 3 1 1 1 —> 4 2 2 2 —> 3 3 3 1 ~ » 4 4 2 —> 5 3 1 1 —> 4 2 2 1 1 —> 3 3 2 2 —> 4 3 3 —> 4 4 1 1 ~ » 4 5 5 2 —> 3 3 2 1 1 —>4321 T h e m a r c h i n g g r o u p is a n e x a m p l e o f a c o n s t a n t p a t t e r n . H e r e w e s e e c o u n t e r s in th e in itial se q u e n c e 3421 c o n v e rg e o n th e ir in a rc h in g f o rm a tio n sim ply by re p e a tin g t h e “s c o o p f r o m t h e left c u p " r u l e t h r o u g h 13 i t e r a t i o n s . J u s t as w e s a w in t w o - d i m e n s i o n a l c e l l u l a r a u t o m a t a , t r a n s i e n t s o f m a n y d i f f e r e n t l e n g t h s c a n b e p r o d u c e d . T r a n s i e n t s o f m a x i m u m l e n g t h a r e u s e d as a n e n d g a m e t a c t i c by i n d i g e n o u s G h a n a i a n p l a y e r s , w h o c a l l it " s l o w m o t i o n ”— a c c u m u l a t i n g p i e c e s o n y o u r side to p r e v e n t your o p p o n e n t from c a p tu r in g t h e m .In n o n l i n e a r d y n a m ic s , th e c o n s t a n t p a t t e r n is c a l l e d a “ p o i n t a t t r a c t o r , ” a n d th e . t r a n s i e n t s w o u l d b e s a i d t o fie in th e "b a sin of a t t r a c t i o n .” T h e m a r c h i n g g ro u p rule c a n also p r o d u c e p e r io d ic b e h a v i o r (a " l im it c y c le " o r “ p e r i o d i c a t t r a c t o r ” i n n o n l i n e a r d y n a m i c s t e r m s ) . H e r e is a p e r i o d - 3 s y s t e m u s i n g only four counters: 2 U - » 22-* 31-» 2 1 J W h i c h lead s to m a r c h i n g groups, a n d w h i c h o n e s iead to p e r i o d i c cycles? T o ta l n u m b e r of counters T h e n u m b e r s w h i c h le a d t o m a r c h i n g g r o u p s — 1, 3, 6 , 10 , 1 5 • - • — s h o u l d l o o k f a m i l i a r t o re a de rs : it ’s t h e t r i a n g u l a r n u m b e r s w e s a w in t a r u m b e r a ! T h e p e r i o d o f c y c l e s in b e t w e e n e a c h m a r c h i n g g r o u p is g i v e n b y o n e p lu s t h e i t e r a t i o n l e v e l o f t h e prev iou s trian gular n u m b e r rea ch ed . ( N o t e : S o m e s e q u e n c e s w ill b e t r u n c a t e d fo r 1 3 , 1 4, a n d 1 5 s i n c e t h e r e a re m o r e c o u n t e r s th a n h oles.) FIGURE 1 2 3 4 5. 6 B ehavior (afrer tran sien ts) ....................M a r c h i n g ................ P e r io d 2 .............. . . M a r c h i n g ................ P e r i o d 3 . P e r io d 3 ....................M a r c h i n g 7.........................P e r io d 4 8...... ................ P e r io d 4 9 ...................... P e r io d 4 10 ....................M a r c h i n g 11 P e r io d 5 12 ................. P e r i o d 5 13 ................ P e r i o d 5 J4 ................ P e r i o d 5 15 ....................M a r c h i n g 7 .1 3 O w ari as o n e-d im en sio n a l ce llu la r a u to m a to n N um eric systems 107 q u i r e d i s t a n t in te r m s o f t h e t e c h n o l o g i e s t h a t p r o d u c e d t h e m , b u t t h e r e m a y w ell b e s o m e s u b t l e c o n n e c t i o n s b e t w e e n t h e tw o . S i n c e c e l l u l a r a u t o m a t a m o d e l t h e e m e r g e n c e o f s u c h p a t t e r n s in m o d e r n . s c i e n t i f i c s t u d i e s o f li v i n g sys­ tem s,, a n d c e r t a i n G h a n a i a n lo g s p i r a h ic o .n s w e r e a ls o i n t e n d e d as g e n e r a l i z e d m o d e l s fo r o r g a n i c g r o w t h , it is n o t u n r e a s o n a b l e t o c o n s i d e r t h e p o s s ib ility t h a t t h e s e l f - o r g a n iz i n g d y n a m i c s o b s e r v a b l e in o w a r i .w ere a ls o l i n k e d t o c o n c e p t s o f b i o l o g i c a l m o r p h o g e n e s i s in t r a d i t i o n a l G h a n a i a n k n o w l e d g e sy s tem s . R a t t r a y ’s c lassic v o l u m e o n t h e A s a n t e c u l t u r e o f G h a n a i n c l u d e s a c h a p ­ te r o n ow a ri, b u t u n f o r t u n a t e l y it o n ly c o v e r s t h e rules a n d stra te g ie s o f t h e gam e. R e c e n t l y Kofi A g u d o a w u ( 1 9 9 1 ) o f G h a n a h a s w r i t t e n a b o o k l e t o n o w a r i “d e d ­ ic a te d to A f r i c a n s w h o a r e e n g a g e d in t h e f o r m id a b l e task o f r e c l a im in g t h e i r h e r ­ it a g e ,” a n d h e d o e s n o t e its a s s o c i a t i o n w i t h r e p r o d u c t i o n : w ari in t h e G h a n a i a n n lan g u a g e Tw i m e a n s “h e / s h e m a r r ie s .” H e r s k o v i t s ( 1 9 3 0 ) , n o t i n g t h a t t h e " a w a r i ” 1 F I GURE 7 . I 4 L ogarith.7nie cttrv es a n d oxvari T he cover of die hinged owari board we saw in figure 7.10 shows concentric circles emanating from the Adinkra icon for the power of god, “Gye Nyame.” A similar icon; without the logarithmic curves, is attributed to a closed fist as a symbol of power. The Gye Nyame symbol thus appears to be a pair of logarithmic curves held in a fist: God Holding the power of life. A fr ic a n fra c ta l matfiemntics io 8 g a m e p la y e d by t h e d e s c e n d a n t s o f A f r i c a n slave s in t h e N e w W o r l d h a d r e t a i n e d som e o f th e p re co lo n ial cu ltu ral a sso ciatio n s from A frica, re p o rts t h a t aw ari h ad a d i s t i n c t “ s a c r e d c h a r a c t e r 1' t o it, p a r t i c u l a r l y i n v o l v i n g t h e c a r v i n g o f t h e b o a r d . O w a r i b o a r d s w i t h c a r v i n g s o f l o g a r i t h m i c s p ira ls (fig.^7.14) c a n b e c o m •of m o n l y f o u n d in G h a n a to d a y , s u g g e s tin g t h a t W e s t e r n s c i e n t i s t s m a y n o t b e t h e o n ly o n e s w h o d e v e l o p e d a n a s s o c i a t i o n b e t w e e n d i s c r e t e s e lf-o r g a n iz in g p a t t e r n s a n d b io l o g ic a l r e p r o d u c t i o n . I t is a b i t v i n d i c t i v e , b u t I c a n ’t h e l p b u t e n j o y t h e th o u g h t o f v o n N e u m a n n , ap o stle o f a m e c h a n is tic N e w W o rld O r d e r th a t w o u ld w ip e o u t t h e i r r a t i o n a l c a c o p h o n y o f l i v in g s y s te m s , s p i n n i n g i n h i s g ra v e e v e ry t i m e we w a t c h a c e l l u l a r a u t o m a t o n — w h e t h e r in p i x e l s o r o w a r i c u p s — b r i n g f o r t h c h a o s in t h e g a m e s o f life. C o n c lu sio n -B oth t a r u m b e t a a n d o w a ri's m a r c J ) i n j : g ro up _d yn a,m ics a r e g o v e r n e d by t h e t r i ­ a n g u l a r n u m bers. T h e r e is n o t h i n g s p e c i a l a b p u t t h e t r i a n g u l a r n u m b e r s e r ie s — s i m i l a r n o nHnjear^gtowth^prQ pe.r.t.ie^_can b e f o u n d i n t h e n u m b e r s t h a t fo r m su c c e s s iv e ly la r g e r r ^ a j y g l e s ^ p e j y t a g o n s ^ o r o t h e j shap.es. N o r is t h e r e a n y t h i n g s p e c i a l a b o u t t h e p o w e r s o f t w o w e f o u n d in d i v i n a t i o n — s i m i l a r a p e r i o d i c p r o p ­ e r t ie s c a n b e p r o d u c e d b y a p p l i c a t i o n s o f m o d 3 , m o d 4 , e t c . W h a t is s p e c i a l is t h e u n d e r l y i n g c o n c e p t o f r e c u r s i o n — t h e w av s i n w h i c h a k i n d o f m a t h e M a t i c a l . f e e d b a c k l o o p c a n g e n e r a t e n e w s t r u c t u r e s in s p a c e a n d ^ n e w d y n a m i c s i n tim e . In t h e n e x t c h a p te r , w e w ill see h o w t h i s u n d e r l y in g p ro c e s s is f o u n d in b o t h p r a c ­ tical a p p lic a tio n s a n d a b stra c t sym bolics o f A fric a n cu ltu res. CHAPTER •Recursion- _ 8 _ R e c u r s i o n j s t h e m o t o r o f f r a c t a l ge o m e try; k j s j h e r e t h a t j h e , 1b as.ic,..transfor' m a t i o n s — w h e t h e r n u m e r i c o r s p a t i a l — a re s p u n i n t o w h o l e c l o t h , a n d t h e p a tt e rn s t h a t e m e r g e o f t e n r e l l j h e s t o r y o f t h e i r w h i r l i n g b i r t h . W e w ill b e g i n by d e f i n in g t h r e e ty p es o f re c u r sio n * 1 W h i l e it is p o ssib le to c a te g o riz e t h e e x a m p l e s in t h i s c h a p t e r s o le ly o n t h e b asis o f t h e s e t h r e e ty p e s , it is m o r e i l l u m i n a t i n g to c o m b i n e t h e a n a ly s is w i t h c u lt u r a l , c a t e g o r i e s . It is in e x a m i n i n g t h e i n t e r ­ a c t i o n b e t w e e n t h e t w o t h a t t h e use o f f r a c ta l g e o m e t r y as a k n o w l e d g e s y s te m , a n d n o t ju s t u n c o n s c i o u s s o c i a l d y n a m i c s , b e c o m e s e v i d e n t . T h e c u l t u r a l c a t ­ e g o rie s b e g i n w i t h t h e c o n c r e t e i n s t a n c e s o f re c u r s iv e c o n s t r u c t i o n t e c h n i q u e s a n d g r a d u a l l y m o v e t o w a r d t h e a b s t r a c t i o n s o f r e c u r s i o n , . a s . s y m b o l i z e d in A f r i c a n ic o n o g r a p h y . T hree types o f r e c u r s io n T h e le a s t p o w e rfu l o f t h e t h r e e , j s c a s c a d e r e c u r s i o n , in w h i c h t h e r e is a p r e - 1 d e t e r m i n e d s e q u e n c e o f s i m i l a r p ro c e s s e s . F o r e x a m p l e , t h e r e is a c h i l d r e n ’s story in w h i c h a m a n buys a C h r i s t m a s t r e e , , b u t d is c o v e r s it is t o o tall for his c e ilin g a n d c u ts o ff t h e t o p . H is do g s fin d t h e d i s c a r d e d t o p , a n d p u t it in t h e i r d o g h o u s e , b u t t h e y t o o d i s c o v e r it is t o o t a l l , a n d c u t o ff t h e to p . F i n a l l y t h e A fr ic a n fractal m athem atics n o m i c e d r a g t h i s t i n y t o p i n t o t h e i r h o l e , w h e r e it fits j u s t f i n e — t h e r e c u r s i o n “ b o t t o m s o u t . ” N o t e t h a t t h e s e w e r e all i n d e p e n d e n t t r a n s f o r m a t i o n s ; it is o n ly by c o in c i d e n c e , so to s p e a k , t h a t th e y h a p p e n e d t o b e t h e s a m e . F ig u r e 8. i a s h ow s t h e n u m e r i c v e r s i o n o f c a s c a d e r e c u r s i o n , in w h i c h w e d i v t d e a n u m b e r by tw o in e a c h p a r t o f t h e s e q u e n c e . TTiis is n o t a v ery p o w e r f u l ty p e o f r e c u r s i o n , fo,r t w o T e n o n s . ' F i r s t , ' i t r e q u i r e s t h a t w e k n o w h o w m a n y t r a n s f o r m a t i o n s we w a n t a h e a d o f t i m e — a n d t h a t is n o t a lw a y s p o ss ib le . I f t h e m o u s e w a s in c h a r g e , h e w p.,uj,dhave sa id “j u s t k e e p d i v i d i n g u n t i l i t ’s s m a l l e n o u g h t o fit in m y . h o le . " . S e c o n d ; w e h a v e to k n o w w h a t t r a n s f o r m a t i o n t o m a k e a h e a d o f t i m e , a n d t h a t is n o t a l w a y s p o s s i b l e , e i t h e r . R e c a l l , fo r e x a m p l e , t h e g e n e r a t i o n o f t h e F i b o n a c c i series w e sa w in c h a p t e r 7 (fig. 8 .1 b ) . A l t h o u g h t h e g e n e r a t i o n is ju s t u s i n g a d d i t i o n , it c a n n o t b e c r e a t e d b y a r e c u r s i v e c a s c a d e , b e c a u s e t h e a m o u n t t o be a d d e d i n e a c h t r a n s f o r m a t i o n c h a n g e s i n r e l a t i o n t o p r e v i o u s re s u lts . G e n e r a t i n g t h e F i b o n a c c i s e r ie s r e q u i r e s a f e e d b a c k l o o p o r, as m a t h e ­ m a t i c i a n s c a ll it, i t e r a t i o n . 0 I n i t e r a t i o n , t h e r e is o n l y o n e t r a n s f o r m a t i o n . p r o c e s s , b u t e a c h t i m e t h e p ro c e s s c r e a t e ^ a n o u t p u t ^ it u s e s t h i s r e s u l t as t h e i n p u t f o r t h e n e x t i t e r a t i o n , as w e ’v e s e e n in g e n e r a t i n g f r a c ta l s . A p a r t i c u l a r l y i m p o r t a n t v a r i e t y o f i t e r a ­ t i o n is “n e s t i n g , ” w h i c h m a k e s u se o f lo o p s w i t h i n lo o p s . H o f s t a d t e r ( 1 9 8 0 , 1 0 3 - 1 2 9 ) n i c e l y i l l u s t r a t e s n e s t i n g w i t h a s t o r y in w h i c h o n e o f t h e c h a r a c t e r s s ta rts to tell a story, a n d w i t h i n chat s t o r y a c h a r a c t e r s t a r ts t o re a d a passag e fro m a b o o k . B u t a t t h a t p o i n t t h e r e c u r s i o n “b o t t o m s o u t " : t h e b o o k p a s s a g e g e ts f i n i s h e d a n d w e s t a r t t o r a s c e n d b a c k u p t h e s t o r i e s . N e s t e d lo o p s a r e v e ry c o m m o n in c o m p u t e r p r o g r a m m i n g , a n d w e c a n i l l u s t r a t e t h i s w i t h a p r o g r a m fo r d r a w i n g t h e a r c h i t e c t u r e j ^ f . M f l k o u l e k . (fig. 8 . 1 c ) , w e ex.arni.ned. in . c h a p t e r . .2.... T h e B a -ila a r c h i t e c t u r e we saw in c h a p t e r 2 c a n a ls o b e s i m u l a t e d t h i s way, using o n e lo o p for th e r i n g s - w it h in - r in g s , a n d a n o t h e r for t h e f r o n t- b a c k s c a lin g g r a d i e n t t h a t m a k e s u p e a c h o f t h o s e rin g s. In c h a p t e r 6 t h e first c o r n - r o w h a i r ­ s t y le (ipciko e le d e ) s h o w e d b r a i d i n g as a n i t e r a t i v e lo o p ; t h e s e c o n d c o r n - r o w e x a m p l e a d d e d a n o t h e r i t e r a t i v e lo o p o f s u c c e s s iv e p e r i m e t e r s o f b r a i d s . 2 U is c o m m o n for c o m p u t e r p r o g r a m s to d o s u c h n e s t i n g s e v e r a l layers d e e p , a n d k e e p ­ i n g t r a c k o f all t h o s e lo o p s w i t h i n l o o p s c a n . b e q u i t e a .c h o re .:T h e t h i r d ty p e o f r e c u r s i o n is “s e l f - r e f e r e n c e . ” W e a r e all f a m i l i a r w i t h t h e w a y t h a t sy m b ols o r i c o n s c a n re fe r t o s o m e t h i n g : t h e s t a r s a n d s t r ip e s flag refers t o A m e r i c a , t h e s k u l l - a n d - c r o s s - b o n e s la b e l re fe rs t o p o i s o n , t h e g r o u p o f l e t ­ ters c - a - t refers to a n a n i m a l . B u t i t ’s a ls o p o s s ib le for a s y m b o l t o re f e r t o itself. K e l l o g g ’s c o r n f l a k e s , fo r e x a m p l e , o n c e c a m e in a b o x t h a t f e a t u r e d a p i c t u r e o f a fa m il y s i t t i n g d o w n to b r e a k f a s t . In t h i s p i c t u r e y o u c o u l d se e t h a t t h e fa m ily h a d a b o x o f K e llo g g ’s c o r n f l a k e s o n t h e i r h re a k f a s t ta b l e , a n d y ou c o u l d see t h a t Recursion ill th is b o x s h o w e d che s a m e p i c t u r e o f t h e family, w i t h t h e s a m e b o x o n t h e i r t a b l e , a n d so o n t o i n f i n i t y ( o r a t le a s t to as s m a l l as t h e K e llo g g c o m p a n y ’s a r t i s a n s c o u ld draw ). S e l f - r e f e r e n c e is b e s t . k n o w n fo r its r q l e j n Jp g ic a ] p a r a d o x . If, fo r e x a m p l e , yo u w ere to a c c u s e s o m e o n e o f lying, it w o u ld b e a n o r d i n a r y statement".”£ u t s u p ­ pose you a c c u s e y ou rself o f lying? T h i s is t h e p a r a d o x o f E p im e n id e s o f C r e t e , w h o d e c l a r e d t h a t “ all C r e t a n s a re liars.” If h e ’s t e l l i n g t h e t r u t h , h e m u s t b e lying, b u t if h e ’s lying, t h e n h e ’s t e l l i n g t h e t r u t h . T h e r o l e o f s e l f - r e f e r e n c e in lo g ic a l input X 2 Nnr e previous 4 2 X 2 X output 2 Nncxt w h ile e -co u n t < 4 do: • d raw en closu re w h ile g-cou n t < 1 2 do: • draw granery • rocate to w a rd c e n te r • s h r in k g r a n e r y size • in crea se g -c o u n t by 1 end o f g - c o u n c ’s l o o p • reset g - c o u n t to 0 • rotate tow ard c e n t e r " • s h r in k e n c lo s u r e size - • in crease e -c o u n t b y 1 end of e - c o u n t ’s loop. FIGURE 8.1 R ecizrsiv e c a sc a d e v e r s u s ite r a tio n (a) A r e c u r s i v e c a s c a d e , i n w h i c h t h e s a m e t r a n s f o r m a t i o n ( d i v i s i o n b y t w o ) h a p p e n s t o b e u s e d m e a c h p a r t o f a s e q u e n c e . T h i s r e q u i r e s k n o w i n g h o w m a n y t i m e s ch e t r a n s f o r m a t i n s h o u l d h ap p e n a h e a d o f t i m e . It a l s o r e q u i r e s t h a t t h e t r a n s f o r m a t i o n is i n d e p e n d e n t o f p r e v i o u s r e s u l t s . {b) T h e F i b o n a c c i s e q u e n c e is p r o d u c e d by a d d i n g t h e p r e v i o u s n u m b e r t o t h e c u r r e n t n u m b e r t o gee t h e n e x t n u m b e r , s t a r t i n g w i t h 1 + 1 = 2 . I n t h e F i b o n a c c i s e q u e n c e w e a d d a d i f f e r e n t a m o u n t ia e a c h i t e r a t i o n — w e c o u l d n o t k n o w h o w m u c h e a c h t r a n s f o r m a t i o n s h o u l d a d d a h e a d o f t i m e , s a a r e c u r s i v e c a s c a d e w o u l d n o t d o t h e j o b . ( c ) I n s o m e c a s e s it is n e c e s s a r y t o p u t a n i t e r a t i v e loop in sid e a n o t h e r i t e r a t i v e l o o p ( " n e s t i n g " ) . H e r e is a n e x a m p l e o f n e s t i n g i n a c o m p u t e r program for d r a w i n g t h e a r c h i t e c t u r e o f M o k o u l e k w e e x a m i n e d i n c h a p t e r 2. It is w r i t t e n i n w h a t p ro g ram m ers c a ll “ p s e u d o c o d e , " a m i x t u r e o f a p r o g r a m m i n g l a n g u a g e a n d o r d i n a r y E n g l i s h . T h e first lo o p d r a w s t h r e e la r g e e n c l o s u r e s , a n d t h e i n n e r l o o p d r a w s 1 2 g r a n e r i e s i n s i d e e a c h e n c l o s u r e . Variable “e - c o u n t ” t r a c k s t h e n u m b e r o f e n c l o s u r e s , a n d g - c o u n t t r a c k s t h e n u m b e r o f g r a n e r i e s . A fr ic a n fra c ta l m athem atics 112 p a r a d o x h a s b e e n i m p o r t a n t for m a t h e m a t i c a l th e o r y , b u t i t h a s a l s o b e e n p u t to prac tical use in c o m p u t e r p ro g ra m m in g . M o s t p r o g r a m m i n g h a s little ro u tin e s called “ p r o c e d u r e s , ” a n d o fte n , a p r o c e d u r e w ill n e e d t o c a ll o t h e r p r o c e d u r e s . I n selfr e f e r e n t i a l p r o g r a m m i n g t h e p r o c e d u r e .calls itself. P r a c t i c a l f r a c t a l s : recu rsio n in c o n s t r u c t i o n t e c h n i q u e s f I n h is d is cu ssio n o f t h e m e t a l - w o r k i n g t e c h n i q u e s o f A f r i c a , D e n i s W i l l i a m s g iv e s \ | a p o e t i c d e s c r i p t i o n o f re c u r s iv e c a s c a d e in t h e e d a n b ra s s s c u l p t u r e s o f t h e I V Y oruba: “T h e im a g e p r o l i f e r a t e s li k e l i g h t s in a b u b b l e : o n e e d a n b e a r s in its lap a n o t h e r , s m a ll e r v e r s io n o f itself, w h i c h b e a rs in t u r n a s m a l l e r in its la p , a n d th is b e a rs a n o t h e r in its lap , e t c . — a s o r t o f s c u l p t u r a l re la y r a c e ” ( 1 9 7 4 , 2 4 5 ) . W h i l e t h e e d a n s c u l p t u r e s a r e u n i q u e t o t h e Y o ru b a , r e c u r s i v e c o n s t r u c t i o n t e c h n i q u e s a re q u i t e c o m m o n in A f r i c a . For e x a m p l e , W i l l i a m s g o e s o n t o n o t e t h a t m u c h A f r i c a n m e t a lw o rk , u n l i k e E u r o p e a n i n v e s t m e n t c a s t in g , uses a “spiral t e c h n i q u e ” t o b u i l d u p s t r u c t u r e s fr o m s in g le s t r a n d s ( w h e t h e r b e f o r e c a s t i n g , as in t h e lost w ax te c h n iq u e , o r a f t e r w a r d s as w ir e ) , r e s u l t in g in “h e l i c a l c o il s f o r m e d fro m s m a l l e r h e l i c a l c o i l s .” A w ig m a d e f r o m m e t a l w ir e s (fig. 8 . 2 a ) s h o w s a s i m i l a r i t e r a t i v e c o n s t r u c t i o n u s i n g c o i l s m a d e o f c o ils . I n c h a p t e r 6 w e s a w s o m e e x a m p l e s o f A f r i c a n h a i r s t y l e s in w h i c h e i t h e r a d a p t a t i o n t o c o n t o u r s o r ab stra c t sp atial tr a n s fo rm a tio n resu lted in a scalin g p a tte r n . T h e fractal braids ; s h o w n in fig u re 8 . 2 b h a v e n o t h i n g to d o w i t h t h e s h a p e o f t h e h e a d ; t h e y a re r a th e r th e result o f su ccessiv e ite ra tio n s t h a t c o m b i n e s tr a n d s o f h a ir in to b ra id s , b r a i d s i n t o b r a i d s o f b r a i d s , a n d so o n . F i g u r e 8 . 2 c s h o w s a n o t h e r w ig, th i s o n e fo r a s c u l p t u r e , t h a t f e a t u r e s b r a i d s o f m a n y sc a le s. T h i s c o l l e c t i o n o f s c u l p t u r e , m e t a l w o r k , a n d h a i r s t y l i n g s o u n d s like., a m o t l e y a s s o r t m e n t , b u t o n c e w e . s t a r t .l o o k i n g f o r r e c u r s i o n w e see a c lo s e r e l a ­ t i o n : a ll e x a m p l e s u s e d a s in g le t r a n s f o r m a t i o n — s t a c k i n g , b r a i d i n g , c o i l i n g — t h a t w a s a p p l i e d s e v e r a l t i m e s . L o o k i n g a t t h e R e la tio n b e t w e e n t h e b a s i c t r a n s f o r m a t i o n a n d its final o u t c o m e c a n h e l p us d is ti n g u is h m n o n g j d i/ f e r e n t types cT r e c u r s i o n . T h e b r a i d i n g p a t t e r n o f fig u re 8 .2 b , fo r e x a m p l e , is b a s e d o n i t e r ­ a t i o n , b e c a u s e t h e w a y e a c h s t a g e is b r a i d e d d e p e n d s o n t h e b r a i d s p r o d u c e d in p r e v i o u s sta g e s; t h e y a r e b r a i d s o f b r a i d s . T h e b r a i d s in fi g u r e 8 . 2 c , o n t h e o t h e r h a n d , a r e o f d i f f e r e n t s c a l e s s i m p l y b e c a u s e e a c h s t a g e u ses d i f f e r e n t a m o u n ts o f sin g le-h air strands— a cascade o f p re d e te rm in e d tran sfo rm atio n s. S i m il a r l y , t h e c o i l s o f c o il s i n d i c a t e i t e r a t i o n , b e c a u s e t h e o u t p u t o f o n e s t a g e b e c o m e s t h e i n p u t for t h e n e x t . R e c u r s i v e c o n s t r u c t i o n .. t e c h n i q u e s a r e a l s o u s e d . f o r t h e d e c o x a t i v e d e s i g n s o f A f r i c a n a r t i s a n s . In o u r d i s c u s s i o n o f t h e f r a c t a l e s t h e t i c in cha p'- Recursion t e r 4, w e e x a m i n e d d e c o r a t i v e p a t t e r n s w h i c h d i d n o t p r o v i d e e v i d e n c e fo r a fo r m a l g e o m e t r i c m e t h o d . T h a t d o e s n ’t m e a n n o f o r m a l m e t h o d c o u l d p ossibly e x is t; i t ’s j u s t t h a t n o n e c o u l d b e r e a d i l y d i s c e r n e d f r o m t h e d e s i g n itse lf, a n d th e a r t is a n s d id n o t r e p o r t a n y t h i n g b e y o n d i n t u i t i o n o r e s t h e t i c ta s te . B u t th e r e a re s o m e d e s i g n s t h a t d o i n d i c a t e a n e x p l i c i t re c u r s iv e t e c h n i q u e fr o m t h e p a t ­ t e r n itself. F ig u r e 8 . 2 e s h o w s a M a u r i t a n i a n t e x t i l e w i t h t w o s u c h s c a l i n g p a t ­ t e r n s . I n t e n t i o n a l a p p l i c a t i o n o f i t e r a t i o n a s a c o n s t r u c t i o n t e c h n i q u e is i n d i c a t e d by t h e w a y t h e X f r a c t a l ’s s e e d s h a p e is s h o w n o n e i t h e r s i d e , a n d by h a v i n g i t e r a t i o n c a r r i e d o u t o n t w o c o m p l e t e l y d i f f e r e n t s e e d s h a p e s in t h e s a m e p i e c e . T h e t r i a n g l e f r a c t a l ( c l o s e t o w h a t m a t h e m a t i c i a n s c a l l t h e “S ie r p i n s k i g a s k e t ” ) is a ls o f o u n d in M a u ' r i t a n i a n - S t o n e w o r k (fig . 8 . 2 f ) . A t h r e e d i m e n s i o n a l v e r s i o n f r o m G h a n a (fig. 8 . 2 h ) m a y h a v e b e e n i n s p i r e d b y t h e s e d e s ig n s . B o t h o f t h e a b o v e a re e x a m p l e s o f a d d i t i v e c o n s t r u c t i o n , as w e s a w i n th e K o c h c u r v e o f c h a p t e r i , b u t s u b t r a c t i v e i t e r a t i o n s , as w e s a w fo r t h e C a n t o r s e t, a r e a ls o f o u n d i n _ A f r i c a n d e c o r a t i v e f r a c t a l s (fig. 8 . 2 i ) . C a r v i n g d e s i g n s in clu d e a p p lic a tio n s o f ite ra tiv e c o n s tru c tio n , p a rtic u la rly for c a la b a sh d e c o ­ r a t i o n s (fig. 8 . 2 I ) . A g e o m e t r i c a l g o r i t h m fo r p r o d u c i n g n o n l i n e a r s c a l i n g t h r o u g h f o l d i n g w as i n v e n t e d b y t h e Y o r u b a a r t i s a n s w h o p r o d u c e d t h e a d ir e c l o t h o f fig u re 8 . 2 n . It is n o t m e r e l y a m e t a p h o r t o re f e r t o a s p e c i f i e d se rie s o f fo ld s as a l g o r i t h m i c ; in f a c t , o n e o f t h e c la s s ic f r a c ta l s , t h e “ d r a g o n c u r v e , ” w as d i s c o v e r e d in i 9 6 0 w h e n p h y s i c i s t J o h n H e i g h w a y e x p e r i m e n t e d w i t h i t e r a t i v e p a p e r f o l d i n g ( G a r d n e r 1 9 6 7 ) . T h e a d i r e c l o t h a l sp .s h e w -s -th g .a p p lic a t i o n o f r e f l e c t i o n ^ yjTuj i e t r y a t e v e ry-scale f r o m s i n g l e - s t i t c h j p w s , w h i c h are r e f l e c t e d o n e i t h e r s i d e o f t h e fo ld e d g e s , t o t K F e n t i r e fa b r ic , w h i c h is c r e a t e d by t h e j o i n i n g o f t w o m i r r o r i m a g e c l o t h s . S o far w e h a v e o n ly discussed t h e t e c h n i c a l m e t h o d e m p l o y e d , b u t o f course c id tiira l m e a n i n g is_.ofl.en a t t a c h ed.tp-t-hegfi. t e c h n i q u e s a s w e ll. R e c u r s i v e h a i r ­ styles, for e x a m p l e , e m b e d layers o f s o c i a l l a b o r w i t h e a c h i t e r a t i o n , a w a y t o invest p h y s i c a l a d o r n m e n t w i t h socJal_.me a n i n g (s u c h as f r i e n d s h i p b e t w e e n sty l­ ist a n d s t y l e e ) . F ig u r e 8 . 3 a s h o w s a F u l a n i w e d d i n g b l a n k e t , i n w h i c h s p i r i t u a l e n e rg y is e m b e d d e d in t h e p a t t e r n t h r o u g h its i t e r a t i v e c o n s t r u c t i o n 1'. P r e s tig e ^ c a n a ls o b e a s s o c i a t e d w i t h i n c r e a s i n g i t e r a t i o n s , as w e f i n d f o r b ra s s c a s t i n g a n d b e a d w o r k in t h e g ra s s la n d a r e a s o f C a m e r o o n (fig. 8 - 3 b ,c ) . T h e s c a l in g iter' a ti o n s in o n e o f t h e b ra s s s c u l p t u r e s (fig . 8 . 3 d ) w as r e p o r t e d t o b e s y m b o l i c as well: it s h o w e d t h r e e g e n e r a t i o n s o f ro y a l ty . B u t k i n s h i p g r o u p s a r e n o t j u s t st a ti c e n t i t i e s ; t h e y c h a n g e a c ro s s t i m e , a n d in t h e f o l l o w i n g tw o s e c t i o n s we will see t h a t A f r i c a n r e p r e s e n t a t i o n s o f s u c h t e m p o r a l p r o c e s s e s o f t e n i n v o l v e 're c u rs io n . I .e, - FIGURE 8 . 2 R e cu rsiv e co n stritctio n te c h m q u e s (a) Coils of coils are used to create this metal wig from Senegal, (b) A scaling cascade of a mask from the Dan societies of Liberinand Cote d'Ivoire, (c) Iterative braiding in tiiis from Yaounde, Cameroon, la t r e s s e cle f l , can be simulated by fractal graphics- (d) Three if of the t r e s s e d e f l simulation. fb, f r o m Barbier-Muelfer 1988.) (/'.far F IG U R E 8 .2 (c o n tin u e d ) I te r a t iv e c o n s tr u c tio n in M a u r i t a n i a n d e c o ra tio n (e) Recursive construction with triangles and X-shapes in'Tuareg leatherwojk. The X-slvape is related to the quincunx discussed in chapter 4. (f) Designs using several iterations of triangles can also be found in Mauritanian stonework. (g) T he use of triangles in this nomadic architecture from Mauritania may be one reason for the popularity of the design. Unlike rectangles, triangles can create a rigid frame using flexible joints— an important feature in a landscape where long poles are scarce and lashing is the most common joinery, (h) A single iteration of a three-dimensional version of the recursive triangle construction, created by Akan artists in Ghana. (e, f r o m Jefferson 1973 ;• f and g, photos courtesy /FAN, D a k a r ; h, f r o m P h i lli p s 1 9 9 5 , f i g 5. J03 ) (fig u re c o n tin u e s ) A fr ic a n fra c ta l m a th e m a tic s FIGURE 8 -2 (continued) S c a lin g p a t t e r n fro m s u b tr a c tiv e i t e r a tio n (i) A F a n te w o m a n p o s in g in fr o n t o f a p a in t e d s t u d io b a c k d r o p , C a p e C o a s t , G h a n a , i 8 6 0 . (j) T h e Fante pattern can be th o u g h t of as tw o iteratio ns o f scaling su btraction ( t h a t is, erasing). Strips are erased from an ail-black background. W h e r e th e th ick strips intersect, we get large squares, and where the r.hin strips intersect we get small squares. (i, photo f r o m the Notional Museum of African Art, Smit/istmfan Institution.) - ( fig u r e .c o n ijv .u R e p r e s e n t i n g r e c u r s io n a s a p ro c e s s in tim e : p a r t I , l u c k a n d age A sim p le e x a m p l e o f A fr i c a n rep rese n t a t i o n for re c u rs io n as a tim e - v n ry ing.orDc.ess is s h o w n in fig u re 8 .4 , w h e r e w e_gee t h r e e d e s i g n s t h a t d e p i c t w is h e s f o r c a t c h e s o f e v e r l a r g e r fish. S i n c e t h e e x p e r i e n c e o f b a d l u c k o r g o o d l u c k in f i s h i n g c a n o c c u r o n a d a ily b asis, it is easy t o s e e h o w a b ig fish c o u l d b e c o m e a n i c o n fo r g o o d luck . B u t in th e s e d e s i g n s t h e a r t i s a n s t a k e t h e c o n c e p t a s t e p f u r th e r. G o o d f o r t u n e is n o t in t e r m s o f a s i n g u l a r c h a n c e e v e n t , as o n e se e s in che m y t h s o f t h e N a t i v e A m e r i c a n t r ic k s t e r .'' T h e w is h is f o r a n i t e r a t i v e p r o c e s s — t h a t e a c h fisiTfs to h e s u c c e s s iv e l y , .larger..t.han t h e j a s t o n e. W h i l e th e s e g o o d lu c k ic o n s a re o f t e n a m o r e in f o r m a l p a r t o f c u l t u r a l p r a c ­ tic e , o t h e r re c u r s iv e p ro c e s s e s a r e t a k e n m u c h m o r e serio u sly . A n t h r o p o l o g i s t s Seed sh iipe, w i t h a c t i v e li ne s in gra y. F ourth iteration en larged, w ith ad a p tiv e scalin g ( m a p p i n g fro m a sph e re to a p la n e ) a p p lie d to m a tc h the a d a p tiv e s ca lin g o f th e calab ash design. F I G U R E 8 . 2 (continued) Itera tio n in c a r v in g s j& O t) T h e B a k u b a o f Z a i r e c r e a t e d s e v e r a l c a r v i n g s t h a t f e a t u r e a s e l f - s i m i l a r d e s i g n . T i t is B a k u b a ,'fcooJen b o t t l e m a k e s u s e o f h e x a g o n s o f h e x a g o n s a s w e l l a s a d a p t i v e s c a l i n g a s it n a r r o w s i n t o c h e n c c L ( I) C h a p p e l ( 1 9 7 7 ) r e c o r d s a w i d e v a r i e t y o f c a l a b a s h d e s i g n s , m a n y w i t h s c a l i n g a t t r ib u t e s . <,T h i s is p r o b a b l y t h e b e s t e x a m p l e o f i t e r a t i v e c o n s t r u c t i o n i n t h e s e c a r v i n g s . T h e d e s i g n J iin iu la t io n n o t o n l y r e q u ir e s r e c u r s i o n b u t a d a p t i v e s c a l i n g a s w e ll - ( m ) S e e d s h a p e a n d f o u r t h I t e r a t i o n ; f o u r t h i t e r a t i o n e n la r g e d , w i t h a d a p t i v e s c a l i n g a p p lie d . ifk, c o u r te s y M u s e e R o y a l d c I A f r i q u e C e n t r a l , B e l g iu m . ) (figure c o n tin u e s ) F I G U R E 8 . 2 (continued) A d i r e c l o t h : s c a l i n g fr o m i t e r a t i v e f o l d i n g (n ) T h is Yoruba adire clo th is actually two separate pieces a tta c h e d along th e horizontal midline. T h e dye pattern is created by sewing alon g folds before dye is applied an d th e n rem oving the th reads so th a t th e w h ite lines are left where th e dye did n o t p e n e tra te , (o) T h e folding m etho d is based o n reflection symmetry across a diagonal. It is easiest to understand by malting a paper model. T h e adire artisans have n o t only developed an algorithm fot generating this n o n lin ear scaling series, b u t have d on e so in a way th a t maximizes efficient production: all folds fall along th e same two edges, so only two edges need be sewn. Your paper m odel c a n im itate this effect by ru n n in g a heavy felt marker along the two edges, so th a t the ink bleeds th rough all th e layers (you can c h eat by inking e a c h fold as you unfold it). N o te th a t th e w hite lines in th e adire are triple— this, too, is created by a reflection symmetry, sewing n ex t to the fold to create the tw o outer lines (on e on each side of the fold), and sewing right on the edge of th e fold to create the ce n te r line. (n, photo from Picton and Mack 1979.) First, cu t o u t a p a p e r rectan g le w ith w id th tw ice d ie h e ig h t, a n d f o ld it in h a lf , m a k i n g a s q u a re . S e c o n d , fold t h e s q u a re a lo n g d ia g o n a l, m a k in g a triangle. T hird, m ark po in ts at Zz a n d Id o f t h e o u t e r sides o f th e triangle. T h e s e p o in ts c a n be d e t e r m i n e d h y f o l d i n g , if o n e w ish es tn m a i n t a i n th e o rig am i e q u iv a le n t o f co m p ass a n d F o u r th , fold f ro m t h e c o r n e r s o u F in ally , f o ld S traig h t-ed g e c o n s tru c tio n , h u t d o in g o p p o s ite sides a lo n g t h e lin e b e tw e e n o v erlap p in g c o rn e r on it hy e y e b a l l w o r k s j u s t fine. the Zz rind Vt m aiks. side. in th e sm n FIG.URE 8 .3 M a k i n g m e a n in g th r o u g h ite r a tiv e c o n s tr u c tio n (a) T h is Fulani w e d d i n g b l a n k e t f ro m M a l i is b a s e d o n d i a m o n d s t h a t s c a le f r o m e i t h e r s i d e as we m ove tow ard the c e n t e r ; a p a t t e r n t h a t is eas ily s i m u l a t e d u s i n g a f rac ta l' ( s e e " d i a g r a m ) : " " T h e w eavers w ho c r e a t e d it r e p o r t t h a t s p i r i t u a l e n e r g y is w oven in to th e pattern, m d t h a t e a c h s u c c e s s i v e i t e r a t i o n s h o w s a n i n c r e a s e in h i s e n e r g y . R e l e a s i n g t h i s s p i r i t u a l e n e r g y is d a n g e r o u s , i n d if t h e w e a v e r s w e r e t o s t o p in t h e m i d d l e t h e y w o u l d isk d e a t h . T h e e n g a g e d c o u p l e m u s t b r i n g t h e w e a v e r o o d a n d k o l a n u t s t o k e e p h i m a w a k e u n t i l it is f i n is h e d , b) T h e prestige b ro n ze o f F o u m h a n , C a m e r o o n , o ften n a k e s u se o f s e l f - s i m i l a r i t e r a t i o n s , ( c ) P r e s t i g e is als o y m b o l i z e d by t h e l a b o r a n d a r t i s t r y r e q u i r e d t o p r o d u c e th e m a n y ite ra tio n s of b ead p a tte n s for this e le p h a n t m ask, (d ) A c c o rd in g to S alefou M b e tu k o m , th e lead in g ca s to r of F o u m b a n , this sc u lp tu re show s th e successio n of k i n g s in t h e r o y a l fam ily . (c,/ro m rig en ce H o a - Q u i . ) A fr ic a n fra c ta l mac/iematics 120 h a v e alw a ys b e e n i n t e r e s t e d in t h e c o n t r a s t b e t w e e n t h e e l a b o r a t e p o l i t i c a l a n d e c o n o m i c h i e r a r c h y o f E u r o p e a n s o c i e t i e s a n d t h e r e l a t i v e l y “c las sles s" ( s o m e ­ t i m e s e v e n r i d e rle s s ) s t r u c t u r e o f m a n y p r e c o l o n i a l A f r i c a n s o c i e t i e s ^ J f it is n o t p o litic a l a n d e c o n o m i c s tr u c tu r e t h a t g o v e rn s th e i r so ciety , t h e n w h a t does? O n e p a r t o f t h e a n s w e r is ag e. A l l h u m a n c u l t u r e s d i f f e r e n t i a t e b e t w e e n c h i l - c f i g u r e 8 .4 I f w is h e s w e r e fish e s (a) Scaling scales: this Bamana tattoo, created with henna, is said to represent the scales of fish, h is good luck, .signifying everlarger fish catches, (b) This is an "abbia,” a carved gambling chip from Cameroon. Given the high stakes of the game, it could be a more aggressive symbolism than just luck, e.g., "just as you have swallowed others, ! will swallow you.” O ther chips appear to carry the iteration out several more levels, although they are less recognizable as fish (c). (d) This print with four iterations of fish is from northern Ghana. It was reported to be a fertility symbol. ( d , p/iofo courtesy o f Traci Roberts a n d A n n C am pbell.) d Recursion 121 c lr e n , adj.iJcs,jind e ld e r s , b u t in m a n y A f r i c a n s o c i e t i e s t h e d iv i s i o n s a r e m u c h m o r e e l a b o r a t e a n d s t r u c t u r e d . I n t h e s e a g e - g r a d e sy s te m s , aH c o m m u n i t y m e m ­ b e r s b o r n w i t h i n a g i v e n n u m b e r o f y e a rs w ill m o v e t o g e t h e r t h r o u g h a se r ie s o f r i t u a l i n i t i a t i o n s . In c h a p t e r 5 w e saw ,.one e x a m p l e in w h i c h t h e s e i n i t i a t i o n sta g e s a p p e a r e d ro be a c c o m p a n i e d by a n i t e r a t i v e s c a l in g g e o m e try , t h e lu s o n a . Figure 8 .5 a show s a n o t h e r g e o m e t r i c v is u a liz a tio n o f ag e-g ra de in itia tio n : a h e x a g ­ o n a l m a s k c r e a t e d by che Bassari o f t h e S e n e g a m b i a n a n d G u i n e a - B i s s a u region . A l t h o u g h t h e niask is o n ly a l i n e a r - c o n c e n t r i c sc a lin g o f h e x a g o n s , a n d th u s n o t a fra ctal, it d o e s suggest a n i t e r a t i v e pro cess, a n d w e m i g h t well s u s p e c t a lin k b e t w e e n sta g e s in a g e -g r a d e a n d s t a g e s in i t e r a t i o n . T h e i n i t i a t i o n p r o c e s s is a c lo sely g u a r d e d s e c r e t, so it is n o t s i m p l y a m a t t e r o f a s k i n g Bassari e x p e r t s , b u t d u r i n g m y v is it w i t h t h e B assari in 1 9 9 4 1 f o u n d t h a t t h e m e a n i n g o f o t h e r m a t h e m a t ic a l p a tte rn s in Bassari c u ltu re c a n b e used to m a k e so m e ed u c a te d guesses a b o u t t h e m e a n i n g o f t h e m as k . D e s p i t e t h e e x t e n s i v e m i g r a t i o n s fr o m t h e v i l ­ lages to c it ie s ( N o l a n 1 9 8 6 ) , t h e r e is s till s t r o n g p a r t i c i p a t i o n in t h e age g ro u p s a n d t r a n s i t i o n rituals. T h e “fo r e st s p i r i t ” A n n a k u d i , fo r e x a m p l e , se e m s t o be u n d a u n t e d by t h e c ity o f T a m b a c o u n d a , w h e r e a lo c al age g ro u p h o s t e d h i m a t a w e l l - a tt e n d e d d a n c e d u r i n g m y stay. I n d e e d , I fo u n d t h e s te re o ty p e o f tr a d i t i o nal e ld ers a n d i r r e v e r e n t y o u t h s to b e s o m e w h a t r e v e r s e d ( w h i c h w as e x p l a i n e d to FIGURE S c a lin g h e x a g o n s in 8 .5 a Bassari m a sk |.U) T h e B a s s a r i i n i t i a t i o n m a s k s f r e q u e n t l y f e a t u r e s c a l i n g h e x a g o n s in t h e c e n t e r . T h i s a p p e a r s t o a lin e a r s c a l i n g , ( b ) O n e o f t h e B a s s a r i e l d e r s d e m o n s t r a t e s t h e t r a d i t i o n a l s t T i n g t a lle y s , w i t h 5ls in g r o u p s o f s ix . I,p fioto f r o m a g e n c e H o a - Q u i/ M ic h e / R e n n n d e m i■) A fr ic a n fractal m athem atics 122 m e as a n e ff e c t o f t h e s t r o n g h i e r a r c h y o f s e c r e t k n o w l e d g e : t h e y o u t h a r e o f t e n m o r e w a ry a b o u t b r e a k i n g t a b o o s b e c a u s e t h e y a re less c e r t a i n a b o u t b o u n d a r i e s a n d c o n s e q u e n c e s ) . T h i s is n o t t o say t h a t t h e r e is a n y o v e r t p r e s e n c e o f fear. In « '1 f a c t , it is t h e p o s i t i v e a s p e c t s o f t h e s e c r e ts t h a t a re stre s s e d , as b e c a m e o b v i o u s \ w h e n e l d e r s g l e e f u ll y re f u s e d m y q u e s t i o n s w h i l e e m p h a s i z i n g t h e w o n d e r f u l '. n a t u r e o f th e in f o r m a tio n th e y c o u ld n o t div u lg e. T h e n u m b er.-S.lx is a p r o m i n a n t f e a t u r e o f B assari m a t h e m a t i c s in m a n y a re a s o f t h e i r life. T h e y h a v e a p o p u l a r g a m e , for e x a m p l e , p l a y e d w i t h p e b b l e s o n a s a n d p a t t e r n , w h i c h m a k e s u se o f tw o a x e s w i t h six h o l e s in e a c h li n e . In t h e i r t r a d i t i o n a l c a l e n d a r t h e r e a r e six m o n t h s p e r year, e a c h o f 3 0 ( 6 x 5 ) days, w i t h a n i n i t i a t i o n a b o u t e v e ry 1 2 (6 X 2 ) y e ars ( t o a t o t a l o f n i n e i n i t i a t i o n s ) . E ach o f th e s e rites of passage in v o lv e s a le n g th y e d u c a tio n in a n e w lev el of t r a d i t i o n a l k n o w l e d g e . T h e m o s t i m p o r t a n t is t h e p a s s a g e t o a d u l t h o o d , w h i c h las ts for six d ay s. I n a d d i t i o n t o t h e s e t i m e m e a s u re s , t h e n u m b e r six a ls o a p p e a rs in t h e B assa ri c o u n t i n g s y s te m . S t r i n g t a l l i e s , t r a d i t i o n a l l y u s e d f o r r e c o r d i n g v a r i o u s c o u n t s , o f t e n u sed k n o t s g r o u p e d b y six. T h e B assari e l d e r w h o d e m o n ­ s t r a t e d t h e s e t a l li e s t o m e (fig. 8 . 5 b ) t o l d tne t h a t h e d i d n o t k n o w m u c h a b o u t t r a d i t i o n a l fo rm s o f c a l c u l a t i o n , b u t h e d id k n o w t h a t in p r e c o i o n i a l t i m e s it was p e r f o r m e d by sp ecialists w h o w e re tr a i n e d in t h e m e m o r i z a ti o n o f su m s. T h i s p r a c ­ tice may e x p la in th e o rig in s o f th e fam ou s A fric a n A m e r i c a n c a lc u la tin g p r o d i g y , T h o m a s F u l l e r . I n 1 7 2 4 , a t t h e a g e o f 1 4, h e w a s c a p t u r e d — q u i t e p o s s ib ly fr o m t h e g e o g r a p h i c a r e a s t h a t i n c l u d e d t h e B a s s a r i^ — a n d s o l d i n t o sla v e ry in V irg in ia , w h e r e h e a s t o n i s h e d b o t h p o p u l a r a n d p r o f e s s i o n a l a u d i e n c e s w i t h h is e x t r a o r d i n a r y c a l c u l a t i n g fe a ts ( F a u v e i a n d G erde.s 1 9 9 0 ). • F in a lly , t h e r e is t h e Bassata d i v i n a t i o n s y s te m . A l t h o u g h t h e c a s t_sh.eHs are i n t e r p r e t e d b y im a g e s r a t h e r t h a n a n y n u m e r i c r e a d i n g , t h e y a re c a s t sixL.times. E a c h c a s t p r o v i d e s t h e a n s w e r t o a sp e c ific q u e s t i o n ( o r v e r i f i c a t i o n o f a p r e v i ­ o us q u e s t i o n ) r e l e v a n t t o t h e c li e n t's p r o b l e m ; t h e final s i x t h c a s t s h o w s t h e p r o b ­ l e m as a w h o l e . If w e c o m p a r e th i s d i v i n a t i o n to t h e i n i t i a t i o n s y s t e m , t h e n u m b e r six c a n b e s e g n as_a m a r k e r for in fo r m a t i o n c lu s te rs ^ a pu.nctuaL i.on p o i n t w h i c h , like t h e ta lly s y s te m , a ll o w s t h e d i s t i n c t i o n s t h a t m a i n t a i n a c o m ­ p r e h e n s i v e s t r u c t u r e . A n d l i k e t h e i n i t i a t i o n , e a c h c y c l e o f six p r o v i d e s a n e x p a n d i n g v ie w o f t h e w h o l e . T h u s it s e e m s lik e ly t h a t t h e s c a l i n g h e x a g o n s o f th e in itia tio n m ask re p re se n t th is six-stage ite ra tio n o f k n o w le d g e . N o n l i n e a r s c a l in g i t e r a t i o n s c a n a ls o b e f o u n d in A f r i c a n i n i t i a t i o n m ask s. F ig u re 8 .6 a s h o w s a B a k w c l e m a s k in w h i c h b o t h size a n d c u r v a t u r e h a v e a n o n ­ lin e a r in c re a s e w ith e a c h stage. M y guess— 1 h a v e n o t fo u n d a n y c u ltu ra l d e s c r i p t i o n s t h a t c a n c o n f i r m t h i s — is r h a t it s u g g e s ts “t o o p e n y o u r e y e s ” a s a m e t a p h o r o f k n o w l e d g e , a n d t h u s m a p s t h e s c a l in g i t e r a t i o n s o f t h e m a s k t o i t e r ­ ; Recursion 123 a t i o n s o f k n o w l e d g e g a i n e d in i n i t i a t i o n stages. F ig ure 8 . 6 b s h o w s a B e m b e m a s k u se d in t h e first o f a t h r e e - s t a g e i n i t i a t i o n f o r a v o l u n t a r y a s s o c i a t i o n , t h e bioami (B ie b u y ck 1973). Before t h e c e re m o n y , t h e m a s k is h i d d e n b e h i n d a sc reen , a n d d u r i n g t h e ritu a l t h e s c r e e n is g ra d u a lly lifte d by a h i g h - r a n k i n g s e n i o r m e m ­ ber. B o th t h e r e l a t i o n b e t w e e n t h e mnnb_er o f eyes in t h e m ask a n d thejaufrTber o f s t a g e s in i n i t i a t i o n , as w ell as t h i s m e t h o d o f v is u a lly e x p o s i n g t h e p a t t e r n as a s e q u e n c e , a g a in su gg est i n t e n t i o n a l use o f a s c a l in g g e o m e t r i c d e s i g n t o r e p ­ r e s e n t s c a l in g i t e r a t i o n s o f k n o w l e d g e . a '•% FIGURE 8 . 6 N o n lin e a r s c a lin g in in itia tio n m a s k s Bcmhe mask, from western Congo, used in the first of a three-stage initiation for a voluntary Kiaiion, the biwmu. (b) Mask used in initiation by the Bakwele of Congo. ' p/ioiii courtesy Gene Isaacson; b, courtesy Musee d e I’Hornme.) 124 A fr ic a n fra c ta l mathematics R e p r e s e n t i n g r e c u r s i o n a s a p r o c e s s in t i m e : p a r t I I , k i n s h i p a n d d e s c e n t If a g e - g r a d e s y s te m s a r e o n e p a r t o f t h e s t a n d a r d a n t h r o p o l o g i c a l e x p l a n a t i o n for h o w " c la s s le s s ” s o c i e t i e s a r e s t r u c t u r e d , k i n s h i p is t h e o t h e r A K i n s h i p sy s­ t e m s a r e p r i m a r i l y b a s e d o n g e n e t i c tie s ( “b l o o d r e l a t i o n s " ) a n c l j f n a r r i a g e , a l t h o u g h m o s t s o c i e t i e s a ls o h a v e - “f i c t i v e ” k i n ( e .g ., a d o p t i o n ) w h i c h a r e ju s t as re a l— k i n s h i p is a c u l t u r a l p h e n o m e n o n . D e s c e n t j s also c u lt u r a l l y b a se d . M o s t W e s t e r n E u r o p e a n a n d A m e r i c a n s o c i e t i e s t h i n k o f d e s c e n t as b io l o g i c a l , p u t t h a t is b e c a u s e m o s t o f t h e m h a v e b i l a t e r a l d e s c e n t , in w h i c h b o t h p a r e n t s a r e u se d t o e s t a b l i s h k i n s h i p . U n i l i n e al d e s c e n t , w h e r e a k i n g r o u p t r a c e s t h e i r li n e a g e t h r o u g h o n e se x o n ly , is a c t u a l l y m o r e c o m m o n ( i n a b o u t 6 0 p e r c e n t o f t h e w o r ld ’s c u lt u re s ) . A "fclan}’ is a u n i l m e a l k i n s h i p g r° u p w h o s e m e m b e r s re p o r t t h a t t h e y a re d e s c e n d e d fr o m a c o m m o n d i s t a n t a n c e s t o r , o f t e n a m y t h o l o g i c a l figure.^Clapls o f t e n h a v e i m p o r t a n t re l ig i o u s a n d p o l i t i c a l f u n c t i o n s , a l t h o u g h t h e y a re ty p i c a ll y s p r e a d o u t a c r o s s m a n Y ^ |] i a g e s i. g n d >ujua]J_y p r o h i b i t m a r r i a g e b_eXiveen_dan'rnein_bers. W e h a v e a lr e a d y s e e n h o w t h e B a m a n a use r e c u r s i o n t o g e n e r a t e a b i n a r y , c o d e in t h e i r d i v i n a t i o n ; h e r e w e w ill l o o k a t t h e i r r e p r e s e n t a t i o n o f d e s c e n t as ; r e c u r s i o n . T h e a n t e l o p e figure in B a m a n a i c o n o g r a p h y is a s s o c i a t e d w i t h b o t h j h u m a n a n d a g r i c u l t u r a l fe r til it y . In t h e clti w a ra a s s o c i a t i o n , w h i c h is o p e n to ! b o t h m e n a n d w o m e n , t h e a n t e l o p e a p p e a r s in a s t r i k i n g h e a d d r e s s (fig. 8 . 7 a ) , !w h ic h represents th e recu rsio n o f re p ro d u c tio n : m o th e r a n d ch ild . W h e n seeing o n e h e a d d r e s s i n d i v id u a l ly , t h e s c a l i n g s e e m s tr iv i a l, b u t w i t h s e v e r a l e x a m p l e s : t o g e t h e r t h e e x t r a o r d i n a r y i n s i s t e n c e o n s e lf- s im i la r it y b e c o m e s a p p a r e n t . T h i s i c o n a c ts as t h e s e e d t r a n s f o r m a t i o n m j m i t e r a t i v e l o o p ^ t h e c h i l d b e c o m e s a m o t h e r , w h o h a s a c h i l d , w h o ' b e c o m e s a m o t h e ' t r ' a n d so o n . F i g u r e 8 . 7 b sh o w s t h e d e s c e n t c a r r i e d to a t h i r d i t e r a t i o n . In c h a p t e r 2 w e sa w s e v e r a l e x a m p l e s in w h i c h d e s c e n t w a s t i e d t o s c a l ­ in g a r c h i t e c t u r e . T h e B a t a m m a l i b a , w h o li v e in t h e n o r t h e r n p a r t s o f G h a n a , B e n in , a n d T ogo, h a v e d e v e lo p e d a n e la b o r a te sy ste m for th is re la tio n s h ip ( B li e r i 9 8 7 ) . F ig u re 8 . 8 s h o w s a d i a g r a m o f t h e i r t w o - s t o r y h o u s e , b a s e d o n che c ir c le o f c irc les f o u n d ir. m u c h o f t h e W e s t A f r i c a n in t e ri o r. I n f r o n t o f t h e h o u se lies t h e first o f tw o s c a l i n g t r a n s f o r m a t i o n s . It is t h e " s o u l m o u n d , ” a c i r c l e o f c y l i n d e r s r e p r e s e n t i n g t h e s p i r i t s o f rho.se c u r r e n t l y l i v i n g in t h e h o u s e a n d p h y s i c a ll y s t r u c t u r e d like a s c a l e d - d o w n v e r s i o n o f t h e h o u s e a r c h i t e c t u r e . As t h e c u r r e n t fa m ily g iv e s w ay t o a n e w g e n e r a t i o n , t h e s o u l m o u n d u n d e r g o e s a s e c o n d t r a n s f o r m a t i o n in w h i c h i t is d i v i d e d i n t o a s i n g l e c y l i n d e r a n d is m o v e d in s id e . A s c a l i n g s e q u e n c e o f t h e s e s i n g l e c y l i n d e r s — o n e fo r e a c h g e n ­ e r a t i o n — c a n be s e e n w r a p p e d a r o u n d r h e c e n t r a ) t o w e r i n s i d e t h e h o u s e . '■ I I FIGURE 8.7 R e c u r s io n a n d re p r o d u c tio n in B a m a n a s c u lp tu r e iM The chi wara figure, used in ritual dances for agricultural fertility, shows a striking self-similarity: fiAlthough the figures vary widely, each one is similar to itself. This can be attributed to the Bamana jtvicw of reproduction as cyclic iterations, (b) Here the cycle is carried out to three iterations. (•: upper le ft , f r o m die de Hevenon Collection, Museum o f A f r i c a n A r t , S m i t h s o n i a n I n s t i t u t i o n ; upper courtesy Muse'e d e I ' H o m m e ; l o w e r , f r o m C a r n e g i e I n s t i t u t e 1970. b, c o u r t e s y Musee d e I ' H o m m e . ) A frican fra c ta l mathematics 126 F IG U R E fig i > I'jK ! (( V 1. .. 8 .8 R e c u r s io n in B a ta m m a lib a a \->v r c h ite c tu r e (a) Diagram or the Batammaliba two-story house. In front of the house lies the “soul mound," representing the spirits of those currently living in the house. (b) Inside the house, single mounds representing ancestors are found in the scaling arrays, with the si2e of the ancestral mounds increasing from youngest to oldest. Here only one such array is shown, but typically there are several in the same household. ( a , fro m B lier 1 9 8 7 .) ii r I ■H ] S J u r* s B lier’s d i a g r a m i n d i c a te s t h a t t h e size o f t h e a n c e s t r a l m o u n d s in c re a se s from y o u n g e s t t o o l d e s t , a n d s h e n o t e s t h a t t h i s reflects t h e B a t a m m a l i b a ’s id e a o f a s p i r i t u a l p o w e r i n p r o p o r t i o n t o ag e. S o far it w o u l d a p p e a r t h a t t h e r e a r e o n ly tw o s c a l in g c a s c a d e s — o n e t o s h r i n k h o u s e s to s o u l m o u n d s , a n d a n o t h e r t o . d i v i d e s o u l m o u n d s i n t o c y l i n d e r ro w s— a n d n o i t e r a t i v e lo o p . B u t if t h e la r g e st m o u n d re p re s e n ts th e o ld e s t, t h e n re c e n t m o u n d s w o u ld b e in c re a sin g ly t h r e a t e n e d by v a n i s h i n g sc a le . H o w w o u ld t h e first d e s c e n d a n t h a v e k n o w n h o w la r g e t o m a k e t h e first m o u n d ? B li e r n o t e s t h a t m a n y o f t h e s y m b o l i c f e a t u r e s o f t h e a r c h i t e c t u r e a r e r e p l a s t e r e d w i t h a d d i t i o n a l la y e r s o f w e t c la y o n r i t u a l o c c a s i o n s , a n d w e c a n s u r m i s e t h a t t h i s a p p l i e s t o t h e a n c e s t r a l m o u n d s as well. T h u s a n i t e r a t i v e lo o p , in w h i c h e a c h n e w . a n c e s t o r a d d s p o w e r to t h e o ld e r o n e s by i n c r e a s i n g t h e i r m o u n d ’s size, w o u l d b e a t w o r k in t h e s c a l i n g s e q u e n c e w e se e a c c u m u l a t i n g a r o u n d t h e c e n t r a l tow er. f t... ' T h e M i t s o g h o so c ie ty o f G a b o n i n c l u d e s s e v e r a l relig io u s a s s o c ia ti o n s t h a t are h o u s e d in che s a m e t e m p l e (e b a n d z a }. F ig u r e 8 . 9 a s h o w s che c e n t r a l p o s t o f a n e b a n d z a fe a tu rin g sc a lin g pairs o f h u m a n figures. A s in. t h e c h i w a ra figure, th e r e is o n ly o n e i t e r a t io n ; t h e s i g n i f i c a n c e j i e s j p t h i s f i g u r e a s t h e se ed t r a n s f o r m a t i o n ^ ^ a d ^ Lil!§Lv.?.PJ°cess. T h e use o f a cross s h a p e m a y b e d u e t o C i u j s n a n influen c e , b u t t h e b ilareral scalin g is q u i t e i n d i g e n o us, as w e see in che classic B a k w e le s c u l p ­ tu r e (fig. 8 . 9 b ) e l s e w h e r e in G a b o n . M o s t i m p o r t a n t , t h e e b a n d z a p o s t p r o v i d e s a v is u a li z a ti o n for t h e i t e r a t i v e c o n c e p t o f d e s c e n t t h a t is w id ely u s e d in t h i s c u l ­ tu r e area. T h i s is b e a u tifu lly d e s c r ib e d by F e r n a n d e z ( 1 9 8 2 ) in a d e t a i l e d e t h n o g ­ r a p h y o f t h e M i t s o g h o ’s n e i g h b o r s a n d c u l t u r a l r e l a ti v e s , t h e F an g. A l t h o u g h t h e F a n g a re p a t r i l i n e a l , t h e y b e l i e v e t h a t t h e a c t i v e p r i n c i p l e o f b i r t h — a ti n y h u m a n ( w h a t w as c a l l e d a “ h o m u n c u l u s ” in early E u r o p e a n m e d ­ ical th e o r y ) — is c o n t a i n e d in t h e fe m a le b lo o d . T h e idea o f t h e n e w existin g w i t h i n t h e o ld , a n d v ic e versa , is a s t r o n g c u l t u r a l t h e m e . F o r e x a m p l e , in o n e r i t u a l t h e m o t h e r p l a c e s a n e w b o r n c h i l d o n t h e b a c k o f h e r o l d e s t s i b li n g t o sy m b o liz e c o n tin u ity o f th e lineage. F e rn a n d e z (1 9 8 2 , 2 5 4 ) n o te s t h a t th e re b irth c o n ­ c e p t is so s t r o n g t h a t “F a n g f a t h e r s o f t e n c a l l e d t h e i r i n f a n t s o n s g t q ...the_ f a m i l i a r f o n n j a / . f a i b e r . ’’ I n m a n y o f t h e F a n g a n d M i t s o g o r e l i g i o u s p r a c t i c e s , t h e s p i r i t is e x p l i c i t l y d e s c r i b e d a s t r a v e l i n g a v e r t i c a l c y c l i c p a t h . A n c e s t o r s rise fr o m t h e e a r t h t o b e c o m e b o r n a g a i n , a n d by p r o p e r l i v i n g t h e y c a n rise h ig h e r w ith each rebirth. T h e s e c y c l i c i t e r a t i o n s a r e v i s u a l i z e d in t h e N g a n g a d a n c e o f t h e B w i ti r e l i g i o n (fig. 8 . 9 c ) . E v e n in C h r i s t i a n - a n i m i s t s y n c r e t i s m , b i b l i c a l c h a r a c t e rs a r e r e i n t e r p r e t e d as c y c li c _ r e b i r t h s : t h e A f r i c a n g o d s Z a m e a n d N y i n g w a n b e c o m e A d a m a n d E v e , w h o b e c o m e C a i n a n d A b e l ( u n d e r s t o o d as m a l e a n d fem ale), w h o b e c o m e C h r is t a n d th e V irg in .M ary . F e rn a n d e z n o te s t h a t th ese c y c le s a re n o t m e r e r e p e t i t i o n , b u t r a t h e r i t e r a t i v e t r a n s f o r m a t i o n s : " T h e s p i r i t u a l - f r a t e r n a l r e l a t i o n o f Z a m e a n d h i s s i s t e r is c o n v e r t e d i n t o t h e c a r n a l relatio n o f A d a m a n d Eve w h ic h d e g e n e ra te s in to th e m aterialistic a n d divisive r e l a t i o n o f C a i n a n d A b e l w h i c h t h e n is r e g e n e r a t e d a s t h e i m m a c u l a t e a n d filial r e l a t i o n s h i p o f M a r y a n d J e s u s ” (p. 3 3 9 ) . A c c o r d i n g t o F e r n a n d e z , t h e s e d e g e n e r a t i o n / r e g e n e r a t i o n d i f f e r e n c e s a r e v i s u a l i z e d as h o r i z o n t a l v e r s u s v e r t i c a l J w h i c h c o u l d e x p l a i n t h e a l t e r n a t i o n in t h e e b a n d z a p o s t s . I n a p p l y ­ in g t h i s c y c l i c c o n c e p t i o n t o t h e e b a n d z a s t r u c t u r e (fig. 8 . pel), w e c a n s e e t h e d e s c e n t m o d e l in its full f r a c t a l e x p a n s i o n . T h e Tabw a, w h o occupy th e ea ste rn s e c tio n of th e D e m o c ra tic R e p u b lic o f C o n g o ( Z a ir e ) , h a v e also d e v e l o p e d s e v e r a l g e o m e t r i c figures t o se r v e as m o d ­ els for t h e i r c o n c e p t i o n s o f k i n s h i p a n d d e s c e n t . M a u r e r a n d R o b e r t s ( 1 9 8 7 , 2 5 ) e x p l a i n t h a t in t h e T a b w a o r i g i n story, a n a a r d v a r k ’s w i n d i n g t u n n e l re s u lts in a b FIGURE 8.9 R e c u r.s w e k in s lu f) i n Q abon (a) T h e central post of t h e ebandza temple in western G a b o n suggests an iterative descent concept. T h is is actually a museum reproduction, (b) Bakwele masks from eastern G a b o n show similar bilateral scaling. f a , f r o m F erro is 19 8 6 ; In left, f r o m Pefrois j 9 8 6 ; rig/u, M etro p o lita n M u se u m o f A r t ; f r o m Z aslavsky 197 .}•) (figure continue*) Recursion 129 F I G U R E 8 . 9 (continued) R e c u rs iv e d e s c e n t in Q a b o n (c) In many of the Fang-and Mitsogo religious practices, the spirit is explicitly described as traveling a vertical cyclic path. Ancestors rise from the earth to be born again, and by proper living they can rise higher with each rebirth. These cyclic iterations are visualized in the Nganga dance of the Bwiti religion, (d) We can apply the explicit mapping of cyclic generations given by the Nganga dance to the iterative posts of the ebandza temple and see the descent model in its full fractal expansion. T h e implication of infinite regress is discussed in chapter 9. (c./rom Fernandez: 1982.) a “b o t t o m l e s s s p r i n g ” fr o m w h i c h e m e r g e s t h e first h u m a n , K y o m b a , w h o s e d e s c e n d a n t s s p r e a d in all d i r e c t i o n s f r o m t h i s c e n t r a l p o i n t . T h i s s p r e a d is v i s u ­ alized by t h e m p an d e, a d is k c u t fr o m t h e e n d o f a c o n e s n a i l, w h i c h is w o r n as a chest p e n d a n t (fig. 8 . 10a). T h e c e n tr a ! p o i n t is d rilled o u t, r e p r e s e n tin g t h e e m e r ­ ge n c e o f K y o m b a fr o m t h e d e e p s p r in g , a n d t h e l o g a r i t h m i c sp iral o f t h e shell .end s y m b o lizes t h e e x p a n s i o n o f k in g r o u p s fr o m t h i s origin.® O n e w ay ro r e p r e s e n t t h e s e e x p a n d i n g i t e r a t i o n s t h r o u g h ti m e is t o ta k e a series of p o r t r a i t s as t h e s t r u c t u r e c h a n g e s : p r o j e c t i o n s a t d i f f e r e n t p o i n t s a l o n g th e ti m e axis. F ig ure 8 . 1 0 b s h o w s t h e first s t e p t o w a r d t h i s d e s ig n : a m o r e li n e a r version o f t h e m p a n d e d isk, in w h i c h a n A r c h i m e d e a n spiral fits b e t w e e n a series i • •'i A fr ic a n fr a c ta l m athem atics o f t r i a n g l e s ( w h i c h r e p r e s e n t t h e w i v e s o f t h e g u a r d i a n o f t h e a n c e s t o r s ) . In fi g u r e 8 . i o c w e s e e t h a t t h e l i n e a r s p i r a l h a s b e c o m e c o n c e n t r i c s q u a r e s , b u t t h e y a r e n o w p o r t r a y e d i n a s c a l i n g s e q u e n c e , s u g g e s tin g a se r ie s o f p o r t r a i t s o f t h e k i n s h i p s p ira l as it e x p a n d s t h r o u g h t i m e . S i m i l a r s c a l i n g s q u a r e s e q u e n c e s , c a r r i e d o u t t o a g r e a t n u m b e r o f i t e r a t i o n s , c a n b e s e e n i n t h e s ta ffs o f t h e i r n o r t h e r n n e i g h b o r s , t h e B a l u b a (fig. 8 . i o d ) . b c d FIGURE 8 . I O T a b w a k i n s h i p r e p r e s e n ta tio n s (a) T h e m p and e shell worn by C h i e f M and a Kaseke Joseph, (b) A more linear version of the m pande disk, in w hich an A rc h im e d e a n spiral fits betw een a series of triangles (w hich represent the wives o f the guardian ancestors), (c) T h e linear spiral has become c o n c e n tric squares, hut they are now portrayed in a scaling sequence, suggesting a series of portraits o f th e kinsh ip spiral as it expands th ro ugh time, (d) Sim ilar scaling of square sequences can he seen in d ie sraffs of their n o rth e rn neighbors, the Baluba. fa-c, from Roberts anti Mai iter 1985; d, Museum f i i r V o lk e v k tm d e , F rcm hfnrt.) Recursion R ecu rsive cosmology I n a ll che d e s c e n t r e p r e s e n t a t i o n s we h a v e e x a m i n e d , k i n s h i p gro u p s tr a c e t h e m ­ selv es to a m y th o lo g ic a l a n c e s t o r a t t h e b e g i n n i n g o f- th e w o rld , a n d t h u s w e m o v e fr o m t h e - o r ig i n s o f h u m a n i t y to t h e o r i g i n s ' b f t h e c o s m o s . A f r i c a n c r e a t io n c o n ­ c e p t s a r e o f t e n b a s e d o n a re c u r s iv e n e s t i n g . T h e b e s t - k n o w n e x a m p l e is t h a t o f t h e D o g o n , as d e sc rib e d by F r e n c h e t h n o g r a p h e r M a r c e l G r i a u l e ( 1 9 6 5 ) . H is w o rk b e g a n d u r i n g t h e 1 9 3 0 D a k a r - D j i b o u t i e x p e d i t i o n , w h e r e h e first m a d e c o n t a c t w i t h t h e D o g o n o f S a n g a in w h a t is n o w M a l i . I n 1 9 4 7 h i s s tu d ie s t o o k a d r a ­ m a t i c t u r n o f e v e n t s w h e n o n e o f t h e D o g o n e ld e rs, O g o c e m m e li, a g re e d to i n t r o ­ d u c e G r i a u l e t o t h e i r e l a b o r a t e k n o w l e d g e s y s te m . C l i f f o r d ( 1 9 8 3 ) p r o v i d e s a d e t a i l e d r e v i e w o f t h e s t r o n g r e a c t i o n s t o G r i a u l e ’s r e s u l t i n g e t h n o g r a p h y . W h i l e m a n y o f che c r i t i q u e s w e re r e a l ly a b o u t t h e fa ilin g s o f m o d e r n i s t a n t h r o ­ p o l o g y in g e n e r a l — t h e t e n d e n c y t o p r e f e r a s t a t i c p a s t o v e r t h e p r e s e n t , o r a s i n g u l a r “t r a d i t i o n ” o v e r i n d i v i d u a l i n v e n t i o n — t h e r e w e r e a ls o t h o s e w h o s i m p l y d id n o t b e l i e v e t h a t s u c h e l a b o r a t e a b s t r a c t i o n s c o u l d b e i n d i g e n o u s . F o r t h e Dogotir t h e h u m a n s h a p e is n o t o n l y a b io l o g i c a l f o r m , b u t m a p s m e a n i n g a t a l P l e v e j s: “T h e fa c t t h a t t h e u n i v e r s e is p r o j e c t e d in t h e s a m e m a n n e r o n a se r ie s o f d i f f e r e n t s c a l e s — t h e c o s m o s , t h e v i l la g e , t h e h o u s e , t h e i n d i v i d u a l — p ro v i d e s a p r o f o u n d ly u n if y in g e l e m e n t in D o g o n life” (D u ly 1 9 7 9 ). . T h e ^ Q g o n h o u s e is p h y s i c a ll y s t r u c t u r e d o n a m o d e l o f t h e h u m a n fo r m , w i t h a la r g e r e c t a n g l e for t h e b o dy , s m a l l e r r e c t a n g l e s o n ^ e a c h , s j d e T o r a rm s , a door, for t h e m o u t h , a n d so o n . T h e D o g o n v i l l a g e , h o w e v e r , r e p r e s e n t s t h e h u m a n fo r m w i t h a s y m b o l i c s t r u c t u r e r a t h e r t h a n a g e o m e t r i c s t r u c t u r e : it is n o t p h y s ­ ically a r r a n g e d as a h u m a n s h a p e , b u t v a r i o u s b u i l d i n g s a r e a s s ig n e d m e a n i n g a c c o r d i n g to t h e i r so cial f u n c t i o n ( t h e s m i t h y s t a n d s for t h e h e a d , t h e m e n s t r u a l • •• lo d g e s as h a n d s y a n d ' s T r o v p T ln r v i s e 'o f ' tw o ' d i f f e r e n t s y s t e m s o f repres..e-n t a t io n p r e v e n t s s e l f - s im i la r it y in t h e p h y s i c a l s t r u c t u r e o f t h e a r c h i te c t u re ^ _ b u t _ s o m e o f t h e D o g o n ’s re l ig i o u s ic o n s d o s h o w h u m a n fo r m s m a d e o u t o f h u m a n fo r m s Tfig. 8.1 i a ) . r A t h r e e f o l d s c a l i n g a p p e a r s in s e v e r a l a s p e c t s o f t h e D o g o n r e l i g i o n , a n d 1 it is h e r e t h a t we f i n d a n i n d i c a t i o n t h a t t h e D o g o n a re u s i n g m o r e t h a n ju s t V a c a s c a d e . G r i a u l e ( 1 9 6 5 , 1 3 8 ) s u m m a r i z e s O g o t e m m e l i ’s c r e a t i o n s t o r y : “G o d . . . h a d t h r e e t i m e s r e o r g a n i z e d t h e w o r l d by m e a n s o f t h r e e su c c e s s iv e W o r d s , e a c h m o r e e x p l i c i t a n d m o r e w i d e s p r e a d in its r a n g e t h a n t h e o n e b e f o r e it." B u t t h e s e r e o r g a n i z a t i o n s a r e n o t m e r e l y l a y e r i n g o n e o n t o p o f che o c h e r ; r a t h e r t h e o u t p u t o f e a c h r e o r g a n i z a t i o n b e c o m e s t h e i n p u t for t h e n e x t . T h e e a r t h g iv e s b i r t h t o t h e first s p i r it s ; t h e s e “N u m m o ” r e g e n e r a t e a n c e s t r a l b e in g s i n t o h u m a n l i k e re p tile s; t h e r e p t i l e - a n c e s t o r s a re a g a in r e b o r n as t h e first tr u e h u m a n s . W i t h i n r e b i r t h , t h e t h r e e f o l d i t e r a t i o n is a g a i n e n a c t e d . In t h e first ( a ) I n t h e D o g o n c o s m o l o g y , t h e s t r u c t u r e o f t h e h u m a n f o r m is c r e a t e d fro m h u m a n form . ( h ) T h e s y m b o l i s m o f tire s t a c k e d p o t s , representing th e b reath o flife, w ith in the feteus, w ith in th e w o m b . W e c a n use an iterativ e d raw in g p ro ced u re to b e tte r u n d e rsta n d ho w this k in d o f scaling can resu lt from a recu rsiv e lo o p . S u p p o s e we h a v e a ro u tin e th a t c a n draw th e circle o f th e p o t given a d iam eter, an d o n e th a t can d r a w a lid. W h i l e d i a m e t e r > m i n i m u m do: D r a w a c i r c l e o f size d i a m e r e r I f size = m i n i m u m , d r a w a lid S h r i n k d i a m e t e r h y 2/3 E n d of “ w h ile " loop. T h i s p r o c e d u r e first c h e c k s t o s e e if w e a r e p ast t h e sm a lle s t d i a m e t e r p o ssib le. If n o t, it d raw s a po t, sh rin k s th e d ia m e te r v alu e b y 2/ 3 S, a n d t h e n g o e s b a c k t o t h e s t a r t o f t h e w h ile lo o p . In o t h e r w o rd s, t h e o u t p u t o f o n e ite ra tio n — a given d ia m e te r— b e c o m e s t h e i n p u t for t h e n e x t ite r a t io n . (c ) D o g o n recursive im age o f m o th e r a n d child. FIGURE 8 . 1 I S c a lin g in D o g o n re lig io u s ic o n s (n, from Lau de 197 J a y C . L e ff.) 3; c o u r t e s y Leste r W io id e r m a n ; c, f r o m C a r n e g i e Institute 19 7 0 ; co u rtesy 0/ Recursion r e g e n e r a t i o n , fo r e x a m p l e , e a c h a n c e s t r a l b e i n g e n t e r s t h e e a r t h 's w o m b , w h i c h t u r n s e a c h o f t h e m i n t o a fetus, w h i c h a ll o w s t h e b r e a t h o f life (n u m m o ) t o e n te r. T h e c o s m o lo g ic a l n a r r a tiv e su g g ests t h a t in th e D o g o n v iew t h e b ir th i n g p ro c e s s e s a t a ll s c a l e s a r e , in s o m e s e n s e , i t e r a t i o n s t h r o u g h t h e s a m e t r a n s f o r ­ m a t i o n , a n d t h a t t h e s e i t e r a t i o n s a r e a c t u a l l y n e s t e d lo o ps. W h y s h o u ld th e D o g o n req u ire s u c h d e e p iterativ e n esting? 1 su sp ect th a t t h e r e a r e tw o m o t i v a t i o n s / F ^ r s t ^ t h e r e is a n i n s i g h t i n t o m o d e l i n g t h e w o rld : r e c u r s i o n is a n i m p o r t a n t f e a t u r e i n b i o l o g i c a l m o r p h o g e n e s i s , as w e l l as in e n v i r o n m e n t a l a n d s o c j a l _ c h a n g e . T h e ;s e c o n d .is t h e c u l t u r a l c o n t e x t o f t h i s T cn ow led ge: e l d e r s n e e d t o e n s u r e t h a t t h e y o u n g e r g e n e r a t i o n r e s p e c t s t h e i r au th o rity , w h ic h c a n o n ly b e d o n e by g iv jn g th e m g radual access to t h e source o f t h i s p o w e r , w h i c h is k n o w l e d g e . A c k n o w l e d g e s y s te m in w h i c h e n d l e s s e x e ­ gesis is p o s s i b l e m a k e s t h e i n i t i a t i o n p r o c e s s a l i f e t i m e a c t i v i t y . B u t h a v i n g so m u c h e x p la n a to ry elb o w ro o m also p re s e n ts a p ro b le m w ith tr a n s la tin g su c h n a r r a t i v e s i n t o m a t h e m a t i c s . ^ W e h a d t o b e c a r e f u l w i t h t r a n s l a t i o n s fo r m o r e fo rm a l p r a c t i c e s , s u c h as i n t e r p r e t i n g t h e B a m a n a d i v i n a t i o n s y s te m as a b in a r y c o d e , o r a d ire c l o t h as a g e o m e t r i c a l g o r i t h m . A n a r r a t i v e is n o t a q u a n t i t a t i v e o r g e o m e t r i c p a t t e r n , a n d its a m b i g u i t y r e q u i r e s ail t h e m o r e ; c a r e i n p r o d u c ­ in g a m a t h e m a t i c a l t r a n s l a d o n ^ t h a t . c l o e s n o t e m b e d i $ h J n d j g e j } O u s c o n c e p t s , firs t, we h a v e to d is tin g u is h b e tw e e n m o d e lin g th e n a r r a tiv e — s o m e th in g a s tru c tu r a l a n th r o p o lo g is t lik e C la u d e L e v i-S tra u ss w o u ld d o — a n d t h e n a r r a ­ t i v e as a n i n d i g e n o u s m o d e l , s u c h as t h e D o g o n ’s s y s te m for r e p r e s e n t i n g t h e i r o w n a b s t r a c t ideas. T h e b e s t way t o li m i t o u r t r a n s l a t i o n to ideas t h a t t h e D o g o n t h e m s e l v e s are t r y i n g t o c o n v e y is t o c o m p a r e t h e s e a b s t r a c t i o n s o f t h e n a r r a ­ ti v e w i t h o t h e r , m o r e f o r m a l D o g o n s y s te m s . T h i s m e a n s m i s s i n g s o m e id e a s t h a t d o n o t h a v e s u c h f o r m a l c o u n t e r p a r t s , b u t it is b e t t e p t o e r r on. t h e safe. s|de. - in t h i s c o n t e x t . . T h e m a t e r i a l d e s ig n s o f t h e D o g o n a re m o r e r e s tr i c te d t h a n ^ t h e . n a r r a ti v e in te r m s o f t h e i r i t e r a t i v e d e p t h . T h e b e s t c a s e is p r o b a b l y in t h e i c o n o g r a p h y of t h e g ra n a r y , w h e r e O g o t e m m e l i e x p l a i n s a s t a c k o f t h r e e p o ts : che la r g e s t rep. re s e n ts t h e w o m b ; t h e o n e o n t o p o f it, c r e a t i n g its lid, r e p r e s e n t s t h e fetus; a n d che lid o f t h a t p o t is t h e s m a l l e s t p o t , c o n t a i n i n g a p e r f u m e t h a t r e p r e s e n t s t h e b r e a t h o f life ( G r i a u l e 1 9 6 5 , 3 9 ) . T h e s m a l l e s t p o t is c a p p e d by a n o r m a l lid; at chis p o i n t t h e r e c u r s io n “b o t t o m s o u t . ” T h i s is n o t m erely a s t a c k o f d i f f e r e n t sizes; in th e D o g o n vie w t h e w o m b c r e a t e s t h e p r e c o n d i t i o n s t h a t give rise t o t h e fetus, w hich is th e p re c o n d i ti o n for t h e e n tr y o f th e b re a th o f life. T h e recu rsion is e m p h a ­ sized in t h e way t h a t e a c h n e w poc b e g in s before che p rev io u s p o t e n d s (fig. 8.1 i b ) , chat is, o n e p o t ’s lid is t h e n e x t p o t ’s b o d y ( G r i a u l e 1 9 65 , 19 9 ). In t h e s c u l p t u r e in figure 8.1 i c t h e m o t h e r ’s b re a s ts b e c o m e t h e c h i l d ’s h e a d — a g a in , a n e w o n e 133 134 A fr ic a n fractal m athem atics b e g in s b e f o r e t h e p r e v i o u s o n e e n d s . A s wc saw in t h e c h i w a r a s c u l p t u r e o f t h e D o g o n ’s B a m a n a n e i g h b o r s , r e p r o d u c t i o n is m o d e l e d as r e c u r s i o n . T h e D o g o n v i e w o f a c o s m o s s t r u c t u r e d as n e s t e d h u m a n - f o r m is q u i t e s im ila r to c e r t a in a n c i e n t E g y p tia n re p r e s e n ta tio n s . F igufe 8 .1 2 sh o w s a re lie f f r o m a t o m b in w h i c h t h e c o s m o s e n c l o s e s t h e sky, w h i c h e n c l o s e s t h e e a r t h . I t is i n t e r e s t i n g t o n o t e t h a t t h e r e a r e a g a i n t h r e e i t e r a t i o n s o f s c a l e . A t h r e e i t e r a t i o n n u m e r i c l o o p is i n d i c a t e d f o r t h e E g y p t i a n g o d o f w i s d o m , T h o t h . H e is re f e r r e d t o as H e r m e s T r i s m e g e s t u s , w h i c h m e a n s “ t h r i c e g r e a t H e r m e s , ’' b u t h e is a ls o re f e r r e d t o a s “ e i g h t t i m e s g r e a t H e r m e s . " W h y b o t h t h r e e a n d e i g h t ? It m a k e s sen se if we t h i n k in te r m s o f t h o s e c o m m o n e l e m e n t s o f A f r i c a n n u m e r ic s y s te m s , r e c u r s i o n a n d b a s e - t w o a r i t h m e t i c . T h r i c e g r e a t b e c a u s e w h i l e a n o r d i n a r y h u m a n m a y rise as h i g h as t h e m a s t e r o f m a s t e r s , H e r m e s T r is m e g e s t u s is t h e m a s t e r o f m a s t e r s o f m a s t e r s ( t h r e e i t e r a t i o n s ) ; t h u s w e c a n s u r m i s e “e i g h t t i m e s g r e a t ” re f e rs t o 2^ = 8. FIG U RE 8 . 1 2 R e c u r s io n in th e c osm olog y o f a n c i e n t Egyf)t Gelt, the Earth, enclosed hy Shu, space, enclosed hy Nut, the stellar canopy. (From Fourier 182 1.) Recursion M a n y o f t h e p r o c e s s i o n a l c ro s se s o f E t h o p i a also i n d i c a t e a t h r e e f o l d i t e r ­ a t i o n (fig. 8 .1 3 ) . A l t h o u g h t h e cro sse s a r e n o w u sed in C h r i s t i a n c h u r c h p r o ­ c e e d in g s , P e rc z e l ( 1 9 8 1 ) r e p o r t s t h a t r e l a t e d d e sig n s c a n b e f o u n d o n o r n a m e n t s e x c a v a t e d f r o m t h e c it y o f A x u m in n o r t h e r n E t h o p i a in t h e s e c o n d h a l f o f th e first m i l l e n n i u m b .c . e ., so w e s h o u l d n o t a s s u m e t h a t t h e t h r e e f o l d i t e r a t i o n was o riginally re la te d to th e C h r is ti a n trinity, a lth o u g h a c o n n e c t i o n m ay h a v e oc curred later (fig. 8 .1 3 b ). C o u l d th e r e be a c o m m o n history b e h in d all th e s e o c c u r­ r e n c e s o f tr i p l e i t e r a t i o n s in t h e re lig io u s i c o n s o f t h e S u d a n a n d N o r t h A fric a ? I t h i n k t h e c o m m o n u se o f r e c u r s i o n it s e l f is d u e to a m u t u a l in f lu e n c e , b u t t h e o c c u r r e n c e o f tr i p l e i t e r a t i o n m a y b e o n l y d u e to t h e s i m i l a r i t y o f c i r c u m s t a n c e s r a t h e r t h a n d if fu s i o n . F o r o n e t h i n g , g i v e n t h e m a t e r i a ls t h e a r t is a n s a r e w o r k ­ ing w i t h , m i n u t e s c a le s a re d if fic u lt, so t h a t t h e t e n d e n c y t o be l i m i t e d t o t h r e e ite ra tio n s m a y sim p ly b e a p ra c tic a l c o n s e q u e n c e o f t h e c ra f t m e t h o d s . It m a y also be t h a t if o n e w is h e s t o g e t t h e c o n c e p t o f i t e r a t i o n across, tw o is t o o f e w ^ w h ile m o re t h a n t h r e e is u n n e c e s s a r y ( w h i c h is w h y m o d e r n m a t h e m a t i c i a n s o f t e n r e p ­ r e s e n t a n in f in i te series by t h e first t h r e e e l e m e n t s , e.g., " 1 ,2 ,3 . . .”). O n t h e o t h e r h a n d , t h e r e a re ca ses w h e r e m a n y s u c h “ u n n e c e s s a r y ” i t e r a t i o n s are m a d e in t h e m o s t d iffic u lt o f c r a f t m a t e r i a ls . F ig u r e 8 .1 4 s h o w s a n a n c i e n t E g y p ti a n d e s ig n , c a rv e d in s t o n e , r e p r e s e n t i n g t h e o rig in m y t h in w h i c h t h e lo tu s flow er (its petalsw ith i n - p e t a ls illu s tra te d by a m u l t i t u d e o f sc a lin g lines) b eg ins che se lf-g e n e r a tin g c r e a t i o n o f t h e m a t e r i a l w o rld . -reference S e l f - r e f e r e n c e is t h e m o s t p o w e r f u l ty p e o f r e c u r s i o n . T h e a b il it y o f a s y s t e m to re fle c t o n it s e l f is a t t h e h e a r t o f b o t h che li m i ts o f . m a t h e m a t i c a l c o m p u t a t i o n as w ell as o u r s u b j e c t i v e e x p e r i e n c e o f c o n s c i o u s n e s s . B u t t h e r e a re r e l a t i v e l y triv ial a p p l i c a t i o n s o f s e l f - r e f e r e n c e as w ell ( o n e c a n a lw a y s use a b l o w t o r c h to lig h t a c a n d l e ) . S e l f - r e f e r e n c e first c a m e to t h e a t t e n t i o n o f m a t h e m a t i c i a n s in s im p le e x a m p l e s o f lo g ic a l p a r a d o x ; for e x a m p l e , t h e “ li a r ’s p a r a d o x ” w e e x a m ­ in ed e a rlie r . T o s e e h o w s e l f - r e f e r e n c e c a n b e m o r e t h a n j u s t a l o g i c i a n ’s j o k e , le t’s e x a m i n e h o w it w o r k s in p r o g r a m m i n g . R e c all t h a t j i s i in p j e c a s c a d e c o u ld n o t b g . u s e d ■if we.,did- n ot.-k no w . h o w ..m a n y j_ t ^ n 5 fp r m a ti o n s _ w e re . n e e d e d .a h e a d of tim e. T h e s a m e p r o b l e m o c c u r r e d for t h e B a t a m m a l i b a a n c e s t r a l m o u n d s ; sin c e t h e first d e s c e n d a n t d i d n o t k n o w h o w m a n y w o u ld b e n e e d e d , t h e s y s te m h a s to allow for i t e r a t i v e resizing. W e a lso saw t h e possibility o f n e s t e d it e ra t iv e loops, illustrated by t h e t w o - l o o p d r a w i n g p r o g r a m fo r M o k o u l e k a r c h i t e c t u r e . B u t su p ­ pose we d i d n ’t k n o w h o w m a n y n e s t e d lo o p s we were g o in g t o n e e d ? I n t h e s a m e way t h a t t h e r e c u r s i v e c a s c a d e c o u l d n o t d e a l w i t h a n u n k n o w n n u m b e r o f iter- 135 S e e d sh a p e ( a ll tin e s are a c tiv e lin es) S e c o n d iteration FIGURE 8 . 1 3 F r a c t a l s in E t h i o p i a n p ro c e s s io n a l c ro sse s (a) Fractal simulations for Ethiopian processional crosses through three iterations(b) Ethiopia converted to Christianity in 333 c.ti., and in the thirteenth century King Lalibela directed the construction of churches to be cut from massive rocks in one of the mountain regions. T he church of St. George {at right) shows a triple iteration of nested crosses: (a, a l l Ethiopian processional crosses f r o m P o r t l a n d Museum in Oregon; pliotos c o u r t e s y o f Cs ilia P e r c z e l , b, photo hy Georg Gerstcr.) Recursion FIGURE 137 8.14 T h e l o t t t s i c o n in a n c i e n t E g y p t i a n cosm ology In the origin story of ancient Egypt the lotus flower was often used as an image of the unfolding of the universe, its petals-within-petals signifying the expansion of scales. This is a very stylized representation used in the capitals of columns in temples. (From F ourier 1 8 2 1 .) a t i o n s , n e s t e d i t e r a t i o n h a s t r o u b l e w i t h a n u n k n o w n n u m b e r o f l o o p s . 10 H e r e is w h e r e s e l f - r e f e r e n c e c a n h e l p o u t . A n e x a m p l e o f s e l f - r e f e r e n c e in p r o g r a m ­ m i n g is il l u s t r a t e d for t h e D o g o n p o t s t a c k in fig u re 8 .1 5 . W e k n o w t h a t t h e D o g o n p o t s t a c k c a n b e d r a w n w i t h a s i n g le i t e r a t i v e l o o p — it d o e s n o t re q u ire s e l f - r e f e r e n c e . B u t t h e ta s k c a n be a c c o m p l i s h e d by s e l f - r e f e r e n c e , a n d w e m i g h t s i m d a r l y ask if t h e r e a re 'c a s e s o f s c a l in g in A f r i c a n d e s i g n s in w h i c h s e l f - r e f e r e n c e p la y s a ro le , re g a r d le ss o f w h e t h e r it is re q u i r e d . In. E u r o p e a n h i s t o r y , s e l f - r e f e r e n c e b e g i n s w i t h t h e s t o r y o f E p i m e n i d e s o f C r e t e , t h e “ li a r ’s p a r a d o x . ” S i m i l a r u t i l i z a t i o n s o f n a r r a t i v e se lf-r e fe re n c e t o c r e ­ a t e u n c e r t a i n t y c a n be f o u n d in c e r t a i n A f r i c a n t r i c k s t e r s to rie s . F o r e x a m p l e , ' iii a n A s h a n t i s t o r y o f A n a n s e ( w h o b e c a m e “A u n t N a n c y ” in A f r i c a n A m e r i ­ c a n f o l k l o r e ) , a m a n n a m e d “ H a t e s - t o - b e - c o n t r a d i c t e d ” is t r i c k e d i n t o c o n ­ tr a d ic tin g h im self. P e ito n (1 9 8 0 , 5 1 ) n o te s t h a t th e a p p lic a tio n o f such s e l f - r e f e r e n t i a l p a r a d o x is a t h e m e in m a n y A n a n s e st o r ie s : “T h u s A n a n s e r e j e c ts t r u t h in f a v o r o f ly in g , b u t o n l y for t h e s a k e o f s p e e c h ; t e m p e r a n c e in fav o r o f g l u t t o n y for t h e s a k e o f e a t i n g ; c h a s t i t y in f a v o r o f la s c i v i o u s n e s s for t h e s a k e o f s e x ." T h e f o l l o w i n g t a l e is n o t n e a r l y a s sp a r s e b u t c a r r ie s t h e f l a ­ v or o f s e l f - r e f e r e n t i a l p a r a d o x q u i t e w ell: O n e of the most com m on of all stories in Africa describes the e n c o u n te r of a man and a hum an skull in the bush. A m ong the N u pe of Nigeria, for instance, they tell of the h u n te r who trips over a skull while in pursuit of game and exclaim s in w o n d e rm e n t, “W h a t is this? How did it get here?" “Talking A fric a n fractal mathematics FIGURE 8 . 1 5 D r a w in g th e D o g o n p o t s ta c k by s e lf-re fe re n c e T h e symbolism of the stacked pots represents th e b reath of life, w ith in th e fetus, w ithin th e wotnB.' W e have aready seen h o w this can be drawn using an iterative loop; now let’s see h ow it ca n be drawn using self-reference. Suppose we h av e a ro u tin e th a t can draw the semicircle of th e p o t given a diameter. Procedure D R A W -P O T If size = m inim um , draw a lid. Else Draw a circle of size diam eter S h rin k diam eter by V3 D R A W -P O T End of "else" clause End o f procedure N otice th a t this procedure first checks to see if we are at the smallest dia m e ter possible. If not, it draws a pot, shrinks the diam eter value it by 2/ss, and th e n calls itself— an application of self-reference. N ow th e program has to execute n D R A W -P O T procedure again. T h e recursion will “bo tto m -o u t" w hen it finally draws a lid. T h e program th e n skips to the “End of procedure" line an d can finally pop hack up to th e place it left off after ex ecutin g the previous D R A W -P O T call. b r o u g h t m e h e r e , ” th e sk u ll re p lie s. N a t u r a l l y t h e h u n t e r is a m az ed a n d q u ic k ly ru n s b ac k to his village, e x c la im i n g a b o u t w h a t h e h a s fou nd . E v e n ­ tu ally t h e kin g h ea rs ab o u t th is w o n d e r an d d e m a n d s t h a t th e h u n t e r ta k e h im / to see it. T h e y re t u r n to th e p la c e in t h e b u s h w h e r e t h e sk u ll is s ittin g , a n d t h e h u n t e r p o in t s ir o u t to h is k in g , w h o n a t u r a l l y w a n t s to h e a r t h e s k u ll’s message. T h e - h u n t e r r e p e a t s .t h e question.:..“H y w d i d you g e t h e re ? ” h u t th e skull says n o t h i n g . T h e k ing, angry now, accu se s t h e h u n t e r o f d e c e p t io n , an d o rders his h e a d c u t off o n th e sp ot. W h e n t h e royal p a rty d e p a rt s , t h e skull speaks o u t, asking th e h u n t e r “W h a t is this? H o w d id you get h e re ? ” T h e he ad replies, “T a lk i n g b ro u g h t m e h e r e ! ” ( A b r a h a m s 1983, 1) S e lf -re f e re n c e is -also visually p o rtra y e d in s o m e A f r i c a n de sign s. Figure 8. i6n s h o w s a n o t h e r a b b i a c a r v i n g fr o m C a m e r o o n , s e e n a ls o in t h e n e s t e d fish e a rl ie r in th i s c h a p t e r . B u t th is a b b ia c a r v i n g is a n ic o n for itse lf— it is a n a b b ia o f a b b ia . A c c o r d i n g to t h e C a m e r o o n C u l t u r a l Review ( i n s i d e c o v e r , J u n e 1 9 7 9 ), its m e a n ­ i n g is “r e p r o d u c t i o n . " A n o t h e r e x a m p l e o f s e l f - r e f e r e n c e f r o m C a m e r o o n is s h o w n in figure 8 . 1 6 b , a life-size b r o n z e s t a t u e o f t h e k i n g o f F o u m b a n . H e r e we s e e t h e k i n g s m o k i n g h is p ip e , t h e b o w l o f w h i c h is a fig u re o f t h e k i n g s m o k ­ in g h is p ip e , t h e h o w l o f w h i c h is a figure o f t h e k i n g s m o k i n g h i s p ip e . L ik e t h e K e l l o g g ’s c o r n f l a k e s b o x d e s c r i b e d e a rl ie r , t h e v is u a l s e l f - r e f e r e n c e i n s t a n t l y leads to in f in ite regress. B u t it c o u ld be m o r e t h a n ju st h u m o r in t h e b ro n z e sculp- Recursion 139 tu r e . S i n c e t h e p ip e is a w e l l - k n o w n s y m b o l o f ro y a l p re s tig e in F o u m b a n , it m ay be t h a t t h e a rtisa n s w ere m a k i n g p u rp o s e fu l u se o f t h e in fin ite regress: “T h e k i n g ’s p o w e r is n e v e r - e n d i n g . ” Figure 8 .1 6 c s h o w s a B a m a n a h e a d d r e s s , t h a t is, a s c u l p t u r e w o r n o n t h e h e a d d u r i n g c e r e m o n i e s . Fag g ( 1 9 6 7 ) s u g g e s ts t h a t t h i s e n a c t s s e l f - r e f e r e n c e : a h e a d d re s s of a p erson w e a rin g a h e a d d re s s of a p erson w earin g a h e ad d ress. O t h e r s (cf. A r n o l d i 1 9 7 7 ) h a v e d e s c r i b e d t h i s as a s y m b o l o f f e r til it y sp irits, b u t th e tw o in te r p r e ta tio n s m ay n o t be m u tu a lly ex c lu s iv e . R e tu r n in g to th e FIGURE 8 . l 6 S e l f 't e f e r e n c e in A f r i c a n i c o n s 1) The abbia carvings from C am e ro o n show a wide variety o f images, but this abbia carving is icon lor itself— it is an abbia of abbia. (b) A life-size bronze statue of the king of Foumban. ere wc sec the king smoking his pipe, .die bowl of w hich is a figure of the king sm oking his pipe. Bamana headdress. Jwu'ing Iwsed on a b b i a p i c t u r e d on iu University Museum o f A f r i c a n th e c o v e r o f A r t.) Cameroon Cultural Review, 1979; c, photo courtesy 140 A fr ic a n fra c ta l ?nacliema£ic5 B a m a n a ’s c lo s e c u l t u r a l r e l a t i v e s t h e D o g o n , w e se e s e l f - r e f e r e n c e s u g g e s te d by O g o t e m m e l l i ’s d e s c r i p t i o n o f h o w t h e e i g h t h a n c e s to r, “w h o w as W o r d itself,” was a b le to use W o r d ( t h a t is, t h e b r e a t h o f life) t o s e l f - g e n e r a t e i n t o ..the n e x t i t e r ­ a t i o n o f h u m a n i t y . I n e x a m i n i n g t h e s e lf-s im ila r it e r a t io n s * o f t h e D o g o n m o t h e r a n d c h i l d in fi g u r e 8 . 1 1 c , w e n o t e d a s t r u c t u r a l c h a r a c t e r i s t i c t h a t c a n b e e x p re s s e d in t h e p h r a s e “ a n e w o n e b e g in s b e f o r e t h e o l d o n e e n d s . " T h i s w o u ld also d e scrib e t h e s t r u c t u r e o f t h e pip e in t h e s t a t u e o f t h e k i n g o f F o u m b a n , w h i c h w e k n o w t o b e e x p l i c i t l y s e l f - r e f e r e n t ia l . P e r h a p s t h e s e l f - r e f e r e n t i a l v e r s i o n o f t h e D o g o n p o t s t a c k w a s t h e c o r r e c t o n e a fte r all. I c o n i c r e p r e s e n t a tio n s o f r e c u r s io n T h e a b b i a o f a b b i a , as a s y m b o l o f ( ^ e p r o d u c t i o n , ’.”>is m o r e t h a n j u s t a n a p p l i ­ c a t i o n o f s e l f - r e f e r e n c e ; it r e p r e s e n t s ~ t h e c o n c e p £ , i t s e l f . I f r e c u r s i o n is r e a l l y a c o n s c i o u s ( t h a t is, s e l f - c o n s c i o u s ! ) a s p e c t o f A f r i c a n k n o w l e d g e s y s t e m s , t h e n w e s h o u l d e x p e c t s u c h r e p r e s e n t a t i o n s , r a t h e r t h a n j u s t i n s t a n c e s in w h i c h t h e c o n c e p t is a p p l i e d . F i g u r e 8 . 1 7 a s h o w s t h e a p p l i c a t i o n o f r e c u r s i o n in t h e t r a - fig u re 8 .1 7 R e /lu x (a) This sketch from the notebook of a nineteenth-century ethnographer in southern Senegal shows an indigenous apparatus for the distillation of liquor from palm wine using a scaling cascade. (b) A ncient Egyptian alchemists drew this snake symbol to represent their reflux technique. A tube comes out of a heated pot and reenters after cooling. This cyclic refinement was used in the creation of dyes and perfumes, bur it also symbolized the alchemists’ goal of refinement of the human soul. (a, pfioto c o u r t e s y /FAN, D a k a r - , b, drawing b a s e d on T o y lor 1 930.) Recursion 141 d i t i o n a l d i s t i l l a t i o n o f p a l m w i n e i n t o l i q u o r in t h e C a s a m a n c e r e g i o n o f S e n e g a ! . S u c h d i s t i l l a t i o n t e c h n i q u e s w e r e d e v e l o p e d t o s o p h i s t i c a t e d le v e ls in a n c i e n t E g y p t, w h e r e t h e p r o c e s s b e c a m e a n i t e r a t i v e l o o p w h i c h m o d e r n c h e m i s t s c a ll a “re f lu x ” a p p a r a t u s . F ig u r e 8 . 1 7 b shovirs-the i c o n i c r e p r e s e n t a t i o n o f t h e reflu x sy s te m in t h e o ld e s t k n o w n a l c h e m i c a l w r i t i n g s (first c e n t u r y c .e .), w h i c h a r e a t t r i b u t e d t o M a r i a ( w h o w r o t e u n d e r t h e n a m e o f M i r i a m , s is te r o f M o s e s ) , C l e o p a t r a ( n o t t h e f a m o u s q u e e n ) , C o m a r i u s , a n d t h e m y t h i c figure o f H e rm e s T rism eg estu s ( T h o t h ) . T a y lo r (1 9 3 0 ) n o te s t h a t a lth o u g h th ese w e r e w r i t t e n in G r e e k , “t h e re l ig i o u s e l e m e n t . . . l i n k s t h e m t o E g y p t r a t h e r t h a n t o G r e e c e , ” a n d h e s u g g e s ts t h a t t h e m o s t li k e ly o r i g i n is f r o m t h e t r a d i ­ t i o n s o f t h e a n c i e n t E g y p ti a n p r i e s t h o o d . 11 I n t h e s e w r i t i n g s w e f i n d t h e re flu x i c o n a s s o c i a t e d w i t h t h e a p h o r i s m “ as a b o v e , so b e l o w , ” r e c a l l i n g t h e selfs i m i l a r s c a l in g c o s m o lo g y w e h a v e s e e n in s u b - S a h a r a n A fric a , as well as its lin k s t o t h e r e c u r s i o n o f s e l f - f e r t i l i z a t i o n . 1^ O f c o u r s e , o n e c a n go t o o far in a t t r i b u t i n g li n k s b e t w e e n a n c i e n t E gy pt a n d s u b - S a h a r a n A f r i c a (see O r i t z d e M o n t e l l a n o 1 9 9 3 ; M a r t e l 19 94; L efk o w itz 1 9 9 6 ). T h e r e is goocf_e_yidence for th e o r i g in s o f t h e E g y p ti a n b a s e - tw o a r i t h m e t i c sy s te m fro m s u b - S a h a r a n A fric a , a n d for t h e p e r s i s t e n t use o f r e c u r s io n in k n o w l ­ e d g e sy stem s acro ss t h e A f r i c a n c o n t i n e n t . B u t it w o u ld b e u n w is e to a ss u m e t h a t o n e c a n a t t r i b u t e m o r e sp ecific f e a t u r e s t o d if fu s i o n . I n p a r t i c u l a r , it is h ig h l y u n l i k e l y t h a t t h e s a m e fig u re o f a s e r p e n t b i t i n g .its ta i l, a p p e a r i n g as a n ic o n for t h e g od D a n in t h e v o d u n r e l i g i o n o f B e n i n (fig. 8 . 1 8 a ) c o u ld h a v e d e r i v e d f r o m tlie E g y p t i a n i m a g e , o r v i c e v e r s a . A s w e s h a l l s e e , t h e m e a n i n g o f t h e v o d u n ic o n h a s n o t h i n g to d o with, t h e E g y p t i a n reflux c o n c e p t . In A u g u s t 1994, t h a n k s to t h e a id o f M a r t i n e d e S o u s a ( o n e o f t h e A f r i c a n d e s c e n d a n t s o f t h e fa m e d F r a n c i s c o d e S o u z a ) , 1 was g r a n t e d a n i n t e r v i e w w ith t h e c h i e f o f t h e D a n t e m p l e in O u i d a h , B e n i n . B o t h t h e c h i e f a n d his wife w e re q u i t e r e s p o n s iv e to my i n t e r e s t in t h e g e o m e t r i c fe a tu re s o f D a n ’s r e p r e s e n t a t i o n s a n d identified th e sinusoidal ico n in ir o n (fig. 8. j 8 b ) as " D a n a t w ork in t h e w o rld ,” p o i n t i n g o u t t h a t h e c r e a t e s o r d e r in w i n d a n d w a te r. T h e c y c li c D a n w as m o r e a b s t r a c t , e x i s t i n g in a d o m a i n w h e r e h e w as in c o m m u n i c a t i o n w i t h o t h e r gods o f v o d u n . M a u p o i l ( 1 9 8 1 , 7 9 ) a is o f o u n d t h a t D a n ( D a n g b e ) w as t h e r e “to ass u re t h e r e g u l a r i z a t i o n o f t h e fo r c e s ,” a n d B li e r ( 1 9 9 5 ) s u m m a r i z e s his ro le as “p o w e r s o f m o v e m e n t t h r o u g h life, a n d n a t u r e ’s b le s s in g s .” R e g u l a r p h e n o m e n a in n a t u r e — t h e p e r i o d i c a s p e c t s o f w e a t h e r , w a t e r w a v e s , b i o l o g i c a l c y c le s, e t c . — a re a t t r i b u t e d t o t h e a c t i o n o f D a n . T h e r e l a t i o n b e t w e e n t h e u n d u l a t o r y D a n “a t w o r k in t h e w o r l d ” a n d _ th e c i r c u l a r fo rm o f D a n as a m o r e a b s t r a c t s p i r i t u a l f o r c e m a p s n e a t l y o n to t h e d i f ­ f e r e n c e b e t w e e n t h e s i n u s o i d a l w a v e s w e s e e in s p a c e a n d t i m e — w a v e s in noise {external temperature changes) input (desired temperature) T h e t h e r m o s t a t t h a t re g u l a t e s t e m p e r a t u r e in a h o u s e is a n e g a t i v e f e e d b a c k l o op . T h e w o r d “ n e g a t i v e " is used b e c a u s e we s u b t r a c t t h e c u r r e n t r o o m t e m p e r a t u r e fro m t h e d e s i r e d t e m p e r a t u r e s e t by t h e t h e r m o s t a t c o n t r o l . O v e r t i m e t h i s wi ll t e n d t o p r o d u c e cy cles o f h e a t a n d co ld . noise {road bumps) D r i v i n g a c a r c a n als o be m o d e l e d by a n e g a t i v e fe e d b a c k lo o p . T h e d r i v e r a t t e m p t s t o stay in t h e c e n t e r o f t h e la n e , a n d will c o r r e c t t o a d j u s t for b u m p s . A g a i n , g i v e n e n o u g h b u m p s ,' w e will t e n d to see cy cles o f s w e rv i n g t o g e t b a c k to t h e c e n t e r . c FIGURE 8 . l 8 T h e v o d u n god D a n I n t h e v o d u n r e l i g i o n o f B e n i n , t h e s n a k e g o d D a n r e p r e s e n t s t h e c y c l i c o r d e r o f n a t u r e . D a n 's s h a p e r e fle c ts t h i s i d e a in t w o w ay s . A s a n a b s t r a c t f o r c e , h e is r e p r e s e n t e d as a f e e d b a c k l o o p ( a ) . A s a c o n c r e t e m a n i f e s t a t i o n , h i s b o d y is a l w a y s o s c i l l a t i n g in a p e r i o d i c w a v e ( b ) . T h i s s a m e id ea o f a p e r i o d i c t i m e s e r i e s f r o m c y c l i c f e e d b a c k is a l s o u s e d in W e s t e r n m o d e l s o f n a t u r e (c ). ( a , p/ioto courtesy J F A N , D akar.) R ecursion w a t e r a n ti c ir r u s c lo u d s , d a il y f l u c t u a t i o n s in h e a t a n d li g h t, t h e b i a n n u a l ra i n y seaso ns, e tc .— a n d th e a b s t r a c t id e a o f a n i t e r a t iv e l o o p t h a t g e n e ra te s th e s e w a v e ­ fo rm s. T h e a s s o c i a t i o n c a n b e d e r i v e d f r o m i:he k i n d o f e m p i r i c a l o b s e r v a t i o n o n e . g e ts in e v e r y d a y o c c u r r e n c e s . A d o p s i d e d w h e e l w ill p r o d u c e u n d u l a t o r y t r a c k s in s a n d ; f r i e n d s w h o p e r i o d i c a l l y g i v e gifts a r e in a “c y c le o f e x c h a n g e , ” a n d so f o r t h . W h a t d id t a k e g r e a t i n s i g h t a n d i n t e l l e c t u a l lab o r, h o w e v e r , was t h e re l ig i o u s p r a c t i t i o n e r s ’ g e n e r a l i z a t i o n o f s u c h o b s e r v a t i o n s i n t o s p e c ific , a b s t r a c t , u n iv e r s a ll y a p p l i c a b l e c a t e g o r i e s , r e p r e s e n t e d by i c o n s w i t h t h e a p p r o ­ priate g eo m etric stru ctu re. ^ T h e m a t h e m a t i c a l e q u i v a l e n t s i n n o n l i n e a r d y n a m i c s a r e li m i t c y c le s a n d p o i n t a t t r a c t o r s — t h e re s u lts o f w h a t e n g i n e e r s c a ll a “n e g a t i v e f e e d b a c k l o o p . ” W e h a v e a l r e a d y s e e n s u c h c h a r a c t e r i z a t i o n s in c e l l u l a r a u t o m a t a a n d o w a r i, w h e r e s p a t i a l p a t t e r n s r e m a i n b o u n d e d w i t h i n a c y c le o r fr o z e n in a s t a t i c p a t ­ te rn . Figure 8 .1 8 c sh ow s s o m e c o m m o n p l a c e e x a m p l e s o f n e g a ti v e fee d b a c k loops, a n d h o w th e y a c t to k e e p th e b e h a v io r o f system s b o u n d e d o r stabilized, e v e n in che p r e s e n c e o f n oise . B u t che v o d u n s y s te m w o u ld n o t b e c o m p l e t e if it c o u ld o n l y a c c o u n t fo r r e g u l a r i t y — w h a t c a u s e s d e v i a t i o n in t h e first p l a c e ? H e n c e t h e ro le o f L g g b a, g o d o f c h a o s . F igu re 8 . 1 9 a s h o w s a n o t h e r i r o n i c o n , t h e fo r k e d p a t h o f L e g b a , “g o d o f t h e c r o s s r o a d s . ” A s e x p l a i n e d to m e by K a k e S . A l f r e d , 1 - ; I------ ------- -..w a d i v i n a t i o n p r i e s t o f v o d u n i n C o t o n o u , B e n i n , L e g b a is r e p r e s e n t e d by t h e fork b e c a u s e " t h e a n s w e r c o u l d b e yes o r n o ; y o u d o n ’t k n o w w h i c h p a t h h e wilt t a k e . ” F or d i v i n a t i o n , in w h i c h a “p a t h ” ( q u e s t i o n ) is o f t e n p u r s u e d fo r f u r c h e r q u e s t i o n s , t h e im a g e b e c o m e s o n e o f e n d l e s s b i f u r c a t i o n s . A t t h e P a l a i s R o y a l in P o r t o N o v o , B e n i n , I w a s t o l d t h a t t h e s h r i n e t o L e g b a w as p l a c e d a t t h e t h r e s h o l d b e c a u s e h is f o r c e w a s so d i s r u p t i v e t h a t it w o u ld u n d o b o c h g o o d a n d e v il, c r e a t i n g a p u r i f i c a t i o n a t the-er.trav.-ce-; K-akc a ls o e x p l a i n e d t h a t w h i l e t h e m u sic o f D a n was slow a n d re g u la r, t h e m u s ic o f L e g b a w as b o t h fast a n d s lo w — sig n ify in g h i s u n p r e d i c t a b l e n a t u r e — a n o b s e r v a t i o n 1 was a b l e to c o n f i r m by r e c o r d i n g t h e d r u m m i n g t h a t w a s u se d t o c a l l e a c h g o d a t t h e t e m p l e o f D a n in O u i d a h . 1-5 A s t h e c o n v e r s e to D a n , t h e b i f u r c a t i n g u n c e r t a i n t i e s o f L e g b a are like a p o s i t i v e fe e d b a c k l o o p , a m p l i f y i n g d e v i a t i o n a n d n o is e (fig. 8 . 1 9 b ) . C o n t r a s t s b e t w e e n a n e g a t i v e fe e d b a c k lo o p , c r e a t i n g stability, a n d t h e p o s­ itive f e e d b a c k o f u n c o n t r o l l e d d i s o r d e r a r e a l s o f e a t u r e d in t h e .ico n ic c a r v i n g s o f t h e Battle. V o g el ( 1 9 7 7 , 5 3 ) n o t e s t h a t t h e B a u le c h i e f is c h o s e n by c o n s e n T iisT a h d t h a t in all i m p o r t a n t d e c i s i o n s h e s e r v e s as m e d i a t o r in p u b li c m e e t i n g s r a t h e r t h a n as a n a u t o c r a t . T h e B a u le c a r v i n g in figure 8 .2 0 a s h o w s t w o c a i m a n s (relativ es o f th e a llig a to r) b i t i n g e a c h o t h e r ’s tails. It is said to r e p r e s e n t t h e c h i e f a n d t h e p e o p l e in b a l a n c e — if o n e b i t e s , t h e o t h e r will b i t e b a c k , k n i c e l y recalls th e k in d s o f n e g a tiv e fe e d b a c k lo op m o d e l s th a t a re o fte n p ro p o s ed in W est- 143 FIGURE 8.19 L egba (a) T h e v o du n god Legba represents th e forces o f disorder. Vodun d iv in a tio n priests explain this icon as'the p a th to the future: w ith Legba there is no way to’h n o w w h ic h p a th will be caken. S in c e o n e crossroad leads to another, th e resulting image is o n e o f bifurcating u n know ns, th e u n certain ty multiplying with ea c h crossroad. n oise (road bum ps) output (n ew road position In contrast to negative feedback, which wili help stabilize a system, positive feedback will destabilize it. A drunken driver, for example, can overshoot the center line and create increasingly large oscillations, eventually running off the road. Here we see positive feedback in an arms race. R ecursion FIGURE 8 .2 0 F e e d b a c k loops in B a u l e ic o n o g r a p h y (a) This Baule carving shows two crocodiles biting e ac h other’s tails. It is a symbol showing the chief and the people in equal power, the idea of social forces in a cycle of balance, (b) Baule door. Holas (1952, 49-50) describes this as a c i r c u i t f e r i n e of f e c o n d i t e (closed tircuir of fecundity); Soppeiasa (1974) and Odica (ii;71) identify these animals as symbols of ‘‘increase.’’ (a,md b, [iliots) courtesy of 1FAN, Dakar.) orn p o li ti c a l th e o r y , h u t t h i s f l o w c h a r t is a p u r e l y in d i g e n o u s i n v e n t i o n . S o , coo, is th e B au le p o s i t i v e f e e d b a c k l o o p o f figure 8 . 2 0 b , s h o w i n g t h a t “ p o w e r c r e a t e s the a p p e t i t e for m o r e p o w e r ”— l i t t l e fish a r e e a t e n by b ig g e r fish, w h o t h e n b e c o m e e v e n b ig g e r fish, T h e f i s h - w i t h i n - f i s h a b b i a f r o m C a m e r o o n we saw earlier m ay h a v e h a d s i m i l a r c o n n o t a t i o n s . [C o n c lu sio n R ecu rsio n c_giibe--found. in a l m o s t e v e r y c o r n e r o f A f r i c a n m a t e r i a l c u l t u r e a n d design, fr o m c o n s t r u c t i o n t e c h n i q u e s t o e s t h e t i c d e s i g n , a n d in c u l t u r a l r e p r e ­ s e n t a t i o n s fr o m k i n s h i p t o c o s m o l o g y . M o s t o f t h e s e a r e s p e c i f i c e n o u g h to allow us to d i s t i n g u i s h b e t w e e n t h e first t w o ty p es o f r e c u r s i o n — c a s c a d e v ersu s I 45 A fr ic a n fra c ta l mat/icniatics i t e r a t i o n — a n d in s o m e c a s e s t h e t h i r d t y p e , s e l f - r e f e r e n c e , is a ls o m a d e e x p l i c i t by t h e i n d i g e n o u s k n o w l e d g e s y s te m . W e h a v e s e e n s e v e r a l c a s e s i n w h i c h t h e i t e r a t i v e lo o p s a r e n e s t e d , b u t t h e s e a r e ra r e ly m o r e t h a n t w o lo o p s d e e p , so it w o u l d n o t a p p e a r t h a t t h e a p p l i c a t i o n o f s e l f - r e f e r e n c e is t n o t i v a t e a by t h e c o m ­ p le x ity o f th e c o m p u t a t i o n . T h e o n l y p o t e n t i a l e x c e p t i o n is t h e c o s m o lo g i c a l n a r ­ r a t i v e o f t h e D o g o n , a n d t h i s n a r r a t i v e is t o o v a g u e t o s e r v e as a m a t h e m a t i c a l f o u n d a t i o n . T h e r e is, h o w e v e r , a n o t h e r r o u t e t o t h e l i m i t s o f c o m p u t a t i o n . A s w e will find in c h a p t e r 10, t h e c o m b i n a t i o n o f n e g a t i v e a n d p o s i t i v e f e e d b a c k in d ic a te d , by c e r t a i n r e c u r s i o n icons, p r o v i d e s / a n o t h e r p a t h t o t l ^ . h e l g h t s . o f c o m ­ p u t a t i o n a l c o m p l e x i t y , o n e we w ill e x p l o r e in d e t a i l . B u t first, w e n e e d t o t a k e a 's K o rt’d e t o u r t h r o u g h ~ m f i n ky. CHAPTER -Infinity- •T h e first t i m e 1 s u b m it te d a j o u r n a l a rt ic l e o n A f r i c a n fractals, o n e re v ie w e r replied t h a t A f r i c a n s c o u l d n o t h a v e “ t r u e ” f r a c ta l g e o m e t r y b e c a u s e t h e y la c k e d t h e a d v a n c e d m a t h e m a t i c a l c o n c e p t o f in finity. O n t h e o n e h a n d , t h a t r e v i e w e r was w ro n g a b o u t fra c ta ls a t a p r a g m a t i c le v e l. If h e o r s h e saw a fra ctal o n a c o m p u t e r s c r e e n it w o u ld b e t a k e n as a “ t r u e ” e x a m p l e , a n d in f a c t n o p h y s ic a lly e x i s t i n g fractal is i n f i n i t e in its scales; a t b e s t i t will h a v e t o b o t t o m o u t i n t o s u b a t o m i c p articles. O n t h e o t h e r h a n d , it raises a n i n t e r e s t i n g q u e s t i o n . I n f in ity h a s b e e n an i m p o r t a n t p a r t o f f r a c ta l m a t h e m a t i c s in E u r o p e ; so h o w d o e s t h a t c o m p a r e to th e use o f in f in ity in A fri c a ? T o t h e a n c i e n t G r e e k s , in f in i ty w as a s s o c i a t e d w i t h w h a t t h e y t h o u g h t (j>f as t h e h o r r o r s o f j n f i n i t e r e g r e s s . A r i s r o t l e t a m e d t h i s p r o b l e m b y r e d e f i n i n g infinity: it w as a li m i t t h a t o n e c o u l d t e n d t o w a r d , b u t it w as n o t c o n s i d e r e d to he a l e g i t i m a t e o b j e c t o f m a t h e m a t i c a l i n q u i r y i n itself. M o s t E u r o p e a n m a t h e ­ m a t ic i a n s k e p t to t h i s d e f i n i t i o n u n t i l t h e C a n t o r s e t, E u r o p e ’s first fr a c ta l , c r e ­ ate d t h e p r o p e r d e f i n i t i o n o f a n i n f i n i t e s e t , t h u s a l l o w i n g in fin ity its e lf t o b e c o n s i d e re d . W e w ill d is c u ss t h i s in m o r e d e t a i l in c h a p t e r 13, b u t fo r n o w it is sufficient to n o t e t h a t t h i s d i s t i n c t i o n d o e s n o t s h a p e A f r i c a n c o n c e p t s o f i n f i n ­ ity. M a n y A f r i c a n k n o w l e d g e s y s t e m s using^ip i m i t y in t h e s e n s e o f a p ro g r e s s i o n w i t h o u t li m i t d o n o t h e s i t a t e to r e p r e s e n t it w i t h i c o n i c s y m b o ls s u g g e s tin g 147 148 A fr ic a n fra c ta l tnmhematics “t h e in f in i te " in its C a n t o r u m s e n s e as a c o m p l e t e d w h o l e . T h i s is h y n o m e a n s a m ore so p h istic a te d o r ela b o ra te d d e fin itio n th a n th a t o f p r e - C a n to r ia n E u ro ­ p e a n m a t h e m a t i c s ; i t is r a r e ly l i n k e d to m u c h m o r e t h a n e i t h e r a . n a r r a t i v e o r a g e o m e t r i c v is u a li z a ti o n . B u t far f r o m b e i n g n o n e x i s t e n t , t h e s e c u l t u r a l l y sp ecific r e p r e s e n ta tio n s sh o w a s tro n g e n g a g e m e n t w ith th e sam e c o n c e p ts t h a t c o u p led in f in i ty a n d fr a c ta ls in c o n t e m p o r a r y W e s t e r n m a t h e m a t i c s . T h e m o s t c o m m o n A f r i c a n v i s u a li z a ti o n s for in f in i ty a r e s n a i l s h e l ls . T h e ( B n lu b a ) fo r e x a m p l e , u s e s p ira l la n d s n a i ls (fig. 9 . 1 ) , a n d t h e r e i n itse t h e spiral e n d o f a sea s n a i l, w h i c h fo r m s a d r i n k i n g c u p t h a t c a n o n l y b e u se d by th e ch ief. U n l i k e t h e a n c i e n t G r e e k a s s o c ia tio n s w i t h t r o u b l i n g p a r a d o x a n d p a th o l o g y , th e Africa_n in f i n ite is typi c a l l y a p o s i ti v e a s s o c ia tio n , in th is c a s e t o in v o k e p ro s p e rity w i t h o u t e n d . I f j h e s e in f in i ty i c o n s w e r e o n ly m e a n t t o c o m m u n i c a t e t h i s d e s ire t h e y w o u l d j i t ' A r i s t o t l e ’s d e f i n i t i o n : a p ro c e s s w i t h o u t e n d . B u t t h e s p i r it u a l e l e ­ m e n t qf_these i c o n s .a d d s a n o t h e r r e q u i r e m e n t : t h e ic o n s n e e d t o c o n v e y t h e se nse t h a t th ex ar.e.d raw .in g o n t h e p o w e r o f in f in ity jt_self{ S n a i l s h e l ls a re used b e c a u s e o f t h e sc a lin g p ro p e r tie s o f t h e i r l o g a r it h m i c spirals; o n e c a n c le a rly see th e p o t e n ­ tial for th e spiral to c o n t i n u e w i t h o u t e n d d espite its c o n t a i n m e n t in a finite space— i n d e e d , it is o n l y b e c a u s e o f its c o n t a i n m e n t in a fi n it e s p a c e t h a t t h e r e is a sen se o f h a v i n g g a i n e d a c c e s s to o r g r a s p e d a t t h e in f in ite ) W e h a v e a lr e a d y s e e n a n o t h e r e x a m p l e o f a n in f i n i t y i c o n in t h e N a n k a n i a r c h i t e c t u r e d is c u s s e d i n c h a p t e r 2. T h e r e t h e c o i l s o f a s e r p e n t o f i n f i n i t e FIGURE 9 . J B a l u b a u s e o f s n a il s h e lls to sy m b o lize in /im ty Davidson (197 1, 120) describes this as a fertilit figure, and notes that the snail shells represent endless growth. ( CoUecfifW Tm;m: T zara, Pnrii; }>lv>ii> hy Elio! Eliso/on.) In fin ity l e n g t h , s c u l p t e d i n t o t h e h o u s e w alls, m a d e u s e o f che s a m e a s s o c i a t i o n b e t w e e n p r o s p e r it y w i t h o u t e n d , a n d a g e o m e t r i c l e n g t h w i t h o u t e n d . ^ h e c o n s c i o u s c r e a t i o n o f t h i s in f in ity c o n c e p t is m o r e c le a r t h a n in t h e c ase o f t h e sn a il sh ells, b e c a u s e o n e c a n n o t a c t u a l l y se e t h e i n f i n i t e c o ils 'tr f-th e s n a k e . A n d u n l i k e t h e n a t u r a l l y o c c u r r i n g sh e lls , t h e p a c k i n g o f t h i s i n f in i te l e n g t h i n t o a fin ite s p a c e ( t h e N a n k a n i d e s c r i b e it as “c o i l i n g b a c k o n i t s e l f i n d e f i n i t e l y ” ) c a n n o t be m i s ­ t a k e n for m e r e m i m i c r y o f n a t u r e ; it is r a t h e r t h e artifice o f f r a c t a l s ^ T h i s s n a k e ic o n d o e s n o t e x is t in is o la t io n ; we saw t h a t - t h e N a n k a n i m a p o u t a s c a l in g p r o ­ g r e s s io n t h a t p a s s e s ' t h r o u g h t h e i r a r c h i t e c t u r e , t h e za h in g a a n d t h e k itm p io , w h i c h p r o v i d e s a re c u r s iv e p a t h w a y to t h i s c o n c e p t o f in finity. In c h a p te r 8 we d iscu ssed th e M i ts o g h o a n d F an g ite r a tiv e m o d e l o f d e s c e n t . F e r n a n d e z ( 1 9 8 2 , 3 3 8 ) n o t e s t h e c o n t r a s t to C h r i s t i a n th e o l o g y : “T h e q u e s t i o n as t o w h e t h e r G o d w a s o n e o r m a n y m a y h a v e b o t h e r e d t h e m i s s i o n ­ aries i n t h e i r c o n t a c t s w i t h F a n g m o r e t h a n t h e F a n g t h e m s e l v e s . H o l d i n g C h r i s ­ ti a n b eliefs in t h e ‘U n c r e a t e d C r e a t o r ’ a n d ‘U n m o v e d M o v e r , ’ m i s s i o n a r i e s w ere c h a l l e n g e d by t h e ‘i n f in i te re g r e ss’ o f t h e g e n e a l o g i c a l m o d e l e m p l o y e d by t h e F a n g — t h e i r b e l i e f t h a t che G o d o f t h i s w o r ld is o n e o f a l o n g li n e o f g o d s a n d like m a n h a s h is o w n g e n e a l o g y .” f T h e F a n g th e o r y o f in f in ite regress is a c o m p l e t e , c o h e r e n t view ; it d o e s n o t n e e d f u r t h e r a m e n d m e n t , for t h e C h r i s t i a n th e o r y o f u n c r e a t e d c r e a t o r is n o m o r e free o f c o n t r a d i c t i o n — a n d p e r h a p s less s o A O f c o u r s e , as F e r n a n d e z h i m s e l f warns, o n e c a n n o t simply p ro c laim t h a t a p a rtic u la r A fric a n n a rr a tiv e is just a n o t h e r w o rk o f t h e o l o g y o r p h i l o s o p h y — or, f o r t h a t m a t t e r , m a t h e m a t i c s . R e c e n t w orks s u c h as M u d i m b e ’s In v e n tio n o f A fr ic a ( 1 9 8 8 ) h a v e s h o w n t h a t s u c h t r a n s l a t i o n s to specific E u r o p e a n d is c ip lin e s are alw ays p a rt ia l , hig h ly in t e r p r e tiv e , a n d in d a n ­ ger o f m i s r e p r e s e n t i n g t h e in d i g e n o u s view. Yet M u d i m b e is also res p e ctfu l o f th e w ork t h a t h a s b e e n d o n e . O f p a r t i c u l a r r e l e v a n c e h e r e a re h is c i t a t i o n s o f A f r i c a n th eo lo g ian E n g elb ert M v en g . Y M v e p g i n c l u d e d s e v e r a l n o t e s o n i n f i n i t y in his s t u d i e s o f t h e r e l a t i o n b e t w e e n t h e A f r i c a n a n d C h r i s t i a n view s. H is b e a u ti f u l te x t, L 'A rt d 'A /r(que N oire ( 1 9 6 4 ) , c o n t a ins d i a g r a m s (p p . 1 0 0 - - 1 0 3 ) s h o w i n g w h a t h e t e r m e d “r a d i a t i o n a m p l i f i c a t i v e ,” s c a l i n g p a t t e r n s in_Afncan_art_a_hd mus_i_c_that he_ i n t e r p r e t e d as r e p r e s e n t a t i o n s o f a t r a n s c e n d e n t a l p a t h to in fin ity . “U n e fois d e plus, n o u s d e c o u v r o n s q u e le m o u v e m e n t r y t h m i q u e , d a n s n o t r e a r t , n ’esc a u t r e c h o s e q u ’u n e c o u rs e v e rs 1'i n f i n i ” ( O n c e a g a i n , w e d i s c o v e r t h a t t h e r h y t h m i c m o v e ­ m e n t in o u r a r t is n o n e o t h e r t h a n t h e p a t h t o w a r d i n f i n i t y ) (p. 1 0 2 ). F a t h e r M v e n g w as a w o n d e r f u l i n s p i r a t i o n d u r i n g m y r e s e a r c h in C a m e r o o n , b o t h fo r his d e e p c u l t u r a l k n o w l e d g e as w ell as fo r his c o u r a g e o u s w o r k as a c ro s s - c u l tu r a l m e d ia to r. D u r i n g o u r last m e e t i n g we d is c u s s e d M u d i m b e ’s b o o k , a n d 1 p ro m is e d 149 A fr ic a n fra c ta l mathematics t o s e n d h i m a cop y- S h o r t l y a f t e r d o i n g s o a r e p l y c a m e f r o m t h e A m e r i c a n C u l t u r a l C e n t e r in Y a o u n d e M v e n g h a d b e e n m u r d e r e d “u n d e r s u s p i c i o u s c i r c u m s t a n c e s ”— a p p a r e n t l y t h e r e s u l t o f o p p o s i t i o n t o h i s c r o s s - c u l t u r a l a c t i v i s m . H e h a s f i n a l l y t a k e n t h e course vers V in f ini. ^ CHAPTER Com plexity IO In o r d i n a ry s p e e c h C ^ c o m p I g ^ ” j u s t m e a n s t h a t t h e r e is a lo t g o i n g o n . B u t for m a t h e m a t i c i a n s t h e t e r m is p r e c i s e l y d e f i n e d , a n d it g iv es us a n e w w ay to a p p r o a c h m a t h e m a t i c s in A f r i c a n m a t e r i a l c u l t u r e . In c h a p t e r 7 we saw h o w c e r ­ ta i n A f r i c a n s y m b o l i c s y s t e m s , l i k e t h e B a m a n a d i v i n a t i o n c o d e , c o u l d be g e n e r a t e d b y a r e c u r s i v e l o o p . S u c h n u m e r i c s y s te m s c l e a r l y t r a n s l a t e i n t o th e W e s t e r n d e f i n i t i o n s o f w h a t it m e a n s t o “c o m p u t e . ” B u t t h e t r a n s l a t i o n w as less c le a r for s o m e of t h e p h y s i c a ll y re c u r s iv e s t r u c t u r e s in A f r i c a n m a t e r i a l c u l t u r e . C a n a s y s t e m o f p h y s ical d y n a m i c s b e s a i d t o “c o m p u t e ” ? M a t h e m a t i c a l c o m ­ p le x i ty t h e o r y , w h i c h is b a s e d o n f r a c t a l g e o m e t r y , p r o v i d e s a w ay to m e a s u r e th e c o m p u t a t i o n e m b e d d e d in p h y s i c a l s t r u c t u r e s , r a t h e r t h a n j u s t s y m b o l sy s­ tem s. By l o o k i n g a t A f r i c a n m a t e r i a l c u l t u r e in t h e f r a m e w o r k o f c o m p le x ity ^ th eory, we c a n b e t t e r im d e rs ta n .d j.h e . p r e s e n c e o f f r a c ta l g e o m e t r y as a n A f r i c a n ) j k n o w ledg e sy stem .. A n alo g c o m p u tin g By th e mid.-,.i_p6ps it w as c l e a r t o m a n y r e s e a r c h e r s c h a t d ig ita l c o m p u t e r s w o u ld be th e w a v e o f t h e fu tu re . B u t be fo re t h e n , a n a l o g c o m p u t e r s h e l d t h e i r o w n , a n d th ey m a y y e t m a k e a c o m e b a c k . I n d i g i ta l sy s tem s , i n f o r m a t i o n is r e p r e s e n t e d by A fr ic a n fra cta l m athematics p h y s i c a ll y a r b i tr a r y s y m b o ls. A s B a t e s o h ( 1 9 7 2 ) said , “T h e r e is n o t h i n g s9.yeni.sh a b o u t t h e n u m e r a l.$ e v .e n .” T h e g e o m e t r i c s t r u c t u r e o f a d i g i t a l s y m b o l h a s li t t l e o r n o t h i n g t o do. w i t h it s m e a n i n g , w h i c h is s i m p l y a s s ig n e d t o it-.-hy s o c i a l c o n ­ v e n t i o n . I n a n a l o g sy s te m s , t h e p h y s i c a l s t r u c t u r e o f t h e 'S i g n a l c h a n g e s in p ro -\ p o r t i o n t o c h a n g e s in t h e i n f o r m a t i o n it r e p r e s e n t s . 1 R a t h e r t h a n b e i n g arb itra ry , t h e p h y s i c a l s m ic .tu re is a d i r e c t r e f le c t io n o f its i n f o r m a t i o n . L o u d n e s s in h u m a n s p e e c h is a g o o d e x a m p l e o f a n a l o g r e p r e s e n t a t i o n . A s I g e t m o r e e x c i t e d , I s p e a k l o u d e r : t h e p h y s i c a l p a r a m e t e r c h a n g e s in p r o p o r t i o n t o t h e s e m a n t i c p a r a m e ter. T h i s is n o t tr u e for t h e d ig ital p arts o f sp e e c h , s u c h as t h e av erag e p i t c h (“ fo m a t f r e q u e n c y ”) o f e a c h w o rd . I n E n g l i s h t h e w o r d “c a t ” h a s a h i g h e r p i t c h t h a n t h e w o r d “d o g ,” b u t t h a t d o e s n o t in fe r a r e l a t i o n in m e a n i n g — in fa c t, t h e d if f e r e n c e is r e v e r s e d in S p a n i s h , s i n c e “g a t o ” h a s a lo w e r a v e r a g e p i t c h t h a n “p e r r o ." T h j s s a m e a n a l o g / d i g i t a l d i s t i n c t i o n o c c u r s in n e u r a j , s i g n a l s . . I n t h e frog r e t i n a , fo r e x a m p l e , s o m e n e u r o n s h a v e a firing r a t e in p r o p o r t i o n to t h e sp e e d o f sm all m o v - ^ i n g im ag es ( G r u s s e r a n d G r u s s e r - C o r n e h l s 1 9 7 6 ) . T h a t is, t h e fa s te r a fly m o v e s C acro s s t h e ey e, t h e f a s te r t h e p u ls e s o f t h e n e u r o n : a n a n a lo g s y s te m . A d i g i t a l e x a m p l e c a n b e fo u n d in th e_rnq tor n e u r o n s t h a t fling o p e n t h e crayfish claw. H e r e p a sp ecific firing p a t t e r n ( o f f - o t v o n - o f f ) s w i t c h e s t h e c l a w t o t h i s d e f e n s e re fle x ( ( W i l s o n a n d D a v is 1 9 6 5 ) . S o far w e h a v e o n l y e x a m i n e d h o w a n a l o g s y s t e m s c a n r e p r e s e n t i n f o r ­ m a t i o n ; fig u re to .1 s h o w s a s i m p l e e x a m p l e o f h o w a n a l o g c o m p u t i n g w o rk s . A l t h o u g h m o s t c o m p u t e r s c i e n t i s t s e v e n t u a l l y s e t t l e d o n d i g i t a l s y s te m s , a n a ­ lo g c o m p u t e r s w e r e q u i t e p o p u l a r u p u n t i l t h e 1 9 6 0 s. E v e n w h e n t h e y b e g a n t o d i e o u t as p r a c t i c a l m a c h i n e s , t h e r e w as a n i n c r e a s i n g j r w a r e n e s s t h a t m u c h o f o u r o w n b r a i n . p p e r a . t e s . b y a n a l o g c o m p u t i n g , a n d t h i s J e d some... s c i e n t i s t s t o w a r d t h e d e v e l o p m e n t o f w h a t .are. n o w c a l l e d .“ n e u r a l , n e t s ”— c o m p u t i n g d e v i c e s t h a t m i m i c t h e a n a l o g o p e r a t i o n s o f n a t u r a l n e u r o n s (fig. 1 0 .2 ) . By t h e m i d - 10 8 0s n e u r a l n e .ts .a n d r e l a t e d a n a l o g d e v i c e s h a d a c h i e v e d e n o u g h s u c c e s s ( a n d dig ital c o m p u te r s h a d ru n in t o e n o u g h b a rrie rs ) to b e g in to c o m p a r e th e t w o . T h e r e w a s a n o d d m o m e n t o f a n a l o g o p t i m i s m , w h e n a few b r a s h c l a i m s w ere m ad e a b o u t th e p o te n tia l su p erio rity o f a n a lo g c o m p u tin g (see D e w d n e y 1 9 8 5 ; V e rg is e t a l. 1 9 8 5 ) , b u t t h e s e a s s e r t i o n s w e r e e v e n t u a l l y p r o v e d i n c o r ­ re c t (B lu m , S h u b , a n d S m a le 1989; R u b e l 1 9 8 9 ). A s it tu r n s o u t, a n a lo g sys­ t e m s h a v e t h e s a m e t h e o r e t i c a l lim its to c o m p u t i n g as d ig i ta l s y s te m s .2 A l t h o u g h t h e s t u d i e s d i d n o t r e s u l t in r e l e a s i n g t h e k n o w n l i m i t a t i o n s , t h e y d id p r o d u c e a n e w f r a m e w o r k for t h i n k i n g a b o u t c o m p u t i n g in p h y s i c a l d y n a m ics: c o m p l e x i t y t h e o r y . Before th is tim e , m a t h e m a t ic i a n s h a d d e fin e d c o m p l e x i t y j n te rm s of ra n d o m n e ss, p rim arily based o n th e w o rk o f S o v ie t m a th e m a tic ia n A . N . C o m p le x ity *53 A n a l o g c o m p x ita tio n Dewdney (1985) shows a great variety of simple physical devices that demonstrate analog computing. This device, created by J. H. Luerh of the U.S. Metals Refining Company, solves the following optimization problem; a refinery must be located to minimize its costs. If transportation in dollars per mile of ore, coal, and limestone are values of O, C, and L, and disrances of these sources are o, c, and !, then the refinery should be located at the point where oO + cC + 1L is at ;t minimum. T he holes through which the strings pass are at the source locations, and the weights on the ends of the strings are proportionate to O, C, and L. T h e brass ring attached tc the strings ijitickly moves to the optimal location on the map. iC o u n e s y A . K. D e w d n e y ) K olm ogorov a n d A m e ric a n s G re g o ry C h a i t i n a n d R ay S o lo m o n o ff. In this d ef­ in i t i o n , t h e c o m R l e x jt jf.o f a s ig n a l ( e i t h e r a n a l o g o r d i g i t a l ) is m e a s u r e d by t h e le n g t h o f t h e s h o r t e s t a l g o r i t h m r e q u i r e d t o p r o d u c e it (fig. 1 0 .3 ) . T h i s m e a n s t h a t p e r i o d i c n u m b e r s ( s u c h as . 2 7 2 7 2 7 2 . . . ) will h a v e a lo w a l g o r i t h m i c c o m 1 plexity. E v e n if t h e n u m b e r is i n f i n i t e l y lo n g , t h e a l g o r i t h m c a n s i m p l y say, “W r i t e a d e c i m a l p o i n t fo l lo w e d by e n d l e ss r e p e t i t i o j i s o f ‘2 7 , ” ' o r e v e n j h o r t e r : “3 / 1 1 . " T r u l y r a n d o m n u m b e rs ( e .g ., a s t r i n g o f n u m b e r s p r o d u c e d by r o l l i n g d i c e ) w ill h a y e j _ h e J x i g h e s t a l g o r i t h m i c c o m p l e x i t y p o s s i b l e , s i n c e t h e i r o n l y a l g o r i t h m is_che n u m b . y . . i t s e l f — f o r . a n j n f i n i t e l e n g t h , y o u g e t i n f i n i t e c o m p l e x i t y . J n a n a l o g s y s t e m s a p e r i o d i c s i g n a l s u c h as t h e v i b r a t i o n fr o m a s i n g l e g u ita r s t r in g o r t h e r e p e t i t i v e s w in g s o f a p e n d u l u m w o u ld h a v e t h e lo w e st a l g o ­ r i t h m i c c o m p l e x i t y , a n d r a n d o m n o i s e s u c h as s t a t i c f r o m a r a d i o t h a t h a s lo s t A fr ic a n fra c ta l m a th em a tics 154 its s t a t i o n ( w h a t is o f t e n c a l l e d “w h i t e n o i s e " ) w o u l d h a v e t h e h i g h e s t a l g o rith m ic co m p lex ity. O n e p r o b l e m w i t h d e fi n in g c o m p l e x it y in te r m s o f r a n d o m n e s s is t h a t it does n o t m a t c h o u r i n t u i t i o n . W h i l e i t ’s t r u e t h a t t h e p e r i o d i c s i g n a l o f a t i c k i n g m e t r o n o m e is so s i m p l e t h a t it b e c o m e s h y p n o t i c a l l y b o r i n g , t h e s a m e c o u l d be said for w h i t e n o i s e — in f a c t , 1 s o m e t i m e s t u n e m y r a d i o b e t w e e n s t a t i o n s if 1 w a n t t o fall a s l e e p . B u t if I w a n t t o s t a y a w a k e 1 l i s t e n t o m u s ic . M u s i c s o m eh o w sa tisfie s o u r j n t u i t i v e c o n c e p t o f c o m p l e x i t y : it is p r e d i c t a b l e e n o u g h t o fo l­ low a lo n g , b u t s u rp ris in g e n o u g h t o k e e p us p l e a s a n t l y a t t e n t i v e . M a t h e m a t i c i a n s e v e n t u a l l y c a u g h t u p w i t h t h e i r i n t u i t i o n a n d d e v e l o p e d a n e w m e a s u r e in w h i c h t h e m o s t c o m p l e x s i g n a ls a re n e i t h e r c o m p l e t e l y o r d e r e d n o r c o m p l e t e l y d i s o r d e r e d , b u t r a t h e r a r e h a l f w a y in b e t w e e n . T h e s e p a t t e r n s ( w h i c h i n c l u d e a l m o s t e v e r y ty p e o f i n s t r u m e n t a l m u s i c ) a ls o h a p p e n to b e f r a c t a l s — in fa c t, as w e ^ l T s e e , th e n ew c o m p le x ity m easu re e x a c tly c o in c j d e s j v it h th e m easure-of fractal d im e n sio n . T h e first ste p in this d i r e c ti o n was t h r o u g h stu dies o f c e llu la r a u to m a ta . R eca ll fr o m c h a p t e r 7 t h a t c o m p u t e r s c i e n t i s t s in t h e early( i g 8 g s-t\ad s t a r t e d t o t h i n k N e u r a l n ets (n) Suppose we balance a ball on a teeter-totter. Unless the ball is at the precise center, the teeter-totter wtl! start to slope toward one side, which will cause the ball t o roll even farther toward that side. In other words, there are two stable states, and anything in between (except ft tiny neutral point) will get caught up in the positive-feedhack loop leading rapidly to a stable st (b) This is an electrical circuit that works much like the teeter-totter. Each triangle is an ampli with two outputs, one normal and the other (black circle) an inverted output. Since the inverte output is connected to the input of the other amplifier in each, they will balance out like the b; at the exact center of the teeter-totter, but rapidly flip to one of the two stable states in which t amplifier is at its maximum (“saturated”). T h a t means that this circuit can solve a simple task: which of two numhers is larger? By putting an initial charge proportionate to one of the two numbers at each inpur, the system rapidly (lips to the saturated stable state favored hy the larget number. Linking thousands of these simple amplifiers Together allows computer scientists to m;i sophisticated machines for pattern recognition and other artificial intelligence tasks. C o m p le x ity 155 a b o u t c e llu la r a u t o m a t a as t h e s i m u l a t i o n o f c o m p l i c a t e d p h y s ic a l d y n a m ic s , s u c h as t h a t j e e n in l i v in g o rg a n i s m s . P h y s i c i s t S t e p h e n W o l f r a m b e g a n to w o n d e r : ju s t h o w c o m p l i c a t e d is it? C le a r l y , li v i n g s y s te m s a r e m o r e c o m p l e x t h a n r a n ­ d o m n o is e , so h e k n e w t h a t t h e o ld c o m p l e x i t y m e a s u r e o f K o l m o g o r o v w o u ld n o t do. B ut W o lfram h a d studied a good d e a l o f c o m p u te r science, a n d h e re a l­ ized t h a t ch e way in w h i c h d i f f e r e n t ty p es o f r e c u r s i o n s a re used t o m e a s u r e c o m p u t in g p o w e r c o u ld a ls o be a p p l i e d t o p h y s i c a l d y n a m i c s . R e c a l l fro m c h a p t e r 8 t h a t we d iv i d e d re c u r s io n in t o t h r e e types: c a sc a d e s , it e ra t io n s , a n d self-reference. FIGURE IO.3 K o n n o g o r o a t - C h u / t m c o m p le x ity m e a s u r e ( j) W h e t h e r it is in d i g i t a l o r a n a l o g s i g n a l s , c o m p l e x i t y c a n b e measured in t e r m s o f t h e i n f o r m a t i o n c o n t e n t . T h e first s u c h measure w as t h a t o f K o l m o g o r o v a n d C h a i t i n , w h o t h o u g h t o f £ co m p lexity in t e r m s o f r a n d o m n e s s . T h e s i n e w a v e is a b o u t as n o n r a n d o m as w e c a n g e t . H e r e it is g i v e n a s a t i m e - v a r y i n g | signal, a l t h o u g h t h e s a m e w o u l d a p p l y t o a s p a t i a l p a t t e r n , s u c h ^ as waves in w a t e r o r s a n d ( i n w h i c h c a s e w e c o u l d m e a s u r e it as w aveleng th, w h i c h is s i m p l y t h e r e c i p r o c a l o f f r e q u e n c y ) , lb) T h e s a m e s i g n a l i n a s p e c t r a l d e n s i t y p l o t . T h i s t e l l s y o u how m u c h p o w e r is a t e a c h f r e q u e n c y . I n t h e c a s e o f t h e s i n e w a v e , all t h e s i g n a l p o w e r frequency, ( c ) W h i t e n o i s e is a c o m p l e t e l y r a n d o m s i g n a l , s u c h as t h a t p r o d u c e d js a t o n e by t h e s o u n d o f bacon fry ing . By t h e K o l m o g o r o v - C h a i r i n d e f i n i t i o n , w h i t e n o i s e is t h e m o s t c o m p l e x s i g n a l . Again, t h is w o u l d a l s o a p p l y t o a s p a t i a l p a t t e r n , s u c h as d u s t s p r i n k l e d o n a t a b l e , ( d ) S p e c t r a l density p l o t for w h i t e n o i s e . B e c a u s e it is c o m p l e t e l y r a n d o m , t h e r e is a n e q u a l l i k e l i h o o d o f a n y wav elen gth o c c u r r i n g a t a n y t i m e , s o t h e s i g n a l ’s p o w e r is e q u a l l y d i s t r i b u t e d a c r o s s t h e s p e c t r u m , (c) In s u m m a r y , t h e K o l m o g o r o v - C h a i t i n c o m p l e x i t y m e a s u r e is s i m p l y a m e a s u r e o f r a n d o m n e s s . Ic.c'uiivtesy R. F. Voss.) A fr ic a n (racial mathematics i5 6 T h e s e co rresp o n d a p p ro x im a te ly to th e th re e form al ca te g o rie s o f recu rsio n use d in c o m p u t e r s c i e n c e , w h i c h w e w ill n o w e x a m i n e in d e t a i l . T h r e e t y p e s o f re c u r s io n : t h e C h o m s k y h ie r a r c h y I n a re c u r s iv e s y s te m , p r e s e n t b e h a v i o r d e p e n d s o n p a s t b e h a v i o r . I t is t h e c a p a ­ b il i t y o f th i s a c c e s s t o m e m o r y t h a t d e fin e s t h e r e l a t i v e d i f f e r e n c e in re c u r s iv e ^ pQAyer. T h e s c a l i n g c a s c a d e , fo r e x a m p l e , c o u l d n o t p r o d u c e t h e F i b o n a c c i ' I s e q u e n c e , b e c a u s e it c o u l d n o t r e c a l l p re v i o u s m e m b e r s o f t h e s e q u e n c e . S i m i \ ^ l a r d i s t i n c t i o n s a r e u se d in c o m p u t e r s c i e n c e to r a n k c o m p u t a t i o n a l p o w e r i n t o t h r e e ty p es o f a b s t r a c t m a c h i n e s , re f e r r e d t o as “C h o m s k y ’s h i e r a r c h y . ” T h e s e a b s t r a c t m a c h i n e s a re c o m p a r e d b y t h e i r . a b i l i t y t o r e c o g n j z e c e r t a i n c a t e g o r i e s o f c h a r a c t e r s t r in g s . A m a c h i n e t h a t c a n r e c o g n i s e p e r i o d i c c h a r a c t e r s t r i n g s s u c h as “a b a b n . . o c c u r s a t t h e lo w e s t le v e l o f t h e h i e r a r c h y , t h e F i n i t e S t a t e A u t o m a t o n ( F S A ) . A n e x a m p l e o f t h e F S A is s h o w n i n figure 10.4. W h a t w o u ld it b e li k e t o be a n F S A ? S i n c e t h e F S A h a s n o m e m o r y s t o r ­ ag e, t h e e x p e r i e n c e w o u ld be s o m e w h a t a n a l o g o u s t o n e u r o s u r g e r y p a t i e n t s w h o h a v e h a d b i l a t e r a l h i p p o c m n ji a X I e s io n s { M i l n e r 1 9 6 6 ) . T h e s e p a t i e n t s a r e fu lly \ ■[ a w a r e a n d i n t e l l i g e n t b u t h a v e lost t h e c a p a c i t y t o t r a n s f e r k n o w l e d g e t o lo n g - I t e r m m e m o ry . T h e ' t d . p p o c a m p a l surg~ery p a t i e n t w h o fin d s h e r s e l f a t t h e e n d o f a b o o k c a n d e d u c e t h a t s h e h a s r e a d its c o n te n ts ., a l t h o u g h s h e d o e s n o t k n o w y w h a t t h e p r e v i o u s c h a p t e r s w e r e a b o u t . A rtJ F S A -'h a s o n l y a n implicit m e m o r y , f 2 * b e c a u s e its p r e s e n t s t a t e c a n n o t r e v e a l a n y t h i n g a b o u t its p a s t , o t h e r t h a n t h e } / f a c t t h a t it m u s t h a v e p a s s e d t h r o u g h o n e o f t h e s e q u e n c e s o f s t a t e s t h a t t e r m i-. ^ ym lL e"ur i h t p re s e n t state. input tape b a a - a b a b b read only Transition table Current symbol on input tape New state s, n Si Si h s2 a S2 • S2 S2 b Si Current state ... FIGURE . , .... . IO .4 T h e fin ite s t a te au to m a to n T h e finite state a u to m a to n (F S A ) has a list of transition rules t h a t tell it how to change from o n e state to th e next, d ep e n d in g o n its curre n t state an d th e symbol it is reading on tire inp ut tape. It has n o mem­ ory, o th e r th a n th a t implied hy its c u re n t state. T h is FSA will en d up in the “a cc ep t” state S, it th e tape ends after an even n u m b er of h’s. 157 Complexity T h e sec o f p a l i n d r o m i c s t rin g s (e .g ., a a b b a a ) is a g o o d e x a m p l e o f t h e l i m ­ i t a t i o n o f_ th e F S A : it l a c k s t h e a b i l i t y to m e m o r i z e t h e first h a l f o f t h e s t r in g a n d t h e r e f o r e c a n n o t c o m p a r e it w i t h t h e s e c o n d . T h e le a s t p o w e r f u l m a c h i n e c a p a b l e o f - t h i s m e m o r y s t o r a g e is t h e P u s h - D o w n A u t o m a t o n f(T O A )\ illus- T t ; ^ t r a t e d in figure 10.5. T h e s t a c k m e m o ry _ o f t h e P D A is u su a lly c o m p a r e d to t h e \ s p r in g -lo a d e d tray s t a c k o f t e n u sed in c a fe terias; o n c e a sy m b o l is read fro m m e m - / 'ory it is g o n e . A s a k n o w l e d g e a n a lo g y , w e m i g h t t h i n k o f a r e a d e r w h o a c c u ­ m u l a t e s s t a c k s o f b o o k s b u t g e ts rid o f e a c h b o o k a f t e r it is re a d . T h i s is a t e m p o r a r y e x p l i c i t m e m o r y , s i n c e t h e P D A ca n ,,m a k e . tw o d i f f e r e n t t r a n s i t i o n s g i y e n t h e s a m e s t a t e . a n d i n p u t , d e p e n d i n g o n its p a s t . It is i m p o r t a n t t o u n d e r ­ s t a n d t h a t g r e a t e r re c u r s iv e c a p a b i l i t y d o e s n o t n e c e s s a r i l y j i p p l y larger m e m ­ ory storage; it m e a n s a n im p ro v e d ability to interact w i t h m em ory. Size on ly m a tte r s in s o fa r as it r e s t r i c t s t h e i n t e r a c t i o n . A l t h o u g h t h e P D A c a n r e c o g n i z e all sets o f s t r in g s r e c o g n i z e d by a n F S A , as w e ll as m a n y o t h e r s , t h e r e a re still ( in f i n i t ely ) m a n y sets o f s t rin g s t h a t jt_ c a n n o t r e c o g n i z e / F o r exam ple;, i t j z a n n o t r e c o g n i z e t h e s e t o f all s t rin g s o f t h e form a ^ b ^ c N ( w h e r e we h a v e N r e p e t i t i o n s o f a, fo l lo w e d by t h e s a m e for b a n d c), b e c a u s e it h a s to w ip e o u t its m e m o r y in t h e p r o c e s s o f c o m p a r i n g t h e n u m b e r o f a ’s a n d b ’s, le a v i n g n o i n f o r m a t i o n fo r c h e c k i n g t h e n u m b e r o f c ’s. ___^ A t t h e t o p o f t h e h i e r a r c h y (fig. 1 0 . 6 ) , t h e T u r i n g M a c h i n b t( T p I ) c a n “ t VA re c o g n iz e a ll c o m p u t a b l e f u n c t i o n s . It is s i m p l y a P D A w i t h ' u n r e s t r i c t e d m e m ­ ory, b u t b e c a u s e o f t h i s c a p a b i l i t y i t , c a n a c h i e v e f u ll s e l f - r e f e r e n c e : t h e a b i l ­ ity tp ,g p.9iyze J.ts.,pw n p r p g r a m . A g a i n , it is n o t a d i f f e r e n c e in m e m o r y size, b u t in m e m o r y a c c e s s — u n l i k e t h e P D A s t a c k , t h e T M m e m o r y i n t e r a c t i o n s c a n o c c u r o v e r a n.y.past s e q u e n c e s o f a n y l e n g t h , a n d it d o e s n o t lose m e m o r y input t a p e a b a a b a b b F IG U R E IO.5 T h e p u s h -d o w n a u to m a t o n rea d only read/write “S ta c k " m em o ry . T h i s a llo w s n e w s y m b o ls to be p o s h e d d o w n o n to p o f th e s ta c k , h u t sym bols c a n b e re a d o n ly by p o p p in g t h e m o f f t h e t o p , a n d e a c h o n e p o p p e d is l o s t . T he push-down automaton (PDA) has a list of transition rules, but these make use of an explicit memory storage as well as internal states. 158 A fr ic a n fra cta l m a them atics input tape a b a a b a b FIGURE b 1 0 .6 T h e T u rin g m a ch in e T he Turing machine has an unconstrained memory; it can implement any algorithm that can possibly exist. a f te r it is re a d . T o c o n t i n u e t h e t e x t a n a lo g y , if t h e F S A is a p e r s o n w h o a c c o m ­ p l i s h e s t a s k s w i t h n o b o o k s , a n d t h e P D A is a p e r s o n w h o s e s i m p l e t a s k s a r e lim ite d to b o o k s t h a t are re m o v e d a fte r th e y are read , t h e n th e T M w o u ld be a b l e t o c o l l e c t a n d r e c a l l a ll b o o k s , i n a n y o r d e r . U n f o r t u n a t e l y t h i s d o e s n o t s o l v e a ll o f o u r p r o b l e m s , b e c a u s e t h e u n b o u n d e d n a t u r e o f t h e T M m e a n s t h a t it fooIishly_acceRt§...s.p.me..tasks. t h a t . r e q u i r e a n i n f i n i t e Jibrary. T h i s is c a lle d t h e “ h a l t i n g p r o b l e m , ” a n d T u n n S h i m s e l f p r o v e d t h a t jt is u n a v o i d a b l e . M a t h e m a t i c i a n R o zsa P e t e r s h o w e d t h a t o n e _ c a n d e f i n e a r e s t r i c t e d s e t o f p r o g ra m s _ th a t^ a r.e _ h a l.t.a b le ( w h i c h s h e c a l l e d t h e s e t_ o f _ “ p r i m i t i v e re c u r s i v e f u n c t i o n s " ) , b u t i n d o i n g s o w e w o u l d a l w a y s s a c r i f i c e s o m e o f t h e T M ’s c o m p u t i n g p o w e r. T h e s e t h r e e m a c h i n e s, F S A , P D A , a n d T M , i l l u s t r a t e t h e a s c e n t u p t h e C h o m s k v _ h i e ta r c h y . T h e y d if fe r in h a v i n g i m p l i c jj ^ m e m o r y , t e m p o r a r y e x p l i c i t m e m o ry, a n d p e rm a n e n t^ x p l .ic j ^ t m e m o r y . By l o o k i n g a t m e m o r y as t h e basis for t h e re c u r siv e lo o p in th e s e s y s te m s— t h a t is, as t h e e l e m e n t t h a t g o v e r n s t h e a b i l ­ ity o f th e system to p e rf o rm in t e r a c t i o n s b e t w e e n its p r e s e n t i n p u t a n d p a s t b e h a v ­ ior— w e c a n see t h a t t h e d i f f e r e n c e s in c o m p u t a t i o n a l p o w e r for th e s e m a c h i n e s d e p e n d s o n t h e d i f f e r e n c e s in re c u r s iv e po w er. M e n s u r i n g a n a lo g c o m p l e x i t y w i t h d ig ita l c o m p u t a t i o n N o w l e t ’s r e t u r n to W o l f r a m a n d h is ce l l u la r a u t o m a t a . A f t e r r u n n i n g t h o u s a n d s o f tr ials, W o l f r a m f o u r u T t h a t all c e llu la r a u t o m a t a g e n e r a l ly d i v i d e d i n t o f o u r s p e ­ cific c lasses. C la s s e s 1 a n d 2 w e r e t h o s e t h a t e i t h e r d i e d o u t , o r w e n t i n t o a p e r i ­ o d i c c y c le . C la s s 3 w as j u s t t h e o p p o s i t e : it w a s u n c o n t r o l l e d g r o w t h t h a t led to a p p a r e n t l y r a n d o m b e h a v i o r , li k e w h i t e n o is e . B u t cla s s 4, w h i c h i n c l u d e d th e “g a m e o f life” c e llu la r a u t o m a t o n , h a d s o m e t h i n g t h a t W o lf ra m d e sc rib e d as “c o m ­ p l e x ” b e h a v io r : n o t as ra n d o m as w h i t e no ise, b u t n o t as b o r i n g as a p e rio d ic cycle. W o l f r a m f o u n d t h a t th is h i g h e s t c o m p l e x i t y a ls o d e m a n d e d t h e h i g h e s t c o m - Complexity p u t a b i li ty : w h ile p u r e o r d e r a n d p u r e d i s o r d e r c o u l d b e r e c o g n i z e d by a n F S A , che p acte rn s_ o f che c o m p l e x b e h a v i o r r e q u i r e d a T u r i n g m a c h i n e . M a c h e m a c ic a l p h y sic ist J a m e s C r u t c h ffe ld (1 9 8 9 ) fo u n d a n e v e n s i m p l e r e x a m p l e o f r e c u r s i v e c o m p l i c a t i o n in a p h y s i c a l s y s t e m . C r u t c h f i e l d u s e d che p o p u l a t i o n e q u a t i o n m a d e f a m o u s b y b i o l o g i s t R o b e r t M a y ( 1 9 7 6 ) : Pn + 1 = P»R (1 - Pn ) ( w h e r e P is a p o p u l a t i o n n u m b e r , sc a le d so t h a t it is b e t w e e n 0 a n d f, a n d R is t h e b i r t h r a t e ) . M a y f o u n d t h a t w h e n R is low , t h e p o p u ­ l a t i o n is s i m p l y a p e r i o d i c c y c l e , s w i t c h i n g b a c k a n d f o r t h b e t w e e n t h e s a m e s e q u e n c e o f le v e l s . A s y o u i n c r e a s e R , ch e l e n g t h o f c h e c y c l e ( t h a t is, th e n u m b e r o f d iffe re n t p o p u la tio n lev els you pass th ro u g h before re tu rn in g t o t h e fir s t o n e ) i n c r e a s e s e x t r e m e l y f a s t . A t R = 3 . 1 , t h e p o p u l a t i o n is in a tw o - l e v e l cy c le , a t R = 3 .4 in a f o u r - l e v e l c y c le , a n d a t R ~ 4 -0 t h e c y c le l e n g t h is a t i n f i n i t y : d e t e r m i n i s t i c c h a o s . C r u t c h f i e l d w a s a b l e t o m e a s u r e t h e c o m ­ p u ta b ility o f th e s e c h a o tic flu c tu a tio n s a n d fo u n d resu lts sim ila r to th o s e of W o lfram : b o th c o m p le te ly p e rio d ic w av es a n d c o m p le te ly d iso rd e re d w aves w e r e c o m p u t a t i o n a l l y q u i t e s i m p l e , b u t t h o s e in b e t w e e n , w i t h a m i x o f o rd e r a n d d iso rd e r, h a d a h ig h d e g r e e o f c o m p u t a ti o n a l c o m p le x ity . T h e sim p le e q u a ti o n e x a m in e d by C r u tc h f ie l d re q u ire d o n ly a P D A , b u t o c h e r research ers (B lu m , S h u b , a n d S m a le 198 9 ) d e m o n s tr a te d t h a t m o re c o m p le x a n a l o g f e e d b a c k s y s te m s w o u l d b e c a p a b l e o f s i g n a l c o m p l e x i t y e q u i v a l e n t t o T M c o m p u ta b ility . F ig ure 10.7 sh o w s h o w t h e s e c o m p l e x w a v e f o r m s , c a ll e d “ 1/ F n o is e ," c o m ­ p a re to p e r i o d i c a n d w h i t e n o i s e w a v e f o r m s . T h i s is e a s i e s t t o see in t h e s p e c - j tra l d e n s i t y p lo ts. A p e r i o d i c s i g n a l h a s all its p o w e r a t o n e w a v e l e n g t h , w h ile! a w h i t e - n o i s e s i g n a l h a s t h e s a m e p o w e r a t a ll w a v e l e n g t h s . 1 / F n o is e is a c o m - j •|iio'inise l)ciw ee iv t h e tw o — b ia s e d so t h a t it h a s t h e g r e a t e s t a m o u n t o f p o w er! a t t h e lo n g e s t w a v e l e n g t h , a n d t h e le ast a t t h e s h o r te s t . For th is reas o n , I / F noise* is fr a c ta l; it h a s f l u c t u a t i o n s w i t h i n f l u c t u a t i o n s w i t h i n f l u c t u a t i o n s . W h e n we t h i n k o f t h e l e n g t h o f t h e s e w a v e f o r m s in t e r m s o f m e m o r y , w e c a n b e g i n to see a c o n n e c t i o n t o c o m p u t a t i o n a l p o w e r. If a sy s te m h a d t h e s a m e b e h a v i o r ov er a n d o v e r a g a in , it w o u ld b e coo f i x e d o n m e m o r y . If it r a n d o m l y p i c k e d a new) b e h a v i o r e v e ry t i m e , t h e n it w o u l d b e t o o free fro m m e m o r y . B u t useful b e h a v - ] ior is g e n e r a l l y a m i x t u r e b e t w e e n t h e t w o . F o r e x a m p l e , t h i n k o f s o m e t h i n g / u n u s u a l you d id t o d a y — m o v i n g so c k s to a n e w side o f t h e d raw er, o r e a t i n g p r e t ; zels i n s t e a d o f c r a c k e r s . W h a t e v e r it w as, c h a n c e s a re it w a s p r e t t y tr iv i a l. If w.e to o k t h e s a m e w h im s ic a l a p p r o a c h t o m a j o r li f e - e v e n ts e a c h d a y — “ to d a y 1 t h i n k I ’ll m o v e to S p a i n , o r g e t p r e g n a n t , o r b e c o m e a p o d i a t r i s t ”— w e w o u l d b e intr o u b le . O u r life is ty p ic a lly a r r a n g e d as 1 / F n o is e : h i g h - p o w e r e v e n t s s h o u l d be' lo n g - te r m c h a n g e s , a n d lo w - p o w e r e v e n t s s h o u l d be s h o r t - t e r m c h a n g e s . 3 In fact, 159 FIGURE I O . 7 C r u t c h / i e l d - S m a l e c o m p le x ity m e a s u r e ( a -b ) P e r i o d i c noise: A simple signal, ( c -d ) White noise: From the viewpoint o f the Crutchficlcl-Smnle measure, this is also of low complexity. An FSA, for example, could define this noise by making all state transitions equally probable, ( e - f ) Fractal noise: The most complex signals in the-Crutchfield-Sinale measure are “scaling fractal noises" in which there are fluctuations within fluctu­ periodic random noise noise noise ations. These signals have the greatest amount of their power in the lowest frequencies (longest wavelength). Since power is rbe reciprocal of frequency, it is often referred to as 1/F noise, (g) In summary, the Ctutchfield-Smale complexity measure is a reflection of the fractal dimension. T he “most fractal” (e.g., dimension of 1.5) will be the most complex, and the function decreases with both higher and lower dimensions. (c a n d e, courtesy R. F. Voss.) Complexity m a n y o f t h e a n a l o g w a v e f o r m s p r o d u c e d by i n t e l l i g e n t h u m a n b e h a v i o r a p p e a r t o b e 1/ F s ig n a ls (V oss 1 9 8 8 ; E g la s h 1 9 9 3 ) . A s m o r e j i c i e m i s t s b e g a n t o t h i n k o f c o m p l e x i t y in te r m s o f c o m p u t a t i o n a n d 1 / F n o is e , t h e y b e g a n t o a c c u m u l a t e e x a m p l e s t h a t s u g g e s te d t h a t t h i s w as w h a t it m e a n t t o h a v e a " s e l f - o r g a n i z i n g ” s y s te m . In t h e e v o l u t i o n o f life, fo r i n s t a n c e , m o s t o f t h e g e n e t i c i n f o r m a t i o n s to re s l o n g - t e r m e v e n t s , s u c h as t h e p h y s i o lo g y t h a t u n d e r w e n t c h a n g e in l i f e ’s e v o l u t i o n fr o m w a t e r t o l a n d . M o r e s h o r t - t e r m a d a p t a t i o n s , s u c h as s k i n co lo r, t a k e up very li t t l e o f t h e g e n e t i c m a t e ­ rial. H e r e a g a in , w e h a v e s o m e t h i n g li k e 1/ F n o is e , w i t h l o n g - t e r m e v e n t s t a k ­ in g u p t h e b u l k o f t h e s y s te m , a n d s h o r t - t e r m e v e n t s t a k i n g u p p r o p o r t i o n a t e l y less. P h y s i c is t s P e r B a k a n d C h a o T a n g ( B a k a n d C h e n 1 9 9 1 ) f o u n d s e v e r a l e x a m p l e s o f s i m p l e p h y s i c a l s e l f - o r g a n iz i n g s y s te m s t h a t p r o d u c e d 1 / F n o i s e . , I n fo rest'fiT eS T 'forex'am p le, v e r y d ry w o o d s w o u ld s p r e a d fire in a n o r d e r l y c ir c le , w h i l e fires in w e t w o o d w o u ld b e t o o s p o r a d i c o r r a n d o m , a n d t h u s d ie o u t. B u t i n - b e t w e e n fires s p r e a d i n a f r a c t a l p a t t e r n , w i t h m o s t o f t h e fire in l o n g - l e n g t h p a t c h e s , less o f t h e fire in m e d i u m p a t c h e s , e v e n less in s m a l l e r p a t c h e s , a n d so o n . In w a t e r we h a v e o r d e r l y c r y s ta ls a n d d i s o r d e r l y li q u id s , b u t in b e t w e e n w e c a n get th e fractal p a tt e r n s o f snow flakes. S in c e we are fam iliar w ith o u r o w n recu rsiv e in te ra c tio n s w ith m em ory, we h a v e a _ g p o d J.n tu i t ive s e n s e f q r w h y 1 / F n o i s e s h o u l d a c c o m p a n y c o m p l e x b e h a v i o r , a n d c l e a r l y it c a n c h a r a c t e r i z e m a n y v a r i e t i e s o f s e l f - o r g a n iz i n g sy s­ tem s— ;p erh ap s all o f t h e m if w e use t h e p r o p e r d e f i n i t i o n . B u t h o w d o e s this h a p ­ p e n ? W h a t is t h e m e c h a n i s m t h a t m a k e s it w o rk ? C o m p l e x i t y t h e o r i s t s h a v e n o t h e s i t a t e d t o su g g e s t i m p l i c a t i o n s o f t h e i r w o r k for-/c u i t u r e ; ) h e r e I w o u l d lik e to su g g est t h e re v e r s e : t h a t c e r t a i n a s p e c t s o f A f r i c a n c u l t u r e c a n p r o v i d e i m p o r ­ t a n t im p l i c a t i o n s for c o m p l e x i t y th e o ry . M o r e so rhan-.any o f t h e p re v i o u s e t h r . c m a t h e m a t i c s m o d e l s w e h a v e s e e n , t h i s p a r t o f m y r e s e a r c h w as m u c h m o r e o f a c o lla b o ra tio n , m u c h clo ser to m y sen se o f th e “p a r tic ip a n t s im u la tio n ” m e t h o d — a l t h o u g h if t r u t h b e k n o w n 1 h a d t o b e d r a g g e d k i c k i n g a n d s c r e a m ­ ing m u c h o f t h e way. C h ristia n S in a D i a t t a : a n A f r i c a n p h y s i c i s t looks at c u l t u r e “R h a b . ” “ P h a n t o m . ” “R h a b ! ” “ P h a n t o m !!" A s t r a n g e d ia l o g flew acro ss t h e c o m ­ p u t e r la b a t t h e I n s t i t u t d e T e c h n o l o g i e N u c l e a i r e A p p l i q u e e a t S e n e g a l ’s U n i ­ v ersity o f ..Dakar- I w as s e a t e d w i t h P r o f e s s o r C h r i s t i a n S i n a D i a t t a , d i r e c t o r o f t h e la b ,^ w atc h in g t h e p u l s a t i n g fo r m s o f c e l l u l a r a u t o m a t a flow a b o u t t h e s c r e e n . D r . D i a t t a l w as t h e lo c a l s p o n s o r f o r r e s e a r c h u n d e r t h e U n i t e d S t a t e s ’ F u l b r i g h t F e llo w s h ip p r o g r a m , a n d w as e a g e r to d iscuss h is o w n id ea s. Mis p h y s i c s la b was A fr ic a n fra c ta l m athem atics a n i n s p ir i n g p l a c e t o be. I h a d a lr e a d y b e e n a b le t o sit in o n a g r a d u a t e s t u d e n t ’s p r e s e n t a t i o n ; a fte r h a v i n g w i t n e s s e d t h e s a m e r i t u a l in t h e p h y s i c s d e p a r t m e n t a t t h e U n i v e r s i t y o f C a l i f o r n i a a t S a n t a C ru z , it m a d e fo r a fascinatin.g: b i t o f crossc u lt u r a l c o m p a r is o n . I tr ie d to m a k e m y s e lf useful by s e t t i n g u p a d e m o o f a n e l e c ­ t r ic a l c i r c u i t t h a t p r o d u c e d d e t e r m i n i s t i c c h a o s ( “C h u a ’s c i r c u i t ”) a n d i n s t a l l i n g variou s types o f so ftw are for s i m u l a ti o n s o f n o n l i n e a r d y n a m i c s . It w as o n e o f th e s e s o f tw a r e d e m o s , R u d y R u c k e r ’s c a l a b , th a t caused our m ultiling ual exchange. A s n o t e d in c h a p t e r 7, s o m e o f R u c k e r ’s m o s t i n t e r e s t i n g p r o g r a m s a re th o s e h e c a lls * |Z h a b o ti n s k y C A s , " w h i c h c a n p r o d u c e p a i r e d log s p i r a ls . I n a d d i t i o n t o t h e tw o s t a t e s o f li v e c e l l a n d d e a d c e l l , t h e s e c e l l u l a r a u t o m a t a r e q u i r e a t l e a s t o n e “gh ost..,state.” S i n c e s o m e o n e h a d p r e v i o u s l y m e n t i o n e d t h e i n d i g e ­ n o u s t e r m for. g h o s t , -rhab, it s e e m e d l i k e a n o p p o r t u n i t y fo r c r e a t i v e t r a n s l a ­ t i o n . 1 e x p l a i n e d (i n F r e n c h , t h e official la n g u a g e o f S e n e g a l ) t h a t a fte r I'eta t m ort ( t h e d e a d s t a t e ) t h e c e ll w e n t t o i ’ e ta t r h a b . T o m y s u r p r i s e , D i a t t a c o r r e c t e d rh a b b a c k to t h e F r e n c h : “p h a n t o m . ” W e w e n t b a c k a n d f o r t h a c o u p l e o f ti m e s b e f o r e I re a l iz e d t h a t i t ' w a s n o t j u s t m y p o o r p r o n u n c i a t i o n . O n l y l a t e r d id I d is c o v e r m y b l u n d e r : D i a t t a was n o t fr o m t h e Is la m i c W o l o f m a j o r i t y ( i n w h o s e l a n g u a g e rha b o c c u r s ) b u t f r o m o n e o f t h e a n i m i s t m i n o r i t y g r o u p s , t h e Jo la. U s i n g W o l o f w as n o m o r e o f a c u l t u r a l t r a n s l a t i o n fo r h i m t h a n i t w o u l d h a v e b e e n t o u se E n g li s h . T h i s w as o n l y t h e s t a r t o f m y m i s t r a n s l a t i o n s . A l t h o u g h D r. D i a t t a w as g reatly e n th u s ia s tic a b o u t my w o rk o n fractals in A fr ic a n a r c h ite c tu r e , he s e e m e d d i s i n t e r e s t e d in t h e f r a c t a l g e n e r a t i o n s o f t w a r e . B u t h e p e r s i s t e n t l y b r o u g h t u p A f r i c a n a r c h i t e c t u r e d u r i n g t h e c e ll u la r a u t o m a t a d e m o s . I f o u n d th is e n t i r e l y t o o f r u s t r a t i n g : t h e w h o l e p o i n t o f rnv r e s e a r c h o n A f r i c a n f r a c t a l s was to e x p lo re th e i n t e n t i o n a l side o f th e s e designs. C e llu la r a u to m a ta c re a te p a t ­ t e r n s n o t b y p r e p l a n n e d j d e s i g n , b u t r a t h e r by t h e i n t e r a c t i o n s o f its a g g r e g a t e c e lls. F r o m m y p o i n t o f v iew , h a v i n g f r a c t a l a r c h i t e c t u r e a s t h e r e s u l t o f a g g r e ­ g a te s e l f - o r g a n i z a t i o n d e s t r o y e d t h e p o s s ib il it y o f i n t e n t i o n a l i t y . By fo c u s in g o n c e l l u l a r a u t o m a t a as a n a r c h i t e c t u r a l m o d e l , D i a t t a s e e m e d t o h e u n d o i n g all m y c a r e f u ll y p r e p a r e d r e s e a r c h . H i s e n t h u s i a s m w a s u n b e a t a b l e , h o w e v e r , a n d 1 b e g a n to s tu d y a e r i a l p h o t o s o f h i s p l a c e o f o r i g i n , t h e J o l a s e t t l e m e n t s s o u t h o f th e C a s a m a n c e R iver. F ig ure 10.8 s h o w s t h e s e t t l e m e n t o f M l o m p , n o t far fro m D i a t t a ’s h o m e t o w n ; its p a i r e d lo g s p i r a l s t r u c t u r e c o u l d h a v e c o m e r i g h t o u t o f R u c k e r ’s Z h a b o t i n s k y C A s . A t r i p to t h e C a s a m a n c e w as c le a r ly c a l l e d for. 1 w as f o r t u n a t e in fi n d in g N fally B a d ia n e , a Jo la g ra d u a te s t u d e n t w h o h a d d o n e h is m a s t e r ’s th esis o n in d ig e ­ n o u s a r c h i t e c t u r e o f t h e s o u t h e r n C a s a m a n c e , as a g u id e . N f a l l y ’s b a c k g r o u n d is ideal for a n a n t h r o p o lo g i s t: raised a m o n g th e I s la m ic m a j o r i ty in D akar, h e is b o t h FIGURE 1 0 . 0 T h e J ola s e t t l e m e n t o f M l o m p , S e n e g a l Ump. (b, Mlomp model generated by com b.n»tior.of s t o ^ s t i c and recursive process, [nstitut Geogmlskique tie Senega!; b. courtesy o f Egondu Onyejelt.ve.) A fr ic a n fra c ta l m athematics s t r a n g e r t o a n d m e m b e r o f t h e Jo la so c ie ty . A s w e t r a v e l e d t h e d e l t a a r e a o f che C a s a m a n c e R i v e t , u s i n g c a t s , tr u c k s , c a n o e s , a n d a n y t h i n g else t h a t m o v e d , h is w a r n in g s a b o u t t h e se c re c y o f J o l a religious k n o w l e d g e w e r e re p e a t e d ly c o n firm e d . S e c u l a r i n fo r m a t i o n a b o u t t e c h n i c a l m e t h o d s o f h o u s e c o n s t r u c t i o n , p r e c o l o n i a l a n d p o s t c o l o n i a l s o c i a l c h a n g e s , k in s h ip , g to p p s ,.,.a .n d .jp a n y o t h e r , a s p e c t s „ p f J o l a s o c i e ty w e r e readilY f o r t h c o m m (E g la s h e t al. 1 9 9 4 ) . ' W e w e r e to ld t h a t t h e c ircu lar b u ild in g c o m p le x e s w ere n o t p re p la n n e d , n o r w e re th e b ro a d cu rv es of t h e s e c o m p l e x e s in e a c h n e i g h b o r h o o d , b u t t h a t t h e y c o u l d n o t te ll us a n y t h i n g a b o u t th e se q u e n c e o f c o n s tr u c tio n because, u n lik e t h e W o lo f, "w e d o n o t h av e a griot (oral h i s t o r i a n ] in J o l a so c ie ty ." T h e s p i r a l s t r u c t u r e v is ib l e in t h e p h o t o w as m a i n l y d u e t o t h e c a r e f u ll y m a i n t a i n e d s a c r e d f o r e s t ^ L n T o u n d i n g e ^ h J _ o c a l n e i g h b o r h o o d . B u t t h e m e c h a n i s m s for c r e a t i n g s u c h c o h e r e n t s t r u c t u r e s o v e r s u c h a n e n o r m o u s r a n g e o f sc a le s r e m a i n e d h i d d e n . A t a n t a l i z i n g g l i m p s e o f t h e J g] a ry, h o w e v e r , led us t o s u s p e c t t h a t t h e r e w a s a ^ c o n sc io u s ele- m e n t t o t h e C A - l i k e s e t t l e m e n t s t r u c t u re. F ir s t, t h e r e w a s t h e s y m b o l i s m o f t h e c h i e f ’s d r i n k i n g vessel: a s p ir a l s h e l l . S eco n d,(K ifalfy ^h ad s e e n t h e i n t e r i o r o f o n e o f t h e s e t t l e m e n t a lt a r s , a n d sa id t h a t it c o n s i s t e d o f a s p i r a l p assag e. T h e b e s t c l u e w e - f o u n d was f r o m D i a t t a h i m s e l f , w h o d e s c r i b e d a lo g s p i ­ ral p a t h in c e r t a i n r i t u a l s t h a t t o o k p l a c e in t h e s a c r e d f o r e s t . B u t h o w t o r e c ­ o n c ile th is self-co n scio u s m o d e lin g w ith w h a t a p p e a re d to be th e e m e rg e n c e o f th e s e t t l e m e n t s tru c tu r e th r o u g h ag g reg ate s e lf-o r g a n iz a tio n ? 1 finally c o n ­ f e s s e d m y d is t u r b a n c e to D i a t t a , a n d a sk e d h i m b o w I m i g h t u n d e r s t a n d t h e a p p a r ­ e n t c o n t r a d i c t i o n . H e s u g g e s t e d y e t a n o t h e r s i m u l a t i o n : t h e J o l a f u n e r a l ritu a l (fig. 1 0 .9 a ) . 'W e h a d b e e n a le r te d to th i s c e r e m o n y as a r e s u lt o f a su spicio us d e a t h d u r i n g c u r - v is i t, b u t w e r e n o t a l l o w e d t o a t t e n d . D i a t t a d e s c r i b e d t h e r i t u a l in d e ta il. T h e body o f th e d eceased was p la c e d o n a p la tfo rm , a n d posts at e a c h o f t h e f o u r c o r n e r s a r e h e l d a lo f t b y p a l l b e a r e r s . If c r i t i c a l k n o w l e d g e is t h o u g h t to h a v e b e e n h e l d b y t h e d e c e a s e d (e.g., a s in t h e c a s e o f a m u r d e r ) , a p r i e s t asks q u e s t i o n s . T i r e p a l l b e a r e r s , r e a c t i n g t o t h e f o r c e o f t h e d e c e a s e d , m o v e che p l a t ­ fo r m t o t h e r i g h t for yes, le ft for n o , a n d f o r w a r d fo r “ u n k n o w n . " T h e s i m u l a t i o n for t h i s ritual..(fig. J0..9.b) is b a s e d o n a n a n a l o g f e e d b a c k n e t w o r k . W e d o n ’t n e e d t o m a k e a n y a s s u m p t i o n s a b o u t w h e t h e r t h e p a l l b e a r ­ ers a r e e x e r t i n g f o r c e d u e t o c o n s c i o u s o p i n i o n s o r s u b c o n s c i o u s b eliefs; it is o n ly n e c e s s a r y t o a s s u m e t h a t t h e y e x e r t f o r c e in p r o p o r t i o n t o t h i s m o t i v a t i o n S i n c e th e y c a n b o t h e x e r t fo rce a n d sen se it f r o m o t h e r s , t h i s w o u ld th e o r e t i c a l l y a ll o w t h e s u m m a t i o n o f k n o w l e d g e a m o n g t h e p a r t i c i p a n t s t o b e e x p re s se d in t h e m o s t e ff e c ti v e w a y p o ss ib le . I n f a c t , t h e t e c h n i q u e is m o r e e f f e c t i v e t h a n a v o t e , s i n c e v o t i n g c a n le a d t o t h e p a r a d o x o f a m i n o r i t y o p i n i o n w in if t h e r e a re m o r e t h a n t w o o p t i o n s . ^ T h e i n f o r m a t i o n e m e r g e d fr o m t h e b o t t o m - u p i n t e r a c t i o n o f Complexity 165 t h e p a rt s , y e t it w as a ls o i n t e n t i o n a l i n t h e s e n s e t h a t t h i s m e c h a n i s m fo r aggreg a te s e l f-o r g a n iz a t io n o f k n o w l e d g e h a d b e e n c o n s c i o u s ly d e s i g n e d . T h i s w a s n o t i n t e n t i o n a l i t y as I k n e w it; it s o u n d e d m o r e lik e t h e d e s c r i p t i o n o f a n e u r a l n e t ­ w o rk j n . c o m p u t e r s c i e n c e : If a p ro g r a m m e r has a n e u r a l n e t w o r k m o d e l o f v is io n , for e x a m p le , h e or she c a n s i m u l a t e che p a t t e r n o f lig h t a n d d a r k fallin g o n t h e r e t in a by a c t i v a t ­ in g c e r t a i n i n p u t n o d e s , a n d t h e n l e t t i n g th e a c t i v a t i o n sp re a d t h r o u g h th e (a) In the Jola funeral ritual four pallbearers hold a platform aloft and move it in response to questions. Since the inform ation (w h e th er o n e believes it to be of spiritual or m u n d an e origin) is held by che pallbearers, w e c a n model che force o f each corner as h a v in g direction and magnitude (a vector) d eterm in ed by the pallbearer’s conviction. Decision making based on a co n tin u o u s range rather th a n on yes/no is called “fuzzy logic” in mathematics. (b) We can th in k of the inform ation processing in the Jola funeral as the equivalent of a neural net (similar to th a t in lig. 10.2) in w hich the sum o f the force vectors of all four pallbearers are inputs to lluee amplifiers, w ith each inverted o u tp u t connected as negative feedback to the oilier two. This would require pallbearers to both exert force as well as sense it, b u t juch force-feedback is actually quite common in motor tasks. unknown FIGURE I O . 9 N e u r a l n e t m o d e l for t h e J o la f u n e r a l r itu a l c o n n e c t i o n s i n t o th e re s t o f t h e n e tw o r k . T h e e ff e c t is a b i t lik e s e n d i n g s h i p lo a d s o f g o o d s i n t o a few p o r t c itie s a l o n g t h e s e a c o a s t, a n d t h e n l e t t i n g a zillion trucks c a r t t h e stu ff .along th e h ig h w ay s a m o n g t h e i n l a n d c ities. But if t h e c o n n e c t i o n s h a v e b e e n p r o p e r ly a rr a n g e d , t h e n e t w o r k w ill so o n s e t t l e i n t o a s e l f - c o n s is t e n t p a tc e r n o f a c t i v a t i o n t h a t c o r r e s p o n d s to a c la s sif ic a ­ t i o n o f th e sce n e . “T h a t ’s a c a t ! ” ( W a l d r o p 1 9 92 , 2 8 9 - 9 0 ) T h e t r i c k y p a r t is “ if t h e c o n n e c t i o n s h a v e b e e n p r o p e r l y a r r a n g e d . ” C l e a r l y i t c o u l d b e a r r a n g e d fo r f o u r p e o p l e , b u t c o u l d it f o r t h i s ' c i t y o f M l o m p , w i t h d o z e n s o f l o c a l n e i g h b o r h o o d s a n d h u n d r e d s o f p e o p l e in e a c h ? A n d M l o m p is n o t a n a n o m a l y . W h i l e w e sa w a m o r e e x p l i c i t f o r m a l s y s t e m in t h e c o n s t r u c t i o n o f se v e ra l f r a c ta l s e t t l e m e n t a r c h i t e c t u r e s in c h a p t e r 2, t h e r e a re also m a n y A f r i c a n s e t t l e m e n t s t h a t h a v e a larg e, diffuse f r a c ta l s t r u c t u r e (see D e n y e r 1978, 1 44). S elf-o rg a n izin g m e c h a n ism s t h a t arran g e su c h vast ag greg atio n s i n t o c o h e r e n t p a t t e r n s w o u l d h a v e t o b e m o r e g l o b a l a n d less e x p l i c i t . O n e k ey m e c h a n i s m i n c o m p l e x i t y t h e o r y is m e m o r y : t h e t h e o r y p r e d i c t s t h a t s e l f - o r g a n iz i n g s y s te m s will u tiliz e 1 / F d.is.tpi.bu.tiQns. in m e m o r y l e n g t h . T h e lu kasa, a visual “m e m o r y b o a r d ” d e v e l o p e d by t h e B a lu b a o f C o n g o (Z a ire ), sh ow s ju s t s u c h f r a c ta l s c a l i n g (fig. 1 0 .1 0 ) . T h e m e m o r y s y s t e m o f t h e l u k a s a is p a rtly b a s e d o n d i g i ta l ( t h a t is, p h y s i c a ll y a r b i t r a r y ) c o d i n g , s u c h as c o lo r , b u t R o b e r t s ( 1 9 9 6 ) n o t e s t h a t m u c h o f t h e lukasa is a “g e o m e try o f id e a s ,” m a p p i n g th e J a e a d e d s p a t i a l s t r u c t u r e t o a n a l o g o u s h i s t o r i c a l e v e n t s . A l t h o u g h t h e r e is c o n s i d e r a b l e i n t e r p r e t i v e a n d c o d i n g v a r i a t i o n , t h e r e is a t e n d e n c y t o h a v e s i n g l e b e a d s r e p ­ r e s e n t i n g i n d i v id u a l s , g r o u p s o f b e a d s r e p r e s e n t i n g r o y a l c o u r t s , a n d la r g e r be ad a r r a n g e m e n t s s h o w i n g t h e s a c r e d fo r e sts t h a t h a v e b e e n g r o w i n g o v e r m a n y g e n e r a t i o n s . T h i s v i s u a l i z a t i o n o f a 1/ F - l i k e d i s t r i b u t i o n o f m e m o r y s u g g e s ts at le a s t t h e p o s s ib ility o f i n d i g e n o u s a w a r e n e s s o f s c a l i n g p r o p e r t i e s in m a i n t a i n ­ in g s e l f - o r g a n iz e d c o m p l e x it y . T h e s t r o n g e s t c a n d i d a t e f o r a m e c h a n i s m u n d e r l y i n g s e l f - o r g a n i z a t i o n is th e c o m p le m e n ta ry p air o f in d ig e n o u s-fe e d b a c k c o n c e p ts, w e e x a m i n e d j n c h a p t e r 8. I n t h e v o d u n r e l i g i o n o f B e n i n , w e f o u n d D a n r e p r e s e n t i n g t h e s t a ­ b i l i z i n g fo r c e o f n e g a t i v e f e e d b a c k , a n d L e g b a r e p r e s e n t i n g t h e d i s r u p t i v e fo r c e o f p o s i t i v e f e e d b a c k . S i m i l a r f e e d b a c k p a ir s w e r e f o u n d in t h e B au le d o o r c a r v i n g s ; t h e c a i m a n s b i t i n g e a c h o t h e r ’s t a i l s a r e a s y m b o l o f n e g a t i v e f e e d b a c k , a n d t h e fish e a t i n g e v e r l a r g e r fish r e p r e s e n t p o s i t i v e f e e d b a c k . T h i s c o m b i n a t i o n o f o p p o s i n g f e e d b a c k l o o p s a ls o a p p e a r s t o h e a t t h e h e a r t o f th e c o n d i t i o n s t h a t s u s t a i n s e l f-o r g a n iz i n g s t r u c tu r e s . O f c o u r s e , in o s t-s e lf-o rg a n iz in g s y s t e m s w ill h a v e m o r e t h a n t w o lo o p s ; b u t in e v e r y c a s e 1 h a v e e x a m i n e d , at le a s t o n e o f e a c h is p r e s e n t , a n d it is t h r o u g h t h i s i n t e r a c t i o n t h a t s u s t a i n e d c o m p l e x i t y c a n a rise. FIGURE 1 0 . 1 0 Lukasa ( F w m R o b e r ts a n d R o b e r ts 1996; photo b y D i c k D e a u lie u x .) African fra c ta l m a th em a tics R e t u r n i n g t o t h e m o s t b a s i c e x a m p l e o f c o m p l e x b e h a v i o r , M a y ’s p o p u l a ­ t i o n e q u a t i o n , w e h a v e tw o c o m p o n e n t s . O n t h e o n e h a n d , t h e r e is p o p u l a t i o n g r o w t h : f*n+1 = Pn R. N e x t y e a r ’s p o p u l a t i o n w ill b e th i s y e a r ’s p o p u l a t i o n tirn es^ " t h e g r o w t h r a t e . A s l o n g as R is a p o s i t i v e n u m b e r , t h i s w ill b e a p o s i t i v e f e e d ­ b a c k lo o p . B u t t h e o t h e r p a r t o f t h e e q u a t i o n , m u l t i p l y i n g b y (1 - Pn ), w as a neg-,; Fa t i v e f e e d b a c k lo o p , a c t i n g lik e a n e p i d e m i c t h a t k ills m o r e p e o p l e w i t h la rg e r p o p u l a t i o n size.-* T o g e t h e r th e y c r e a t e d e t e r m i n i s t i c c h a o s : t h e p o s i ti v e f e e d b a c k \ k e e p s e x p a n d i n g t h e p o p u l a t i o n , a n d t h e n e g a t i v e f e e d b a c k k e e p s it w i t h i n j b o u n d s . T h i s w o r k s for o t h e r c h a o s e q u a t i o n s as w e ll. F ig u r e •' i o .i j show s a c h a o s e q u a t i o n c a l l e d t h e " R o s s le r a t t r a c t o r ’’ m o d e l i n g a c a r w i t h tw o d riv e rs . ^ ^ O n e is d r u n k a n d o v e r c o m p e n s a t e s by s t e e r i n g t o o far w i t h e a c h c o r r e c t i o n ; t h e C ) o t h e r is s o b e r a n d p u lls it b a c k o n t h e r o a d w h e n t h e d r u n k e n o s c i l l a t i o n s g e t \ . { t o o larg e. B e c a u s e it a lw ay s s t e e r s b a c k t o a s l ig h t ly d i f f e r e n t p o s i t i o n , t h e o scil- \ ) l a t i o n s n e v e r r e p e a t — d e t e r m i n i s t i c c h a o s .^ W e c a n see th e sam e c o m b in a tio n o f n e g a tiv e a n d po sitiv e feed b ack cre ­ a t i n g s e l f - o r g a n i z a t i o n in a g g re g a t e s y ste m s. T h e “ g a m e o f life" c e l l u l a r a u t o m a ­ t o n offers a p a r t i c u l a r l y c l e a r i l l u s t r a t i o n o f th i s p h e n o m e n o n . If we g iv e a ru le s e t t h a t m a k e s b i r t h t o o easy (e.g ., t h e c e ll-g o es t o t h e “l i v e " s t a t e if t h e r e is o n e o r m o r e n e a r e s t n e i g h b o r s a l i v e ) , t h e n t h e r e is t o o m u c h p o s i t i v e f e e d b a c k a n d w e g e t a r a p i d ly s p r e a d i n g d is k . If w e m a k e d e a t h t o o e a sy (e.g., t h e c e l l g oes to t h e “d e a d ” s t a t e if t h e r e is o n e o r m o r e n e a r e s t n e i g h b o r s a l i v e ) , t h e s c r e e n g oes FIGURE 10 .1 1 R o s s le r a t t r a c t o r as fe e d b a c k i n a u t o m o b i l e d r iv in g T he Rossler attractor is a set- of three simple equations whose output is derprminisric chaos, thar_i.s«. a signal with variable oscillations which remain bounded but never repeat the exact same pattern. How can such a simple system produce infinite variation? A n automobile driving model can help us see what these equations are doing. (a) Positive f e e d b a c k . First, there is a part of the system that provides a positive feedback loop; this acts like a drunken driver who swerves farther and farther off the road. N ote that the car is not properly aligned with the direction of travel; this skidding is the nonlinear relationship between road position X and steering angle Y. (b) N e g a t i v e f e e d b a c k . T he other part of the system is a negative feedback loop; given a swerving input, this cautious driver steers back toward the center of the road. “C aution” is represented by the third variable, 2. (c) Combination o f n e g a t i v e a n d positive f e e d b a c k . Here we see the complete Rossler system at work. T he "caution" variable Z controls the facial expression (diameter of eyes and mouth, angle of eyebrows). Note that after the oscillation gets large enough, the negative feedback kicks in, and we go hack toward the center of the road. Because the car never steers back to exactly the same position on the road, the behavior never repeats. If, for example, you looked at the number of increasing oscillations that occur before the negative feedback dampens it hack toward the center, it would appear to he completely random, with no predictable pattern. Yet the pattern is entirely deterministic (that is, determined only hy this set of equations); it could be predicted if you knew the initial conditions with infinite precision. Driver (mis)observation noise road poslllon Driver observation: hos car been deviating? L YES Move steering angle in ( Y) new steering angle (V) a Positive feedback Driver ooservnllon Increase Driver observation: - --- * cauttonl is cor position >setpolnt ? (2) Decrease _a steering (V) angle new poslllon on road (X) x b Negative feedback 0.1 5 y of negative and positive feedback x A fr ic a n fra cta l macliematics b l a n k in a few g e n e r a t i o n s . T h e “classic” life rule set,.(found by J o h n H o r t o n C o n ­ w ay in 1 9 7 0 ) is o f t e n re f e r r e d t o as “ 3 - 4 ’’ life b e c a u s e it t a k e s 3 n e a r e s t n e i g h ­ b o rs to g iv e b ir th , b u t 4 results in d e a t h . C o n w a y d is c o v e re d t h a t th is c o m b i n a t i o n o f n e g a tiv e a n d po sitiv e feed b ack m axim ized th e c o m p le x ity of b e h av io r. S im ­ ilarly, w h e n P e r B a k f o u n d e m p i r i c a l d a t a for s e l f - o r g a n i z a t i o n i n p h y s i c a l sys­ t e m s — fo r e st fires, e a r t h q u a k e s , a v a l a n c h e s , e t c . — h e n o t e d t h a t it o c c u r r e d o n ly a t a “critic a l s t a t e ” in w h i c h t h e r e was a b a l a n c e b e t w e e n n o i s e - s u p p r e s s i n g m e c h ­ a n i s m s — w h i c h w o u ld c o r r e s p o n d t o n e g a t i v e f e e d b a c k — a n d t h e p o s i t i v e f e e d ­ b a c k o f n o i s e - a m p l i f y i n g lo op s. " It is u n f o r t u n a t e t h a t so m u c h o f th e classic re s e a rc h o n A f r i c a n social m e c h ­ a n i s m s c a m e fro m f u n c t i o n a l i s t a n t h r o p o l o g y , s i n c e t h e y m a d e a n a l m o s t e x c l u ­ siv e e m p h a s i s o n t h e r o l e o f n e g a t i v e f e e d b a c k in a c h i e v i n g e q u i l i b r i u m . W h e n it c o m e s t o c o n s c i o u s k n o w l e d g e s y s te m s , A f r i c a n s o c i e t i e s d o n o t e x c l u s iv e l y fo cu s o n b a la n c e , h a rm o n y , a n d stasis. T h e c o m p l i m e n t a r y roles o f D a n a n d Legba, o f o r d e r a n d d is o rd e r, are m u c h m o r e c o m m o n , as w e s e e in t h i s passage: “I n t h e m i n d o f t h e B a m b a r a s t h e air, w i n d a n d fire . . . a r e i n d i s p e n s a b l e e l e m e n t s o f t h e w o r l d ’s o n w a r d m o v e m e n t . B u t as t h e s e p r i n c i p l e s m a y b e a c t i v e in a n u n c o n t r o l l e d , t h a t is, u n r u l y a n d o f t e n e x c e s s i v e m a n n e r , N y a l e is c o n s i d e r e d t o b e a p ro f u s e a n d e x t r a v a g a n t b e i n g . . . . S o by h e r v e r y n a t u r e N y a l e is, to a c e r t a i n e x t e n t , a f a c t o r o f d is o r d e r . T h a t is w h y it is s a i d t h a t B e m b a . . . t o o k , a w a y h e r ‘d o u b l e ’ t o e n t r u s t i t t o F a r o . . . w h o s e e s s e n t i a l a t t r i b u t e is e q u i l i b ­ r i u m " ( Z a h a n 1 9 7 4 , 3 ). A s i m i la r p a ir in g o c c u rs in t h e D o g o n r e l ig i o n , w h e r e A r n m a , th e h i g h god, c r e a t e s t h e N u m m o t o e n a c t o rd e r , a n d a c c i d e n t a l l y c r e a t e s t h e d i s o r d e r l y O g o ; t o g e t h e r t h e t w o g e n e t viic life as w e k n o w it. I n t h e r e p e r t o i r e o f d y n a m - • ical c o n c e p t s o c c u r r in g in se v e r a l A f r i c a n k n o w le d g e system s, t h e r e is r e c o g n i t i o n o f th e .u s e fu l t e n s i o n b e t w e e n e q u il ib r iu m a n d d i s e q u i l i b r i u m , t h e d a n c e b e t w e e n o r d e r a n d c h a n c e t h a t r e s u l t s in s e l f - o r g a n iz e d c o m p l e x i t y . A n d ju s t as S t u a r t K a u f f m a n h a s s h o w n a bias t o w a r d o r d e r in e v o l u t i o n ’s “e d g e o f c h a o s , ” t h e h i g h \ / g o d e n s u r e s t h a t t h e tr ic k s t e r c a n a c t o n ly sp o r a d ic a lly , t h u s c r e a t i n g m o r e p o w e r j y t o w a r d l o n g - t e r m o r d e r in t h e s e A f r i c a n c o s m o l o g i e s . A l t h o u g h f r a c ta l s r e s u l t i n g f r o m g e o m e t r i c a l g o r i t h m s :a r e u s u a lly s e e n as s t a t i c s t r u c t u r e s , t h e y t o o c a n b e v i e w e d as t h e c o m b i n a t i o n o f f e e d b a c k lo o p s . A s e e d s h a p e w i t h a h u g e n u m b e r o f t i n y l i n e s e g m e n t s (fig. 1 0 . 1 2 a ) w_ii_l t e n d t o b e s h a p e - p r e s e r v i n g u n d e r s e l f - r e p l a c e m e n t i t e r a t i o n s ; h e r e d e v i a t i o n s due_tp r e p l a c e m e n t a r e d a m p e d —-(the d i f f e r e n c e b e t w e e n a l i n e s e g m e n t a n d t h e s e e d s h a p e is u s u a l ly n o t i m p o r t a n t ' ( a n d t h e r e s u l t i n g g r a p h w ill h a v e a lo w f r a c ­ ta l d i m e n s i o n , i.e., t e n d i n g t o w a r d j . o ) . B u t fo r s e e d s h a p e s m a d e u p o f o n l y a few la r g e l i n e s (fig. 1 0 . 1 2 b ) , t h e d i f f e r e n c e b e t w e e n a l i n e s e g m e n t a n d its FIGURE l o . 1 2 F r a c t a l g r a p h ic s a s fe e d b a c k A fr ic a n fra c ta l m athem atics 172 r e p l a c e m e n t s h a p e w ill b e v e r y i m p o r t a n t . L a r g e d e v i a t i o n s t e n d t o b e a m p l i ­ fied in a q u i c k p o s i t i v e f e e d b a c k , s o m e t i m e s e x p l o s i v e l y g r o w i n g o u t o f b o u n d s in o n l y a few i t e r a t i o n s . F i g u r e 1 0 . 12 b h a s ' b e e n s c a l e d d o w n t o fit o n t h e p a g e , b u t t h e a c t u a l f r a c t a l g r a p h w ill q u i c k l y g r o w o u t o f b o u n d s a n d b l a c k e n t h e s c r e e n e n t i r e l y (i .e . , a f r a c t a l d i m e n s i o n c l o s e t o 2 . 0 ) . F i g u r e 1 0 . 1 2 c s h o w s a f r a c ta l d i m e n s i o n c lo s e tjo 1.5, t h e " m o s t f r a c t a l " m e a s u r e , w h i c h r e s u l t s fr o m a b a l a n c e b e t w e e n t h e n e g a t i v e f e e d b a c k o f s m a ll s e g m e n t s h a p e p r e s e r v a t i o n a n d t h e p o s i t i v e f e e d b a c k o f la r g e s e g m e n t r e p l a c e m e n t d e v i a t i o n . T h e r e is n o q u a n t i t a t i v e m e a s u r e o f f r a c t a l d i m e n s i o n in p r e c o l o n i a l A f r i c a n k n o w l e d g e sy s te m s . B u t t h e i d e a o f a s p e c t r u m p r o g r e s s i n g f r o m m o r e o r d e r ly t o less o r d e r l y is v i v i d ly portraye r! in c e r t a i n m a t e r i a ! d e s i g n s . T h e best e x a m p l e s a re in t h e Taffia p a l m t e x t il e s o f t h e B a k u b a (fig. 1 0 .1 3 a ). T h e se t e n d to s h o w p e r i o d i c t i h n g a l p n g o n e axis, a n d aperiodic.t.i.ling,— o f t e n m o v i n g fr o m o rd e r to d iso rd er— a lo n g th e o th e r. S im ila r g e o m e tric v is u alizatio n s o f th e sp e c tru m f ig u r e 1 0 .1 3 F r o m o r d e r to d is o r d e r in a B a k u b a c l o t h ( a ) T h e B a k u b a o f t e n c r e a t e c l o t h d e s i g n s t h a t s t a y fa i rl y c o n s t a n t a l o n g t h e v e r t i c a l a x i s , b u t g r a d u a l l y c h a n g e a l o n g t h e h o r i z o n t a l a x is . I n m a n y c a s e s, t h e h o r i z o n t a l t r a n s f o r m a t i o n s u g g e s t s a n o rd er-d iso rd er range, (b ) C o m p u t e r sc ie n tist C liffo rd P ic k o v e r c r e a te d th is p a t t e r n to sh o w h o w a s p e c tru m from o r d e r to d is o rd e r c o u ld b e v isu a lize d by a llo w in g a r a n d o m v a r ia b le to h a v e i n c r e a s i n g i n f l u e n c e o n t h e g r a p h ’s e q u a t i o n . T h u s it, t o o , m a k e s u se o f p e r i o d i c r i l i n g a l o n g t h e v e r tic a l axis a n d a p e r i o d i c a l o n g t h e h o r iz o n ta l. (a, from Me u r a n t 1986, b y perniis.u'on o f the author; b , f r o m Pick over 1990, b y permission o f the author.) C o m p le xity fro m o r d e r to d i s o r d e r h a v e b e e n u se d in c o m p u t e r s c i e n c e (fig. 1 0 .1 3 b ). A s far as I c a n tell, t h e B a k u b a w e a v in g s n e v e r r e a c h m o r e t h a n halfw ay across t h e s p e c ­ tr u m — th e y a r e ty p ically m o v i n g b e t w e e n j a n d 1.5, t h a t is, fr o m p e rio d ic t o fractal, r a t h e r t h a n s t r e t c h i n g all t h e w ay tg vp u r e diso rd er.^ 1 k n o w o f o n l y o n e A f r i c a n t e x t i l e t h a t cakes t h i s la st s t e p , a n d t h a t is t h e b lo c k p r i n t s h o w n in figure 10.14. T h i s p a t t e r n is r e m in i s c e n t o f t h e title o f N i g e r ia n a u t h o r C h i n u a A c h e b e ’s f a m o u s n o v e l , T h in g s F a ll A p a r t. G i v e n t h e a n t i c o l o n i a l c o n t e x t o f A c h e b e ’s w r i t i n g , it m i g h t b e t e m p t i n g t o re a d it as a n i n d i c a t i o n t h a t w h i t e n o i s e o n l y c o m e s w i t h w h i t e p e o p l e , b u t a t le ast in te r m s fig u re 1 o . 14 B lo c k p r in t t e x tile This print from West Africa suggests the full spectrum from order to disorder. (From S i e b e r 1972.) *73 A fr ic a n fra c ta l mathematics 174 o f t h e i n d i g e n o u s k n o w l e d g e s y s t e m s u c h a s s u m p t i o n s a r e u n f o u n d e d .® T h e r e is, fo r e x a m p l e , a f o r m o f m u s ic i n d i g e n o u s t o N i g e r i a t h a t h a s s o m e t h i n g lik e a w h i t e n o is e d i s t r i b u t i o n of-sounds. A k p a b o t (-1975) d e sc rib e s “ t h e r a n d o m m usic o f t h e B i r o m , ” a f l u te e n s e m b l e d e s i g n e d to a llo w e a c h m u s i c i a n t o e x p r e s s i n d i ­ v i d u a l fe elin g s t h r o u g h w h a t e v e r i d i o s y n c r a t i c n o i s e ( o r e v e n s ile n c e ) _ h e o r sh e c h o o s e s , r e s u l t i n g in " a n i n d e t e r m i n a t e p r o c e s s [in w h i c h ] t h e s o u n d s p r o d u c e d / . b y t h e p la y e rs a re n o t o b s t r u c t e d by a c o n s c i o u s a t t e m p t t o o rg a n i z e t h e r h y t h m s ^ a n d h a r m o n i e s ” ( p . 4 6 ) . P e l t o n ( 1 9 8 0 ) re fe rs t o t h e N i g e r i a n ( Y o r u b a ) t r i c k s t e r E s h u as t h e “lo r d o f r a n d o m , ” a n d n o t e s t h a t t h e r e is a c o u p l i n g b e t w e e n t h e o rd erly w o rk of O lir u n a n d th is u n p r e d ic ta b le sp irit, sim ila r to t h e n e g a tiv e fe e d b a c k /p o sitiv e feed b ack c o m b in a tio n s we n o te d earlier. T h e c h a ra c te riz a ­ ti o n o f e x tr e m e d is o rd e r m i g h t w ell b e a p p lie d to t h e e x p e r i e n c e o f c o lo n ia l rule, b u t w e s h o u ld n o t assu m e t h a t t h e c o n c e p t was u n k n o w n b e fo re th e n . A s u m m a r y o f s e l e c t e d A f r i c a n c o m p l e x i t y c o n c e p t s is s h o w n i n f ig u re 1 0 .1 5 ; n o t e t h a t t h e c e n t r a ! p e a k o f s p i r i t u a l p o w e r is a n a l o g o u s t o t h e c e n t r a l p e a k o f c o m ­ p u t a t i o n a l p o w e r in t h e C r u t c h f i e l d - S m a l e c o m p l e x i t y m e a s u r e . C o n c lu s io n T h i s c h a p t e r is o n l y t h e b a r e o u t l i n e o f w h a t I h o p e w ill b e f u t u r e a r e a s o f r e s e a r c h , e x a m i n i n g t h e r e l a t i o n s b e t w e e n t e c h n i c a l , c u J t u raJ j . a,p d - P 9 ]it i? ? i s y s te m s t h r o u g h t h e n e w f r a m e w o r k s o f f e r e d b y c o m p l e x i t y , t h e o r y . F o r t h e m o m e n t , w e w ill h a v e t o l i m i t o u r s e lv e s t o t h e few f r a g m e n t s t h a t m y S e n e g a le s e c o l l e a g u e s p o i n t e d o u t so d i l i g e n t l y i ^ F i r s t ^ t h i s d o e s n o t . n e g a t e t h e p r e v i o u s e x a m p l e s o f e x p l i c i t a l g o r i t h m i c d e s i g n in A f r i c a n fractals,,?..buj..j_t..docs su g g e s t t h a t a t le a s t in t h e c a s e o f se ttle_ m en t a r c h i t e c t u r e t h e y c a n . a r i s e f r o m a n o t h e r s o u r c e as w ell. T h e c r e a t i o n o f f r a c ta l s e t t l e m e n t p a t t e r n s t h r o u g h a g g re g a t e self­ o rg a n iz a tio n , w h ile u n li k e t h e p l a n n e d s tru c tu r e s we saw in c h a p t e r 2, d o n o t seem t o b e t h e r e s u lt o f u n c o n s c i o u s s o c i a l d y n a m i c s (a s w e sa w j b r t h e u r b a n sp r a w l o f E u r o p e a n c itie s in c h a p t e r 4 ). T h i s m a y b e d u e t o a differ e r v c ^ b e t w e e n A.fr i ca n c o n c e p t s o f i n t e n t i o n , w h i c h c a n a p p ly t o a g r o u p p r p j e c t_ c r e a te c l.p v e r . s e v e r a l g e n e r a t i o n s , v e rs u s t h e W e s t e r n fo c u s o n a n i n d i v i d u a l p e r f o r m i n g i m m e d i a t e a c t i o n in d e f i n i n g i n t e n t i o n a l i t y . M o s t i m p o r t a n t , t h e r e a r e i n d i c a t i o n s t h a t th is p a t t e r n c r e a t i o n t h r o u g h g r o u p a c t i v i t y is s u p p o r t e d by c o n s c i o u s m e c h a n i s m s s p e c i f ic t o s e l f - o r g a n i z a t i o n as d e f i n e d in c o m p l e x i t y t h e o r y . B o t h t h e s c a l i n g ( ' d i s t r i b u t i o n o f i n t e r a c t i o n s w i t h m e m o r y a n d che s p e c t r u m fr o m o r d e r t o d i s ­ o r d e r h a v e a t le a s t s o m e g r a p h i c c o u n t e r p a r t s in A f r i c a n d e s i g n s . T h e b e s t c a n ­ d i d a t e for a c o n s c i o u s m e c h a n i s m is t h e c o m b i n a t i o n o f n e g a t i v e a n d p o s i t i v e fe e d b a c k . W e d id n o t e x a m i n e e v ery possible case o f d e te r m i n is t ic c h a o s an d ^ order fractal disorder Akan (G hana): Ananse t h e trickster ' \ J \ S \ S \ , I c o n f o r '“c a l m w a t e r s ” N y a m e ' s p o w e r o f lif e; tu r b u le n t waters o f Tanu V od un (B e n i n , Nigeria, African L e g b a , Eshu che tricksters diaspora): D an M a w u (acts th rough lo w e r gods, e.g., th e b ifu rca tin g doublings o f Shango) U'gon (Mali): Ogo th e trickster N u m m o (d raw in g based on p h o t o o f r i t u a l s t a f f in I m p e r a t o 1 97 8) A m i n a ( d e s c r i b e d as a n e x p a n d i n g spiral, lik e a w h irlw in d) FIGURE 1 0 . 1 5 A f r i c a n c o m p l e x i t y c o n c e p ts i n r e l i g i o n 176 A fr ic a n fractal m a th em a tics a g g r e g a t e s e l f - o r g a n i z a t i o n , b u t it w o u l d a p p e a r t h a t t h e c o m b i n a t i o n o f n e g ­ a t i v e a n d p o s i ti v e fe e d b a c k lo o p s , w h i c h fo r m t h e basis o f s e v e r a l A f r i c a n k n o w l ­ e d g e s y s te m s , a ls o f o r m ' a k e y m e c h a n i s m o f g e n e r a l s e l f - o r g a n i z i n g s y s te m s . A s n o t e d in t h e first c h a p t e r , it is j u s t as i m p o r t a n t t o f i n d w h a t is m i s s ­ i n g as it is to f i n d w h a t is p r e s e n t . W h i l e fo u r o f t h e fiv e b a s i c c o n c e p t s o f f r a c ­ t a l g e o m e t r y — s c a l i n g , s e l f - s i m i l a r i t y , r e c u r s i o n , a n d i n f i n i t y — a r e a ll p o t e n t aspiects o f A f r i c a n m a t h e m a t i c s , a q u a n tita tiv e m ea su re o f d im e n sio n (th e H a u sd o rfB e s ic o v itc h m e a s u r e ) is c o m p le te ly a b se n t. T h e r e is a w e a k s e n s e o f a c o m p l e x i t y sp e c tru m o f order-disorder, w h ic h w o u ld co v ary w ith th e H a u sd o rf-B c s ic o v itc h m e a s u r e , b u t t h a t s p e c t r u m is n e i t h e r q u a n t i t a t i v e n o r ( t o m y k n o w l e d g e ) e v e r c o m p a r e d to a c o n c e p t o f d i m e n s i o n in any. i n d i g e n o u s A f r i c a n s y s te m . T h i s is a n e n o r m o u s g ap in t h e A f r i c a n k n o w l e d g e o f f r a c t a l g e o m e t r y , e s p e c i a l l y s i n c e t h e d i m e n s i o n a l m e a s u r e is o f t e n c o n s i d e r e d t h e m o s t v a l u a b l e c o m p o n e n t by c o n t e m p o r a r y r e s e a r c h e r s i n t h e field... O n t h e o t h e r h a n d , w e a ls o n e e d t o a p p r e c i a t e all k n o w l e d g e s y s te m s in th e i r o w n rig h t, a n d A f r ic a n fractals h a v e a s u rp ris in g ly s tr o n g u tiliz a tio n o f r e c u r s i o n . I n d e e d , in M a n d e l b r o t ’s s e m i n a l t e x t , T h e F ra c t a l G e o m e tr y o f N a tu r e ( 1 9 7 7 ) , t h e i n d e x lis ts “ r e c u r s i o n ” o n l y t w i c e , a n d t h e t e r m s i t e r a t i o n , self-, r e f e r e n c e , s e l f - o r g a n i z a t i o n , a n d f e e d b a c k a r e e n t i r e l y . a b s e n t - A s w e w ill s e e , t h i s a b s e n c e is n o a c c i d e n t ; it.re fle c ts a E u r o p e a n h i s t o r i c a l tr e n d - B u t w h y h a v e E u r o p e a n s t r a d i t i o n a l l y p la c e d s u c h li t t l e i m p o r t a n c e o n r e c u r s i o n , a n d w h y w as i t so s t r o n g l y e m p h a s i z e d in A f r i c a n fracta ls? In p a r t m o f t h i s b o o k w e will t a k e u p s u c h c r o s s - c u l t u r a l c o m p a r i s o n s in d e t a i l . Implications C H A P T E R —T heoretical— —frameworks— ------------- j n ------------- cultural studies-o f knowledge- ~J P a r t s 1 a n d 11 o f t h i s b o o k e m p h a s i z e d t h e g e o m e t r i c , s y m b o li c , a n d q u a n t i t a - ^ [ tiv e a s p e c t s o f A f r i c a n fracta ls. S o m e c ase s w e r e m o r e s p e c u l a ti v e t h a n o t h e r s — j / a d i f f e r e n c e t h a t 1 h o p e w as c l e a r l y i n d i c a t e d — b u t e v e n in t h e u s e 'o f m y t h i c ] \ ' J n a r r a ti v e , 1 g e n e ra lly r e s tr a i n e d c o n c l u s i o n s to th o s e t h a t h a d g e o m e t r i c o r q u a n - i | t i t a t i v e c o u n t e r p a r t s . In o t h e r w o r d s , t h e c l a i m s m a d e in p a r t s i a n d n s h o u l d i ■, be fulsifiah le in t h e s e n s e o f K a r l P o p p e r ; t h e d a t a e i t h e r s u p p o r t s t h e h y p o t h e - j \ sis o r re fu te s it.* B u t th e c h a p t e r s in th i s last s e c t i o n w ill s w i t c h t o to p ic s in c u l- ' t u r a l p o l i t i c s a n d o t h e r h u m a n i t i e s . T h e s e is s u e s a r e t o o c o m p l e x a n d m u l t i d i m e n s i o n a l to b e r e d u c e d t o f o r m a l r e p r e s e n t a t i o n s ; t h e y c a n o n l y b e a p p r o a c h e d b y e x p l o r i n g t h e i r j u u e r p r e t a r i y e d e p _ th s (P o e tr y ' a n r e v e a l as m u c h t r u t h a b o u t t h e w o r l d as a n y s c i e n c e ; w e o n l y n e e d t o k e e p in m i n d t h a t it is a d i f f e r e n t w ay o f g o i n g a b o u t it. W h i l e t h e p h i l o s o p h y , p o l i t i c s , a n d p o e t i c s o f c u l t u r e a r e n o t s t r ic t ly fa jsjfia b le, t h e y c a n o f t e n a p p r o a c h t h e j i r e a s o f l i f e j h a t P o p p e r i a n p o s i t i v i s m c a n n o t - — a r e a s w e c a n n o t li v e w i t h o u t . G i v e n t h a t o n e c a n m a k e a g o o d c a se fo r a t lea st f o u f o f t h e five b a sic e le - ^ m encs o f f r a c t a l g e o m e t r y in A f r i c a n m a t h e m a t i c s , w h a t s h o u l d we m a k e o f it \ in te r m s o f c u l t u r e ? T o ask t h i s q u e s t i o n e f f e c t i v e l y w e n e e d t o a v o j d tw o p i t ­ falls. T h e first is t h e p o s s ib il it y o f “o v e r d e t e r m i n e d ” e x p l a n a t i o n s fo r A f r i c a n fractals, e x p l a n a t i o n s t h a t s e e m t o b e w a i t i n g f o r us b e f o r e w e ’v e e v e n b e g u n i8 o Im plications t o e x a m i n e t h e e v i d e n c e . T h e s e c o n d is t h e d i f f i c u l t y o f s u s t a i n i n g s k e p t i c i s m in a r a c i a l l y c h a r g e d e n v i r o n m e n t . j t h e p o s s i b i l i t y t h a t w e m i g h t sh y a w a y fr o m c r i t i q u e o v e r fe ars t h a t e x p r e s s i n g a n e g a t i v e v i e w c o u l d b e t a k e n <is h a v i n g a n e t h n o c e n t r i c o r r a c i s t m o t i v a t i o n ^ B o t h fa ilin g s a r e e q u a ll y d a m a g i n g . R e c e n tl y , re s e a rc h e rs h a v e d r a w n a t t e n t i o n to t h e w ays t h a t th e o r ie s o f k n o w le d g e ( e p i s t e m o l o g y ) c a n s n e a k u n e x a m i n e d i n t o c u l t u r a l p o r t r a i t s . If we a r e to a v o id t h e t r a p o f s e e i n g A f r i c a n f r a c t a l s as a n i n d i c a t i o n t h a t A f r i c a n s a r e “c l o s e r to n a t u r e , ” o r c o n c r e t e r a t h e r t h a n a b s t r a c t t h i n k e r s , o r u n i f i e d in a s i n g l e h o m o ­ g en eo u s c u ltu re, t h e n we n e e d to k n o w ..a .b it a b o u t th e orig in o f th ese m is­ c o n c e p t i o n s . T h e fir s t s t e p i n th a t\p v o c e s s ,.is t o e x a m i n e t h e e p i s t e m o l o g i c a l f r a m e w o r k s t h a t a re a p p l i e d t o t h e s t u d y o f c u l t u r e . T h e u n i t y j d i v e r s i t y d e b a t e a n d t h i n d e s c r ip t io n A c c o r d i n g to M u d i m b e j r i 9 8 8 ) , t h e c o n c e p t o f a u n i t a r y , t r a d i t i o n a l “A f r i c a n c u l ­ t u r e ” is a n i n v e n t i o n c r e a t e d first b y c o l o n ia l is ts , w h o s o u g h t t o r a t i o n a l i z e t h e i r c o n q u e s t w i t h t h e m y t h o f th e p rim itiv e, a n d s u b s e q u e n tly by a n ti c o lo n i a li s ts s e e k ­ ing to c o n so lid a te t h e i r o p p o sitio n . A s im ilar c ritiq u e is p ro v id e d b y (A p p ia h (1 9 9 2 ) , ’w h o su gg ests t h a t t h e d if fe re n c e s a m o n g v a r i o u s A f r i c a n s o c i e t i e s w e re m u c h to o b r o a d to a llo w a n y g e n e r a l i z a t i o n s (p . 2 5 ) : “S u re l y d i f f e r e n c e s in re lig io u s o n t o l ­ ogy a n d r i tu a l , in t h e o rg a n i z a ti o n o f p o l i t i c s a n d t h e fam ily, in r e l a ti o n s b e t w e e n t h e sex es a n d in a rt , in sty les o f w a rfa re a n d c u is in e , in la n g u a g e — surely all th e s e are f u n d a m e n t a l k i n d s o f d if f e r e n c e s ? ” A p p i a h a n d M u d i m b e p r o m o t e v a r i o u s k i n d s o f s o l i d a r i t y in c o n t e m p o r a r y A f r i c a ( a s w e ll as i n t e r n a t i o n a l l y in t h e d i a s p o r a ) ; t h e y o n l y c a u t i o n t h a t t h i s c u l t u r a l u n i t y is o f r e l a t i v e l y r e c e n t o r i g i n , a n d t h a t a t t e m p t s to s e e a n A f r i c a n ‘‘e s s e n c e ” o r a u n i f i e d A f r i c a n c u l t u r e p r e c e d . i n g . m a j o r E u r o p e a n i n t e r v e n t i o n (i.e., p re v io u s to t h e First W o r l d W a r ) will e v e n t u a l l y h a v e t o fall b a c k o n racjaMy d e f i n e d c a t e g o r i e s , w h i c h is c e r t a i n l y a s e l f - d e f e a t i n g b a s i s f o r a n t i r n c i s t m o v e m e n t s . F r o i n A p p i a h ' s a n t i e s s e n t i a l i s t p o i n t o f v i e w o n e c a n n o t d is c u s s p r e c o l o n i a l “A f r i c a n c u l t u r e ; " o n l y “A f r i c a n c i i j t u r e s . ” O n t h e o t h e r e x t r e m e o f t h e u n i t y v e r s u s d i v e r s i t y d e b a t e 1ies t h e A f r o c e n t r i c p o s i t i o n . W h i l e its p r o p o n e n t s a ls o a g r e e t h a t t h e r e was_n.o_.single, homogerie.0115 A fricarT o T lT u T e, t h e y e m p h a s i z e ^ t h e ^ h j p x e d - . e l e m e n t s . A s a n t e a n d A s a n t e ’s A fr ic a n C u l t u r e : R hythm s o f U n ity ( 1 9 8 5 ) , f o r e x a m p l e , b e g i n s by ti. s t a t i n g t h a t w h ile b la c k u n it y c a n n o t be b ased o n g e n e t i c g r o u n d s , b ro a d ly s h a r e d c u l t u r a l u n d e r c u r r e n t s w e r e f o u n d t h r o u g h o u t t h e d i v e r s e s o c i e t i e s o f p re - / co lo n ia l A frica: T heoretical fra m e w o rk s in cultural studies A l t h o u g h th e p r e c is e a c tio n s an d id eas m a y d iffe r w ith in th e a c c e p ta b le ra n g e a n d still r e m a in s q u a r e ly in t h e c a t e g o r y o f A f r i c a n c u ltu re , th e r e are s o m e b e h a v ­ io rs a m o n g s o m e A f r i c a n e t h n i c g r o u p s w h i c h m a y h a v e t h e o p p o s it e m e a n ­ i n g a m o n g o t h e r s . T w i n n e s s is c o m m o n l y c O A s i d e r e d a p o s i t i v e c h a r a c t e r i s t i c in A f r i c a n s o c i e t i e s , y e t t h e r e a r e s o m e e t h n i c g r o u p s w h i c h a c c e p t t w i n n e s s as a n e g a t i v e c h a r a c t e r i s t i c . . . .Y e t t h is p a r t ic u l a r is t ic e m p h a s i s w o u l d n o t m a k e th e e th n ic g ro u p u n re la te d to th e o th e rs . P a tte rn e d b e h a v io rs by A f r ic a n e t h ­ n ic g ro u p s are c u ltu r a l, n o t rigid o r fix e d , b u t rela ted to h is to r y a n d e x p e r ie n c e . C u l t u r e c a n v a r y o v e r t i m e , b u t in t h e c a s e o f A f r i c a n c u l t u r e , it w i l l a l w a y s b e a r t i c u l a t e d in t h e s a m e w ay. T h e r e is a lo t g o in g o n in th i s p a ra g r a p h , b u t t h e cru cia l p o i n t fo r m y a n a l y ­ sis is A s a n c e a n d A s a n t e ’s d i s t i n c t i o n b e t w e e n t h e .sur/ace p a r t i c u l a r i t i e s o f v a r i o u s e t h n i c g ro u p s , w h i c h m a y d iffer, a n d d eep er c u l t u r a l s e n s i b i l i t i e s o r p a t ­ tern s o f a r tic u la tio n (w h ic h th e y la te r illu strate w ith “ th e th r e e tr a d itio n a l v a lu e s : h a r m o n y w i t h n a t u r e , 2 h u m a n e n e s s , a n d r h y t h m ” [p. 7]). In t h i s A f r o c e n t r i s m , it is o n ly a t t h e d e e p le v e l in w h i c h we fin d i m p o r t a n t c u l t u r a l a t t r i b ­ u te s h e l c H n c o m m o n . p p p i a h also m a k e s t h i s d i s t i n c t i o n b_etween m v i a l su rfa c e ^ a n d t h e ‘T u nd a m e n t a T ‘ d e p t h s . T h e o n l y d i s a g r e e m e n t b e t w e e n h i m a n d t h e A s a n t e s is w h e t h e r o r n o t t h e d e p t h s r e v e a l d i f f e r e n c e s . O n e way a r o u n d t h i s q u e s t i o n is in t h e “t h i c k d e s c r i p t i o n ” p r o p o s e d by a n t h r o p o l o g i s t C l i f f o r d p e e r t z O1973). G e e r t z w a s m o t i v a t e d in p a r t b y h i s d i s s a t i s f a c t i o n w i t h t h e w ay s t h a t C l a u d e L e v i - S t r a u s s ’s s t r u c t u r a l i s m s e e m e d t o r e d u c e s y m b o li c c u l t u r e to a flat, m e c h a ­ n is ti c s y n t a x . F o r p e e r t j , c u l t u r a U y m b o l s s h o u l d b t ^ i i i ^ , l g n i ^ f j j ^ n a m i c play, a n d t h e e t h n o g r a p h e r s h o u l d s h o w t h e i r t u r b u l e n t e x p a n s i o n t h r o u g h layers o f m e a n i n g , n o t t h e i r r e d u c n o n to jpsinj^le fixed s t ru c tu r e . G e e r tz d e fin e d th e s e d e e p e l e m e n t s , w h i c h t e n d to b e m o r e s u b j e c t i v e a n d literary, as specific to a p a r t i c ­ u lar c o m m u n i t y . F o r h i m j t w o u ld b e e x t r e m e l y d ifficult _to c o m p a re d e e p e l e ­ m e n t s fro m o n e lo c a t i o n to t h e n e x t , b e c a u s e t h e d e e p e l e m e n t s a re the_resu.lt o f lo cal i n t e r p r e t a t i o n s . T a k e n t o t h e e x t r e m e , G e e r t z ’s c h ic k d e s c r i p t i o n w o u ld sim p ly reply t h a t t h e q u e s t i o n A p p i a h a n d t h e A s a n t e s a r e a s k i n g c a n n o t b e a n sw e re d . T h e f r a m e w o r k 1 h a v e u s e d in p a r t s 1 a n d u o f th i s b o o k , w h i c h is t h a t o f e t h n o m a t h e m a t i c s in g e n e r a l , m i g h t b e re f e r r e d to as ch in d e s c r i p t i o n ^ stu d y o f th e su rfa c e p a r t i c u l a r i t i e s , s u c h as m a t e r i a U l e s i g n s a n d s y m b o li c f o r m u la s . A s th e A s a n t e s p o i n t o u t , a m a t h e m a t i c a l e l e m e n t like d o u b l i n g (“ tw in n e s s " in t h e i r q u o t a t i o n ) is ju s t a su rfa c e fe a tu re . W h e t h e r o r n o t it h a s d e e p e r m e a n i n g s — a n d th u s t h e e n t i r e A f r o c e n t r i s m / a n t i e s s e n t i a l i s m d e b a t e — is a q u e s t i o n o u t s i d e o f t h i n d e s c r i p t i o n . F o r t h i s r e a s o n , t h e t h i n d e s c r i p t i o n use o f A f r i c a n i c o n s to 181 Implications r e p r e s e n t s p e c i f ic m a t h e m a t i c a l c o n c e p t s o r s t r u c t u r e s ( e .g ., t h e t r i c k s t e r = d i s o r d e r ) is n o t n e c e s s a r i l y in c o n f l i c t w i t h t h e j h i c k d e s c r i p t i o n o f t h e s e s y m ­ b o ls i n t h e i r d e e p s e m i o t i c d y n a m i c s . P e l t o n ( 1 9 8 0 ) s e t s u p j u s t s u c h a c o n f l i c t , a n d p e r h a p s r i g h t l y s o — t h e r e h a s i n d e e d b e e n - a t e n d e n c y fo r s t r u c t u r a l i s t s to c l a i m t h a t t h e y h a d r e d u c e d c u l t u r e t o its t r u e e s s e n c e . T h e i r e r r o r w as to in s is t t h a t t h e s e b a r e - b o n e s s t r u c t u r e s w e r e t h e t r u l y d e e p m e c h a n i s m s o f c u l ­ t u r e , a n d t h a t t h e d i s c u r s i v e p la y o f m e a n i n g s h o u l d b e d i s r e g a r d e d as s h a l l o w d i s t r a c t i o n . A s l o n g as w e k e e p t h e t h i c k s t u f f as t h e d e e p , a n d t h e p a r e d d o w n s t r u c t u r e s as t h e s u r f a c e , t h e r e is n o c o n f l i c t . W h i l e t h e l a c k o f A f r i c a n u n i t y in “ t w i n n e s s ” is n o t a p r o b l e m fo r t h o s e . c o n c e r n e d o n l y w i t h d e e p e r m e a n i n g s , w o u l d n ’t it p r e s e n t a p r o b l e m f o r t h i n l y "7 d e s c r ip ti o n ? T h a t is, if d o u b l i n g is s u p p o s e d to b e a n i m p o r t a n t f e a t u r e o f A f r i c a n y m a t h e m a t i c s , t h e n h o w d o e s o n e e x p l a i n t h e A f r i c a n s o c i e t i e s t h a t d o n o t u se it? I n d e e d , h o w is f r a c ta l g e o m e t r y s u p p o s e d to b e a n _ A f r i c a n k n o w l e d g e s y s te m if t h e e x a m p le s o f its use are so d isparately s c a tte r e d across t h e c o n t i n e n t ? T o a n sw er th is q u e s tio n , we n e e d to c o n sid e r w h a t W i t t g e n s te in c alled a “family re s e m b la n c e .” W h e n w e lo o k a t t h e p h o t o g r a p h o f a la r g e fa m ily w e c a n s e e t h a t e v e r y o n e is r e l a t e d , e v e n if t h e r e is n o s i n g l e c h a r a c t e r i s t i c t h a t t h e y all s h a r e ( s o m e h a v e b ig n o s e s a n d s o m e s m a ll, s o m e l i g h t h a i r a n d s o m e d a r k , e t c . ) . I n t h e s a m e way, it is n o t u n c o m m o n for a g ro u p o f m a t h e m a t i c a l id e a s t o s h a r e m a n y c o m m o n ­ a l i ti e s w i t h o u t a s i n g u l a r e s s e n c e . I n J a m e s G l e i c k ’s ( 1 9 8 7 ) h i s t o r y o f c h a o s th e o r y , fo r e x a m p l e , h e sh o w s t h a t t h e e m e r g e n c e o f n o n l i n e a r d y n a m i c s as a dis­ c i p l i n e w as d u e t o a s lo w g a t h e r i n g o f m a n y d i f f e r e n t s t r a n d s o f m a t h e m a t i c s — s t r a n g e a t t r a c t o r s , f r a c ta l g e o m e t r y , c e l l u l a r a u t o m a t a , a n d s o o n . I n o r d e r for s c i e n t i s t s t o c o l l a b o r a t e o n t h i s d e v e l o p m e n t , t h e r e w a s a l o n g p e r i o d in w h i c h s e v e r a l r e s e a r c h e r s w o r k e d h a r d to p o i n t o u t t h e fa m ily r e s e m b l a n c e o f t h e s e d i s ­ p a r a t e m a t h e m a t i c a l to o l s , a n d m a n y a s p e c t s o f t h e i r r e l a t i o n s h i p s a r e still u n c e r t a i n today. Sim ilarly, A fr i c a n fractal g e o m e try is n o t a s i n g u la r b o d y o f k n o w l ­ ed g e , b u t r a t h e r a p a t t e r n o f r e s e m b l a n c e t h a t c a n b e s e e n w h e n w e d e s c r i b e a w i d e v a r i e t y o f A f r i c a n m a t h e m a t i c a l i d e a s a n d p r a c t i c e s . A n d as w e sa w in t h e c a s e o f B a n n e k e r ' s q u i n c u n x , it is n o t t h e o n l y p a t t e r n p o ss ib le . P a rticip a n t s i m u la tio n W h e t h e r o n e b e l i e v e s i n G e e r t z 's t h i c k d e s c r i p t i o n o r in s o m e o t h e r m e t h o d for re s e a r c h in g t h e d e e p e r m e a n i n g s o f a lo cal c u ltu re , a n d n o p o l o g i s t s g e n e r a l ly agree t h a t it re q u ire s lo n g - te r m local e t h n o g r a p h i c study. M y thin, d e s c r i p t i o n field w ork ^ l a s t e d o n l y a y e a r a n d m o v e d t h r o u g h S e n e g a l , M a l i, B u r k i n a F a so , T h e G a m ­ b ia , C a m e r o o n , B e n i n , a n d G h a n a . T h i s d is p e r s e d i n v e s t i g a t i o n is q u i t e u n l i k e T heoretical jYameivorks m cultural studies w h a t is u n d e r t a k e n b y m o s t a n t h r o p o l o g i s t s , w h o o f t e n s p e n d a c o u p l e o f years in o n e v illa g e a l o n e , u sin g “p a r t i c i p a n t o b s e r v a t i o n ” to tr a v e r s e t h e d e p t h s o f th e lo c a l c u l t u r e by a c t i v e l y l i v i n g it. T h e r e is, h o w e v e r , a n i m p o r t a n t j h f f e r e n c e : I w as n o t t r y in g t o u n d e r s t a n d h o w t h e Y o ru b a e x p e r i e n c e grief, o r t o d e t e r m i n e ' t h e i n n e r m e a n i n g o f c o m m u n a l sp irit a m o n g t h e B ak a. M y i n t e r e s t was p rim a rily in t h e f o r m a l p r o p e r t i e s o f d e s i g n , in m e t h o d s o f c o n s t r u c t i o n , a n d in o t h e r t e c h n i c a l q u e s t i o n s t h a t c o u ld o f t e n b e a n s w e r e d in a d i r e c t a n d s i m p l e fa s h io n . M a n y o f t h e A f r i c a n s 1 s p o k e w i t h w e r e c l e a r l y r e l i e v e d t o h e a r t h a t 1 w as a m a t h e m a t i c i a n . - * O f c o u r s e 1 w as s t il l f a c e d w i t h s e v e r a l o f t h e s a m e p r o b l e m s i n v o l v i n g e t h n o g r a p h i c a c c u r a c y a n d a u t h o r i t y ( s e e C l i f f o r d 1 9 8 3 ). B u t e v e n th e s e w ere s o m e tim e s d ifferen tly p o sed. In p a rtic u la r, 1 b e g a n to t h i n k o f my m e t h o d o l o g y n o t as p a r t i c i p a n t o b s e r v a t i o n , b u t r a t h e r as p articip an t simulation, se e k in g to c o l l a b o r a t i o n m a th e m a tic a l a n aly sis a n d v ir tu a l r e c o n s tr u c tio n w ith m y A fricari co lleag u es. P a r t i c i p a n t s i m u l a t i o n w as c a r r i e d o u t to c o n c l u s i o n o n ly in t h e r e s e a r c h w ith C h r i s t i a n S i n a D i a t t a , h u t 1 tr ie d t o m a i n t a i n t h e p ra c t ic e a t s o m e level w ith e v e r y o n e I h a d t h e o p p o r t u n i t y t o w o r k w i t h . T h a t m e a n t h a u l i n g d ia g r a m s of f r a c ta l g r a p h i c s w i t h m e i n t o t h e e q u a t o r i a l r a i n f o r e s t a n d a c ro s s t h e s a v a n n a h , a n d d i s r u p t i n g r e s e a r c h t i m e w i t h m a t h le c t u r e s , b u t in t h e e n d it w as w ell w o r t h i t 7 T h e r e w as t h e p o t e n t i a l p r o b l e m t h a t s o m e o n e w h o k n e w w h a t 1 was afte r m i g h t f a b r i c a t e w h a t 1 w a n t e d t o h e a r (as in Sc. L o uis, S e n e g a l , w h e n o n e o f t h e local c h i l d r e n h e a r d nJe t a l k i n g a b o u t B e n j a m i n B a n n e k e r a n d c l a i m e d t o k n o w h i m p e r s o n a l l y ) , / v ’m o r e p r e s s i n g p r o b l e m w as m y r e s i s t a n c e t o t h e i r sugg estio ns, as o c c u r r e d in m y in itia l d i s a p p o i n t m e n t w i t h tire la c k o f p la c e v a lu e n o t a t i o n in che B a m a n a d i v i n a t i o n c o d e , o r h e a r i n g che d e s c r i p t i o n o f t h e o s c i l ­ la to ry s n a k e as “ D a n a t w o r k ” (a ll 1 c o u l d t h i n k o f a t t h e t i m e w as a r o a d c o n ­ s tr u c ti o n s ig n ). O f c o u rs e , th e r e are a lw a y s th e a fte re ffe c ts — S e n e g a le s e s o c i o lo g i s t F a to u S o w sa id “ if t h e r e a r e n o t f r a c ta l s in A f r i c a n o w , t h e r e surely will h e by t h e t i m e y o u l e a v e ”— h u t t h e n t h a t is a fe a t u r e o f all e t h p o g r a p h y ; j u i d p a r t i c i p a n t s i m u l a t i o n is a b o ur t u r n i n g thac_inco _ a n .a d v a n t a g e . T h e r e a s o n c o l l a b o r a t i v e a p p r o a c h e s ..like p a r t i c i p a n t s i m u l a t i o n w e r e n o t t r a d i t i o n a l l y u s e d in e t h n o g r a p h y c o m e s f r o m c o n c e r n s o v e r a c c u r a c y — t h e desire to-ol 5 t a i n a n o b j e c t i v e a c c o u n t — a n d c o n c e r n s o v e r a u th o r it y , a s u s p ic io u s m o t iv e in tj^e c o l o n i a l c o n t e x t o f m o s t t r a d i t i o n a l a n th r o p o lo g y . C liffo rd ( 1 9 8 3 ) d e s c r ib e s t h e m o v e t o w a r d c o l l a b o r a n v e ’t e c h n i q u e s as b o t h t h e a n t h r o p o l o g i s t s ’ o w n s e l f - c r it iq u e o f a u t h o r i t y a n d as a g r o w in g r e c o g n i t i o n t h a t s i n c e t h e e t h n o ­ g r a p h e r h a s as m u c h m o t i v a t i o n as t h e i n f o r m a n t d o e s, a c c u r a c y a n d o b j e c t i v ­ ity c a n b e b e t t e r a p p r o a c h e d hy s h a r i n g a u t h o r i t y w i t h i n d i g e n o u s v o i c e s t h a n by u sing t h e m in a k i n d o f v e n t r i l o q u i s t a c t. S i m p l y p r o c l a i m i n g a c o l l a b o r a t i v e Implications 184 a p p r o a c h is o f c o u r s e n o g u a r a n t e e t h a t y o u w ill h a v e o n e , a n d j i a r t i c i p a n t s i m ­ u l a t i o n is p e r h a p s e v e n m o r e s u s c e p tib l e t o m a n i p u l a t i o n d u e t o . t h e ro le o f te c h n o lo g i c a l^ e x p e r tj s e , O n t h e o t h e r h a n d , s i n c e d i e c re a t i o n o f v i r t u a l w o r l d s — s i m u l a t i o n s — is irk so m e w ays t h e p r o d u c t i o n o f s o m e t h i n g fake, p a r t i c i p a n t s i m u l a t i o n d o e s h a v e t h e a d v a n t a g e o f a v o i d i n g s o m e o l d - f a s h i o n e d c o n c e p t s o f a u t h e n t i c i t y . It w as, ^ a f t e r all, t h e c r e a t i o n p f a n “a u t h e n t i c n a t i y e ” (s e e A p p a d u r a i 1 9 9 5 ) t h a t h e l p e d / c o l o n i s t s t o ja il r e b e l s a m o n g b la c k S o u t h A f r i c a n s a n d N a t i v e A m e r i c a n s ; a n d ! o n e c o u ld e v e n h e a r t h e o c c a s i o n a l g u il t- r id d e n l a m e n t a m o n g t h e c o l o n i a l rulers | t h a t t h e y t h e m s e l v e s w e r e to b l a m e for h a v i n g a c c i d e n t a l l y p o l l u t e d t h e n a t u r a l | p u r i t y o f th e s e “c h i l d r e n o f t h e f o r e s t” w i t h t h e i r o w n t r o u b l i n g a rtific e ( s e e t h e ^ a p a r t h e i d c u l t u r e c o m e d y , T h e G o d s M u s t Be C r a ^ ) . L o c a t i n g i n d i g e n o u s a c ti v 1ity in v ir tu a l w o rld s c a n , if d o n e p roperly, c o u n te r ^ th i s h a b i t u a l t e n d e n c y to p la c e a r t ifi c ia l o n t h e W e s t e r n s i d e a n d n a t u r a l o n t h e i n d i g e n o u s sid e. . D o in g it p r o p e r ly re lie s o n t h e o t h e r ro o t, w h ic h c o m e s fro m t h e o ld f a s h i o n e d — a n d , 1 t h i n k , s t il l c r u c i a l — m e t h o d o f p a r t i c i p a n t o b s e r v a t i o n . P a r tic ip a n t o b s e r v a tio n re c ru its a k in d o f re s p o n sib ility t h a t c a n be sadly la c k ­ in g in v i r t u a l e t h n o g r a p h i e s . T a k e , f o r e x a m p l e , t h e g r o w i n g fie ld o f c y b e r ­ e t h n o g r a p h y , in w h i c h a n t h r o p o l o g i s t s s t u d y t h e v i r t u a l c o m m u n i t i e s o f t h e I n t e r n e t . S i n c e “l u r k i n g " ( o b s e r v i n g t h e e l e c t r o n i c e x c h a n g e s w i t h o u t p a r t i c i ­ p a t i n g ) is s o easy, t h e r e h a v e b e e n a n u m b e r o f s t u d i e s in w h i c h t h e e t h n o g ­ r a p h e r is r e d u c e d t o e a v e s d r o p p e r o r spy, w i t h n o a t t e m p t t o w o r k w i t h t h e c o m m u n i t y in e i t h e r o f f - l i n e o r o n - l i n e liv e s . O n t h e o t h e r h a n d , r e c r u i t s c a n in c lu d e b o th d ra fte e s, w h o h a v e little real in te re s t in w o rk in g c o lla b o ra tiv e ly , a n d f a n a t i c s , 'w h o a t e all t o o i n t e r e s t e d in w h a t G a y a t r i S p i v a k ( 1 9 8 7 ) c a ll s t h e “b e n e v o l e n c e o f t h e w e s t e r n g a z e .” T h u s p a r t i c i p ^ ^ s i m u l a d o n . js_an_atternpt to t a k e t h e b e s t o f bo th ^ a p p ro a ch es, a n d t o u se t h e m in a k i n d o f c h e c k s - n n d - b a l a n c e s s y s te m . By in s i s t i n g o n p a r ­ t i c i p a t i o n w e .c a n h e l p a y o i d _ g l i b j r e s p o n s i b i l i t y ; and . by u s i n g s i m u l a t i o n w e c a n strive to a v e rt th e p o lic in g o f b o u n d a rie s a r o u n d c o n s m ic t ip n s o f a u th e n tic ity a n d re a lis m . F r o m t h i s p o i n t o f v ie w w e d o n o t n e e d t o e m p h a s i z e t r a d i t i o n o v e r i n v e n t i o n ; t h e m a t h e m a t i c a l c r e a t i o n s o f a s i n g l e i n d i v i d u a l a re s till e x a m p l e s o f i n d i g e n o u s m a t h e m a t i c s , e v e n if s h e is t h e o n l y o n e w h o k n o w s t h e y e x is t. In ten tto n aliP y a n d e th n o m a th e m a tic s T h e r e a r e c l e a r a d v a n t a g e s to a m e t h o d o l o g y t h a t c a n c r e d i t t h e i n v e n t i o n s o f a single in d iv id u a l, b u t w h a t a b o u t th o s e c r e a t io n s t h a t d o n o t h a v e a single i n v e n ­ tor? A s we s a w in t h e c a s e o f c o m p l e x i t y in c h a p t e r i o , it is p o s s ib le t o e r r o n T heoretical fra m e w o rk s in cu ltu ra l studies I che o c h e r s i d e by i n s is t in g c h a t c o n s c i o u s c r e a t i o n s c a n o n l y c o m e f r o m s i n g u ­ lar i n v e n t o r s . A b e t t e r u n d e r s t a n d i n g o f tht's p r o b l e m £ a n b e g a i n e d t h r o u g h t h e c o n tr ast b e tw e e n e th n o m a c h e m a tic s a n d m a th e m a tic a l, an th ro p o lo g y . M a th e .rn.atical a n t h r o p o l o g y is g e n e r a l ly fo c u s e d o n r e v e a l i n g p a t t e r n s t h a t are n o t c o n ' sc io u sly d e t e c t e d by its s u b j e c t s o f s tu d y . I n p a r t t h i s is d u e t o a c o n v i c t i o n t h a t m a n y o f t h e u n d e r p i n n i n g s o f s o c i e ty a re f o r c e s u n n o t i c e d by its m e m b e r s — n o t o n ly b e c a u se s u c h forces o p e r a t e d a t le v e ls b e y o n d in d iv id u a l a w a ren ess; b u t a ls o „ b e c a u s e r e g u l a to r y m e c h a n i s m s w o u l d h a v e t o be c o v e r t , o b s c u r e d , o r o th e r w i s e ' p r o t e c t e d fr o m m a n i p u l a t i o n a n d c o n s c i o u s r e f l e c t i o n . For th e s e rea s o n s, m a t h e ­ m a t i c a l a n t h r o p o l o g y m a k e s g o o d s e n s e , a n d it h a s i n d e e d p r o d u c e d w o n d e r f u l i n s ig h t s . B u t its e m p h a s i s o n u n c o n s c i o u s p r o c e s s a ls o aro se fr o m i m i t a t i o n o f t h e r e s e a r c h e r - o b j e c t r e l a t i o n in t h e n a t u r a l s c i e n c e s : if a n t h r o p o l o g i s t s w e re s i m p l y r e p o r t i n g i n d i g e n o u s d i s c o u r s e , t h e n t h e y w o u ld n o t c o u n t as s c i e n t i s t s . T h i s p ro b le m o f m e re re p o r tin g is in d e e d t h e c a se for “ n o n - W e s t e r n m a t h e m a t i c s , ” w h i c h is m a i n ly fo c u s e d o n d i r e c t t r a n s l a t i o n s f o r C h i n e s e , H i n d u , a n d M u s lim , m a t h e m a t i c s a n d t h u s c o n s i d e r e d a s u b j e c t for h i s t o r i a n s . H e n c e m a t h e m a t i c a l a n t h r o p o l o g y ’s t e n d e n c y t o a v o i d i n t e n t i o n a l i c y c a n b e p r o b l e m a t i c . T h e T n t e n t i o n a l i c y p r o b j e m in m a t h e m a t i c a l a n t h r o p o l o g y c a n b e s e e n in K o l o s e i k e ’sM-J-9.74) m o d e l fo r m u d t e r r a c e c o n s t r u c t i o n in t h e lo w h i l l s o f E c u a d o r. ^ o l o s e i k t J b e g a n w i t h t w o h y p o t h e s e s : e i t h e r t h e I n d i a n s l e a r n e d fro m th e ln c a s t o n e te rra c e s in t h e h i g h m o u n t a i n s a b o v e , o r th e y w e re u n i n t e n t i o n a l b y - p r o d u c ts o f c u l t i v a t i o n o n h il ls id e s . H e t h e n m a d e a list o f n i n e o b s e r v a t i o n s t h a t w e re r e l e v a n t t o d e c i d i n g bet< veen t h e tw o . O f p a r t i c u l a r i n t e r e s t a re t h e fo llo w in g : . 3. T h e sam e hillside soil is u sed in r a m m e d - d i i t b ouses a n d fen ce walls, a n d th e s e sta n d for years. 4. B u t 1 n e v e r saw a terra ce b e in g c o n s t r u c t e d , n o r did peo ple talk a b o u t su c h a project. 5. S m a ll cav es are o fte n d u g in t o t h e te r ra c e face for s h e l te r d u ri n g r a i n ­ storms- T h a t this p o ten tially w e a k e n s t h e terrace face does n o t seem to c o n ­ c e rn people. (i9 7 4 , 29-30) K olo seik e c o n c lu d e s th a t th e s e te rra c e s are th e u n i n t e n ti o n a l re s u lt o f \ / a n a c c r e t i o n p r o c e s s fr o m t h e c o m b i n a t i o n o f c u l t i v a t i o n a n d e r o s i o n , a n d \ th e n p ro c e e d s to d e v e lo p a m a t h e m a t i c a l m o d e l for th e ra te o f t e r r a c e g r o w th . M y p o i n t is n o t in q u e s t i o n i n g t h e a c c u r a c y o f t h e m o d e l , b u t r a t h e r th e way t h a t i n d i g e n o u s i n t e n c i o n a l i t y is p o s i t i o n e d as a n o b s t a c l e t h a t m u s t be o v e r c o m e b e f o r e m a t h e m a t i c s c a n b e a p p l i e d . E v e n a s m a l l d e g r e e o f a w a re n e s s — b e i n g a w a r e t h a t a c a v e d u g i n t o a t e r r a c e fa c e m i g h t w e a k e n it— m u s t be e l i m i n a t e d . . Jmpliaiticms i8 6 I n a d d i t i o n , it rev eals a p a r t i c u l a r c u lt u r a l c o n s t r u c t i o n o f t h e s u p p o s e d un iI v ersal a t t r i b u t e o f “ i n t e n t i o n . ” A s a W e s t e r n e r , K o l o s e i k e is u s e d t o a s o c i e t y in a h u rry . P r o j e c t s t o b e . d o n e m u s t g e t d o n e , a n d a lw a y s w i t h s o m e o n e in c h a r g e . ! T h e i d e a o f a l o n g - t e r m i n t e n t i o n a l p r o j e c t , p e r h a p s ‘e x t e n d i n g o v e r s e v e r a l ■ g e n e r a t io n s , o r t h e c o n s t i t u t i o n o f c o ll e c ti v e i n t e n t i o n a l i t y r a t h e r t h a n in d iv id u a l i n t e n t , is n o t b r o u g h t u n d e r c o n s i d e r a t i o n . I t m a y w e ll b e t h a t t h e m a t h e ­ m a t i c a l m o d e l K o lo s e i k e o ff e re d w a s n o t o n l y a c c u r a t e , h u t a ls o h a d a n in d i g e 1 nous co u n terp art. E t h n o m a t h e m a t i c s , in c o n t r a s t , h a s e m p h a s iz e d t h e p o s s ib il it ie s fo r i n d i g e ­ n o u s i n t e n t i o n a l i t y in m a t h e m a t i c a l p a t t e r n s . "For e x a m p l e , G e r d e s ' . ( i 9 9 i ) used t h e L u s o n a s a n d d r a w i n g s o f t h e C h o k w e p e o p l e o f n o r t h e a s t e r n A n g o l a to d e m o n s t r a t e i n d i g e n o u s m a t h e m a t i c a l k n o w l e d g e . W h i l e it w o u ld h a v e b e e n p ossible to a t t r i b u t e this p r a c t ic e to a n u n c o n s c i o u s so c ia l p ro cess, s u c h as t h e reg­ u l a t i o n o f a u th o r it y , G e r d e s c h o s e to focus o n t h e i r propertjes_as.£p.Qscjous_indigen o u s i n v e n t i o n s . A s c h e r ( 1 9 9 1 ) n o t e s t h e s a m e cype o f E u l e r i a n p a t h d ra w in g s in t h e S o u t h Pacific, a n d s h o w s t h e m t o be p r i m a r i l y m o t i v a t e d hy s y m b o l i c n a r ­ ra t iv e s , in p a r t i c u l a r t h e i r use b y t h e M a l e k u l a i s la n d e r s as a n a b s t r a c t m a p p i n g o f k i n s h i p r e l a ti o n s . A g a i n , t h i s is in s t r o n g c o n t r a s t t o t h e t r a d i t i o n o f m a t h e ­ m a t i c a l a n t h r o p o l o g y , w h e r e k i n s h i p a lg e b r a w as c o n s i d e r e d a t r i u m p h o f W e s t ­ e r n a n a ly s is (arid e v e n a s o u r c e o f m a t h e m a t i c a l s e l f - c r i t i q u e ; K ay t i 9 7 1 ] h a r s h ly n o t e s t h e a n t h r o p o l o g i s t s ’ t e n d e n c y to i n v e n t a n e w " p s e u d o - a l g e b r a ” for v a rio u s k i n s h i p sy s te m s r a t h e r t h a n a p p ly o n e u n i v e r s a l s t a n d a r d ) . , A sch er); d e sc rip tio n o f th e N a tiv e A m e r ic a n g a m e of D is h show s th is c o n tr a s t Tn a m o re su b tle form, i n t h e C a y u g a v e rs io n o f t h e g a m e , six p e a c h stones, b l a c k e n e d o n o n e sid e, a re to s se d , a m X th e n u m b e r s l a n d i n g b l a c k sid e o r b r o w n sid e u p w e re re c o r d e d . T h e t r a d i t i o n a l C a y u g a p o i n t s c o r e s fo r e a c h o u t c o m e are ( t o t h e n e a r e s t i n t e g e r v a l u e ) in v e r s e ly p r o p o r t i o n a l t o t h e p r o b a b i li ty . A s c h e r d o e s n o t p o s i t a n i n d i v i d u a l C a y u g a g e n iu s w h o d i s c o v e r e d p r o b a b i l i t y th e o ry , n o r d o e s s h e e x p l a i n t h e p a t t e r n as m e r e ly a n u n i n t e n t i o n a l e p i p h e n o m e n o n o f r e p e a t e d a c ti v it y . R a t h e r , h e r d e s c r i p t i o n (p . 9 3 ) is f o c u s e d o n h o w t h e g a m e is e m b e d d e d in c o m m u n i t y c e r e m o n i a l s , s p i r i t u a l b e l i e f s , a n d h e a l i n g ri tu a l s , s p e c ific a lly t h r o u g h t h e c o n c e p t o f “c o m m u n a l p l a y i n g ” i n w h i c h w i n n i n g s 3 re a t t r i b u t e d t o t h e g ro u p r a t h e r t h a n t o t h e i n d i vidual''p.lnve.r. J u x t a p o s i n g th i s c o n t e x t w7 t h T l 'e t a il e e r a t te n t io n t o a b s t r a c t c o n c e p t s . o f r a n d o m n e s s a n d p r e d i c ta b i li ty in a s s o c i a t i o n w i t h t h e g a m e — in p a r t i c u l a r rite id e a o f “e x p e c t e d .v alu es " a s s o ­ c i a t e d w i t K s u c c e s s iv e to sses— h a s t h e e f f e c t o f a t t r i b u t i n g t h e i n v e n t i o n o f p r o b a b i l i t y a s s i g n m e n t s to c o l l e c t i v e i n t e n t . A t th e s k e p tic a l e x t r e m e in e t h n o m a t h e m a t i c s , D o n a l d C r o w e has re frain ed fr o m m a k i n g a n y i n f e r e n c e s a b o u t i n t e n t i o n a l i t y a n d in s is t s t h a t h is s t u d ie s o f Theoretical fra m e w o rk s in cultural studies 187 s y m m e t r y in i n d i g e n o u s p a t t e r n c r e a t i o n s (s e e W a s h b u r n a n d C r o w e 1 9 8 8 ) are sim ply e x am p les o f ap p lied m a th e m a tic s . B ut sin ce C ro w e has re s tric te d h is / /’ w o rk to o n ly t h o s e p a t t e r n s w h i c h c o u ld b e a t t r i b u t e d to c o n s c i o u s d e s ig n ( p a i n t - ing, c a r v i n g a n d w e a v i n g ) , it c re a t e s t h e o p p o s i t e '^ f f e c t o f m a t h e m a t i c a l a n t h r o - ) .-1 p o l o g y ’s a t t e m p t to e l i m i n a t e i n d i g e n o u s i n t e n t . T h i s is e v i d e n c e d b y C r o w e ’s / d e d i c a t i o n t o t h e use o f th e s e p a t t e r n s in m a t h e m a t i c s e d u c a t i o n , p a r t ic u l a r ly h is C / ! C, t e a c h i n g e x p e r i e n c e in N i g e r i a d u r i n g t h e la t e 1 960s, w h i c h g re a t ly c o n t r i b u t e d , ' ’ ' | t o Z a s la v s k y ’s ( 1 9 7 3 ) s e m i n a l t e x t , A fr ic a C o u n t s . W h i l e n o n - W e s t e r n m a t h e m a t i c s is e x c l u s i v e l y f o c u s e d o n d i r e c t t r a n s ­ l a t i o n s ( s u c h as H i n d u a lg e b r a o r M u s l i m g e o m e t r y ) , e t h n o m a t h e m a t i c s c a n b e o p e n t o a n y s y s t e m a t i c p a t t e r n d i s c e r n a b l e t o t h e r e s e a r c h e r . In f a c t , e v e n t h a t d e s c r i p t i o n is t o o r e s t r i c t i v e : , b e f o r e G e r d e s ’s s t u d y t h e r e . w a s n o W e s t e r n c a t e ­ gory o f “re c u r siv e ly g e n e r a t e d E u le r i a n p a t h s ” ; it was o n ly in t h e a c t o f t h e i r p a r - C " t i c i p a n t s i m u l a t i o n t h a t G e r d e s — a n d t h e C h o k w e — C re a te d t h a t h y b r i d . A n d u n lik e m a th e m a tic a l a n th ro p o lo g y , e t h n o m a th e m a ti c s p uts a n e m p h a s is o n t h e a t t r i b u t i o n o f c o n s c i o u s i n t e n t t o t h e s e p a t t e r n s . A t t h e s a m e t i m e , it d e m a n d s q u a n t i t a t i v e o r g e o m e t r i c c o n f i r m a t i o n t h a t is l a c k i n g i n t h e p u r e l y i n t e r p r e t i v e a p p r o a c h o f N e w A g e m y s ti c is m , s u c h as t h a t o f F r i t j o f C a p r a ’s T a o o f P h y sic s (s e e c r i t i q u e s in R e s t i v o 1 9 8 5 ) . C l a i m s t h a t a n c i e n t k n o w l e d g e sy s­ t e m s r e v e a l t h e s t r u c t u r e o f t h e a t o m o r t h e e q u i v a l e n c e o f m a .tte r a n d e n e r g y d o m o r e h a r m t h a n g o o d — first b e c a u s e th e y a re w ro n g , a n d s e c o n d b e c a u s e t h e r e is n o m e a n s by w h i c h su c h k n o w le d g e c o u ld be o b t a i n e d . S u c h m y s tific a tio n d a m ­ ages c r e d i b l e r e s e a r c h in i n d i g e n o u s k n o w l e d g e sy s te m s , a n d r e m o v e s t h e a ttr.b b u t i o n o f i n t e n t i o n a l i t y a n d i n t e l l e c t u a l l a b o r fr o m t h e p u t a t i v e k n o w e r s . Lvehiflev. -is o. brish a n d n o t a la d d e r: the c u ltu r a l lo c a tio n o f A f r i c a n fr a c ta ls W e a re in c r e a s i n g ly s u r r o u n d e d by e x p l a n a t i o n s j a a s e d o n b io l o g i c a l d e t e r m i n ­ ism, a n d t h e r e is n o n e m o r e v i r u l e n t t h a n r a c i s m } E v e n in t h e s u p p o s e d li b e ra l c lim a te o f U .S . a c a d e m ja , m y le ctu re s o n fractiilsTn A fric a are f r e q u e n tl y follow ed by a qu ,estio n .ab o q t n e u ro s c ie n C e rT y p ic a lly this is a n i n n o c e n t re m a rk c o n c e r n i n g N o a m ( b h o m s k y ’s id e a s o n u n i v e r s a l c o g n i t i v e s t r u c t u r e , b u t e v e n so, it is q u i t e te llin g t h a t a l e c t u r e o n E u r o p e a n f r a c ta l s in v o k e s q u e s t i o n s a b o u t t h e g e n i u s o f in d iv id u a ls , w h i l e A f r i c a n fr a c ta ls a r e c o m p u l s i v e l y a t t a c h e d to bio logy. T h e m y t h o l o g y o f r a c e is to o c o m p l e x t o r e c o u n t h e r e (s e e ^ n o te 6 ) , b u t it is useful to d i s t i n g u i s h b e t w e e n t w o c a t e g o r i e s o f racism .( P r i m i t i v i s t racism..^ o p erates by m a k i n g a g ro u p o f p e o p le t o o c o n c r e t e , a n d j . h u s “clo ser_to n a t u r e ”— n o t re ally a c u l t u r e a t all, b u t r a t h e r b e in g s o f u n c o n t r o l l e d e m o t i o n a n d d i r e c t ’ LT.l T I ?! rrrw^i •» Implications b o d ily s e n s a tio n , r o o t e d in a n e d e n ic ecology. O r i e n t i a l i s t ra c ism o p e r a t e s by m a k - / in g a g r o u p o f p e o p l e t o o j r b s t r a c t , and__thus “a r a b e s q u e ”— n o t r e a l ly a n a t u r a l ] h u m a n , b u t o n e w h o is d e v o id o f e m o t i o i v c a r i n g only, f o r m o n e y o r- a n in s c ru ta b le sp iritu al tra n sc e n d e n c e . T h e a l t e r n a t i v e t o b i o g e n e t i c e x p l a n a t i o n is s o c i o c u l t u r a l , a n d h e r e t h e c a t e g o r i e s o f p r i m i t i v e a n d o r i e n t a l c a n b e m u c h m o r e c o m p l e x . H is t o r i c a l l y , m a n y research ers w h o stro n g ly o p p o se d b o th racism a n d e th n o c e n tr a c is m h av e b e e n l o c a t e d in i n s t i t u t i o n s w i t h t i t l e s l i k e “ M u s e u m o f P r i m i t i v e A r t s ” or “ D e p a r t m e n t o f O r i e n t a l i s t S t u d i e s , ” a n d it w o u l d b e u n w i s e to s i m p l y s n e e r a t t h e i r w o rk , p a r t ic u l a r ly c o n s i d e r i n g t h e a n t i r a c i s t c o n t r i b u t i o n s by b l a c k a n t h r o ­ po lo gists s u c h as Z o ra N e a l e H u r s t o n o r J o m o K e n y e tta . T h e r e is v a lu e t o b e f o u n d ; in e v e n t h e w e a k e s t o f t h e s e o p p o s i t i o n a l t h e o r i e s , a n d p r o b l e m s in e v e n t h e i strongest. I n g e n e r a l t h e s e t h e o r i e s c a n j j e g r o u p e d i n t o t w o s t ra te g i e s : s a m e n e s s a n d ^ d iffe re n c e . S a m e n e s s c a n u su a lly a v o i d o r i e n t a l i s m a n d p r i m i t i v i s m , s i n c e it a r g u e s t h a t w h a t is i m p o r t a n t a b o u t a n o n - W e s t e r n c u l t u r e a r e th o s e t h i n g s h e l d i n c o m m o n w i t h t h e E u r o - A m e r i c a n s , a n d w h a t is d i f f e r e n t is ( i n t h i s c o n t e x t ) tr iv i a l. C l a u d e L e v i-S t ra u s s , fo r e x a m p l e , a r g u e d t h a t t h e “s a v a g e m i n d ” is b a s e d o n system s o f sy m b o lic s tru c tu r e s , j u s t.l ik e t h e E u ro p e a n m in d , so t h a t a n A f r i c a n w o r k i n g w i t h a s y s te m o f m y t h o l o g i c a l s y m b o l s is p e r f o r m i n g t h e s a m e c o g n i t i v e o p e r a t i o n s as a E u r o p e a n w o r k i n g w i t h a s y s te m o f c o m p u t e r c o d e s y m ­ bo ls. O n e . d r a w b a c k o f s a m e n e s s is t h a t w e b e c o m e p la y e r s i n a g a m e c r e a t e d by s o m e o n e else: “ 1 a m w o r t h w h i l e o n ly in s o far as 1 a m t h e s a m e as y o u . ” D iff e re n c e c a n a v o id t h i s tr a p , a l t h o u g h it h a s m o r e t j p u b l e a v o i d i n g p r i m i t i v i s m a n d o r i ­ e n t a l i s m . F o r e x a m p l e , A i'm e C e s d i r e ’s n e o l o g i s m " n e g r i t u d e ”'bega'!i a s a w ay o f ' s p e a k i n g a b o u t t h e d i f f e r e n c e o f A f r i c a n c u l t u r e in o p e n - e n d e d , d y n a m i c , c r e ­ a t i v e te r m s , b u t la tp r ( i n t h e h a n d s o f o t h e r s ) t h e c o m p a r i s o n ' w a s fr o z e n i n t o a s e t o f b i n a r y o p p o s i t i o n s ( i n t u i t i v e vs. a n a l y t i c , c o n c r e t e vs. a b s t r a c t , e t c .) .^ In o t h e r w o rd s , b o t h s a m e n e s s a n d d i f f e r e n c e h a v e m o m e n t s o f f a ilu re as w ell as m o m e n t s o f su c c e s s .__________________ _ T h e r e c e n t focus o n a n c ie n U E g y p t) in c e r t a i n c ir c le s o f A f r i c a n s t u d i e s h a s ce rta in ly s e e n _ b o t h ju o m e n ts . M o t iv a t e d by c o n s id e ra b le scho larly w ork (e.g., D rak e 1 9 8 4 ) , it h a s also b e c o m e a t t a c h e d t o s o m e d i s r e p u t a b l e a n d q u e s t i o n a b l e cla im s (s e e c r i t i q u e s in O r i t z d e M o n t e i l a n o 1 9 9 3 ; M a r t e l 1 9 9 4 ; L e fk o w itz 1 9 9 6 ) . It is w o r t h n o t i n g , h o w e v e r , t h a t s o m e o f t h e c r i t i q u e s h a v e b e e n e q u a l l y l a c k i n g in t h e i r r e s t r a i n t . In his r e v i e w o f t h e P o r t l a n d B a s e l i n e Essays, for e x a m p l e , R o w e ( 1 9 9 5 ) — w h i l e ri g h tl y p o i n t i n g t o a n u m b e r o f u n s u p p o r t e d a s s e r t i o n s — im p lie d t h a t c l a i m s for a n a n c i e n t E g y p t i a n g li d e r s h o u l d h e d is m i s s e d b e c a u s e t h e a u t h o r w as m e r e l y a n a e r o d y n a m i c s t e c h n i c i a n r a t h e r t h a n a P h . D . R o w e was T heoretical fra m e w o rk s in cu ltu ra l studies I q u i t e r i g h t in o b j e c t i n g t o t h e w ild le a p f r o m e m p i r i c a l te s ts o f a s m a ll w o o d e n c a r v i n g to t h e a u t h o r i t a t i v e c la i m s fo r a n c i e n t E g y p t i a n s flying fr o m p y ra m id s ; b u t to im ply t h a t sim p le e x p e r i m e n t s a re a u t o m a t i c a l l y s u s p e c t b e c a u s e t h e y w e re m a d e by a t e c h n i c i a n is n o t h i n g h u t cln ssist p r e j u d i c e . O n t h e o t h e r h a n d , th e fact t h a t t h i s r e s e a r c h e r w as a t e c h n i c i a n r a t h e r t h a n a P h D sp e a k s to t h e u n d e r ­ lyin g c a u s e for th e s e p ro b l e m s : t h e l a c k o f i n s t i t u t i o n a l r e s o u rc e s a n d p re c a r io u s e c o n o m i c s a m o n g m a n y b la c k e d u c a t i o n a l c o m m u n i t i e s . A p p e a ls to a n c i e n t E gypt c a n also e n c o u n t e r p ro b le m s as a s trateg y o f s a m e ­ ness. O n t h e o n e h a n d , a n c i e n t E g y p t ’s s t a t u s as a s t a t e e m p i r e d i r e c t l y o p p o s e s p r i m i t i v i s t a s s u m p t i o n s t h a t A f r i c a c o n s i s t s o f n o t h i n g b u t tr i b a l v illa g e s . O n t h e o t h e r h a n d , it r e i n f o r c e s t h e v ie w t h a t t h e k n o w l e d g e s y s te m s o f n o n s t a t e in d ig en o u s societies are n o t c o m p a r a b le to th o s e o f s t a te societies. T h i s vie w cotn es fr o m t h e o ld id e a o f c u l t u r a l e v o l u t i o n as a l a d d e r , a u n i l i n e a l .. p r o g r e s s io n fr o m “p r i m i t i v e ” t o “a d v a n c e d . ” I n t h e l a d d e r m ^ d e L i h g „ s m a l l - s c a l e d e c e n t r a l i z e d (“b a n d ”) s o c i e ti e s w o u ld b e o n t h e b o t t o m ru n g , t h e m o r e h i e r a r c h i c a l ( “ tr i b a l ” ) s o c ie tie s w o u ld be o n t h e n e x t r u n g , a n d t h e m o s t h i e r a r c h i c a l ( “st a te " ) s o c ie tie s w o u ld b e o n th e t o p ru n g . O f c o u rs e , sim p ly p o s i ti n g t h a t t h e s o c ie tie s w i t h c o m ­ p l e x so c ia l o r g a n i z a t i o n (e.g., l a b o r s p e c i a l i z a t i o n a n d p o l i t i c a l h i e r a r c h y ) h a v e g r e a t e r t e c h n o l o g i c a l c o m p l e x i t y is n o t i n h e r e n t l y d e m e a n i n g ; b u t it is n o t /"'entirely a c c u r a t e . A n t h r o p o l o g i c a l r e s e a r c h h a s p e r s i s t e n t l y s h o w n t h a t n e i t h e r ^ I s o c i a l s t r u c t u r e s n o r t h e i r k n o w l e d g e s y s t e m s c a n b e c o n s i s t e n t l y r a n k e d in a ' J u n i l i n e a ! s e q u e n c e ; f o r e x a m p l e , m o n o t h e i s t i c r e l i g i o n s t e n d t o o c c u r in b a n d ^ a n d s t a t e s o c i e t i e s m o r e t h a n in t r i b a l . J u s t as b i o l o g i c a l e v o l u t i o n h a s b e e n re v is e d f r o m Lo.vejQ.y’j “g r e a t c h a i n o f b e i n g ” to. G o u l d ’s “c o p io u s ly ,j> ra n c h i n g b u s h ,”^ so t o o c u l t u r al e v o l u t i o n is n o w ty p ic a lly p o r t r a y e d as a b r a n c h i n g d i v e r ­ sity o f fo rm s. T h e r e is n 6 r e a s o n t o f o c u s - o n s t a t e s o c i e t i e s o v e r n o n s t a t e s o c i ­ e ti e s in t h e p u r s u i t o f a n t i p r i t m t i v i s r p o r t r a i t s . T h e d if fic u ltie s o f t h e o r e t i c a l f r a m e w o r k s in the. e p i s t e m o l o g y o f n o n s t a t e s o cieties h a v e b e e n m u c l i m q r e mixecl. A p p i a h ( 1 9 9 2 ) p ro v i d e s a n e x t e n s i v e d is ­ c u s s io n o f t h i s i n t e r s e c t i o n , s t a r t i n g w i t h e th n o p h i l o 's o p h y . H is a n a ly s is w e a v e s ( b e t w e e n t h e positions, o f W ire d 'q ( 1 9 7 9 ) , w h o c r i t i q u e s t h e fo cu s o n c o m p a r i s o n to W e s t e r n s c i e n c e r a th e r'tlV a n r e l i g i o n ( n o t i n g t h a t it le a v e s t h e s u p e r s t i t i o n s [ a n d folk p h i l o s o p h i e s o f t h e W e s t u n e x a m i n e d ) , a n d E ^ u n t o n d | i ) ( i 9 8 3 ) , w h o ( \ a r g u e s a g a in s t a n y m i m e t i c c o m p a r i s o n , s u g g e s tin g t h a t etKiTO'pFfilosophy a n d i t s \ laities a re d r e s s in g E u r o p e a n m o t i v a t i o n s in a u t o c h t h o n o u s g a rb . B o t h c r i t i q u e s ) ^could c e r t a i n l y b e a p p l i e d to A f r i c a n f r a c ta l s . B u t lik e M u d i m b e ’s ( 1 9 8 8 ) F o u cau ltian analysis o f A fric a n epistem ology, a n d G i l r o y ’s (1 9 9 3 ) fractal history ( w h ic h we wit! e x a m i n e in t h e f o l lo w in g c h a p t e r ) , A p p i a h ' s d i a l e c t i c a l c o n t o u r m a p s A f r i c a n e p i s t e m o l o g y as a n h i s t o r i c a l p r o c e s s r a t h e r t h a n a n o b j e c t o f s t r i c t l y Implications 190 p re - o r p o s t - W c s t e r n p r e s e n c e . T h e c a u t i o n s o f W i r c d u , H o u n t o n d j i , a n d o t h ­ ers a re serio u s r e m i n d e r s t h a t A f r i c a n fra ctals c a n o n l y s u c c e e d as p a r t i c i p a n t s i m ­ u l a t i o n , n o t as I n d i a n a - J o n e s d i s c o v e r i n g - a n o t h e r l o s t t e m p l e . ( G i v e n th o s e p r e c a u t io n s , it m a k e s s e n s e to see A f r i c a n fr a c ta ls as ju s t a n o t h e r m o m e n t in a h i s t o r i c a l s e q u e n c e . O n e c o u l d , f o r e x a m p l e , p l a c e t h e m i n M u d i m b e ’s h i s t o r y o f e t h n o p h i l o s o p h y , o r Z a s l a v s k y ’s ( 1 9 7 3 ) h i s t o r y o f research o n A fric a n m a th . B ut th e re are o th e r rese a rc h e rs w h o h a v e p o in ted o u t so m e o f th e fractal c h a ra c te ris tic s o f A f ric a n d e sig n s a n d p ra c tic e s , a n d i t is u s e f u l t o e x a m i n e t h e m as a g r o u p , e v e n if t h e y l a c k t h e c l e a r h i s t o r i c a l tr a je c to ry o f o t h e r c ate g o ries. W e h a v e a lre a d y m e n t i o n e d t h e o b s e r v a tio n of n o n l i n e a r s c a l i n g by B n t i s h ’a r t h i s t o r i a n W i l l i a m ^ a g g ^ c h a p t e r 6 ) , a n d tjhe i n t e r p r e t a t i o n o f s c a l i n g d e s i g n s as s i g r d f i e r s o f t j T f i m . t y j j i th e .,w p rk .-.o f C a m e y p o m a n t h .^ o lo g tj p u E n g e l b e r t M v e n g ( c h a p t e r 9 ) . L e o p o l d S e n g h o r , t h e d i s t i n g u i s h e d m a n o f l e t t e r s w h o b e c a m e S e n e g a l ’s first p r e s i d e n t , a ls o h a d a n e y e fo r A f r i c a n f r a c t a l s . H is t e r m w a s “d y n a m i c s y m m e t r y , ” w h i c h h e t o o k f r o m arc h i s t o r i a n s . B u t S e n g h o r ’s m o t i v a t i o n w a s p r i m a r i l y id e o l o g y ; d e f i n ­ in g a “ n e g r i t u d e ” t h a t w o u l d e n c o m p a s s t h e k i n d o f c u l t u r a l p o l i t i c s h e saw as n e c e s s a r y t o i n d e p e n d e n c e . M o s t r e c e n t l y , H e n r y L o u i s ^ a t f i ^ ( 1 9 8 8 ) e x p l o r e d t h e d o u b l i n g p r a c t i c e s o f v o d u n d i v i n a t i o n in t e r m s o f j a j i t e r a r y v e r ­ sion o f d e te rm in is tic ch ao s; here th e re c u rsio n g e n e ra te s a c u ltu ra l u n c e rta in ty tK a tT re e s g e n d e r i d e n t i t y f r o m s t a t i c b o u n d a r i e s : “T h e F o n a n d Y o ru b a e s c a p e t h e W e s t e r n v e rs io n o f d iscu rsiv e se x ism th r o u g h t h e a c ti o n o f d o u b lin g th e d o u b l e ; t h e n u m b e r 4 a n d its m u l t i p l e s a r e s a c r e d in Y o r u b a m e t a p h y s i c s . E s u ’s ' t w o s i d e s ‘d i s c l o s e a h i d d e n w h o l e n e s s , ’ r a t h e r t h a n c l o s i n g o f f u n i t y , t h r o u g h t h e o p p o s i t i o n , t h e y s i g n if y t h e p a s s a g e f r o m o n e t o t h e o t h e r as s e c t i o n s o f a subsum ed w h o le .” W h i l e all four h a v e h i t u p o n m a t h e m a t i c a l a s p e c ts o f A f r i c a n fractals, n o n e o f th e s e a u th o r s h a v e focused o n re p r e s e n ta t io n s o f m a t h e m a t ic a l k n o w led g e. M v e n g , t h e t h e o l o g i a n , p r o v i d e s a t h e o l o g i c a l i n t e r p r e t a t i o n . Fagg, t h e a r t i s t , c o n c l u d e s w i t h a c o m p a r i s o n t o D ’A r c y T h o m p s o n ’s f a m o u s n a t u r e d r a w in g s . S e n g h o r , t h e s t a t e s m a n , se e s h is d y n a m i c s y m m e t r y as a s i g n o f c u l t u r a l — a n d t h u s n a t i o n a l — id e n tity . A n d G a t e s , as a li te r a ry c r i t i c , sees it as d is c u rs iv e t e c h ­ n i q u e . S u r e l y m y in s i s t e n c e o n i n d i g e n o u s m a t h e m a t i c s is n o less a n i m p o s i t i o n o f s e e i n g t h e w o r ld t h o u g h m y o w n le n s e s , b u t s i n c e t h a t is n o d i f f e r e n t f r o m t h e o t h e r e x p l a n a t i o n s , w h y d o e s e t h n o m a t h e m a t i c s a p p e a r t o b e so m u c h m o r e c o n t r o v e r s i a l ? I t is b e c a u s e a p o r t r a i t o f m a t h e m a t i c a l s o p h i s t i c a t i o n in n o n s ta te societies creates a strong c o n flic t w ith th e o ld lad d er m odel of cu ltu ral e v o l u t i o n , a m o d e l t h a t is itse lf o v e r d u e fo r e x t i n c t i o n . Theoretical fra m e w o rk s in cu ltu ra l studies ConclMsion S o fa r w e h a v e o u t l i n e d s e v e r a l t h e o r e t i c a l f r a m e w o r k s t h a t c o u l d r a i s e p r o b ­ l e m s f o r A f r i c a n f r a c t a l s . O n t h e o n e h a n d , t h e r e ^ a r e t h e o r i e s in w h i c h t h e d e s i g n s c o u l d b e d is m i s s e d as u n c o n s c i o u s b i o l o g i c a l o r s o c i a l p r o c e s s . O n t h e o t h e r h a n d , g reat care m u st be t a k e n to a v o id e it h e r in flated claim s or a m a th e m a tic a l v e rsio n o f n e g ritu d e . W i t h th e e x c e p tio n o f b io lo g ical d e te r ­ m in is m , n o n e o f th e fra m e w o rk s re v ie w e d h e r e are necessarily g o o d o r bad. T h e r e a r e c a s e s in w h i c h m a t h e m a t i c a l a n t h r o p o l o g y is m o r e a p p r o p r i a t e t h a n t h e e t h n o m a t h e m a t i c a l a p p r o a c h , o r w h e n s a m e n e s s is a b e t t e r s t r a t e g y th a n d ifferen ce, o r w h e n a t t e n t i o n to a n c ie n t Egypt n e e d s to su p ersede a t t e n ­ t i o n t o s u b - S a h a r a n A f r i c a — j u s t as t h e r e a r e c a s e s in w h i c h t h e o p p o s i t e is tr u e . O u r g o a l is n o t t o f i n d t h e o n e t r u e f i n a l f r a m e w o r k — it d o e s n o t e x i s t — . b u t t o k e e p a w e l l - s t o c k e d t o o l b o x a n d k n o w h o w t o p i c k t h e r i g h t t o o l fo r t h e r i g h t j o b . N o w t h a t w e a r e w e l l p r e p a r e d f o r c o n s t r u c t i v e ta s k s , it is t i m e to m o v e t o p o l i t i c s . CHAPTER — T h e ---politics o f —A frican— —fractals™ G i v e n t h e p o s s ib le d a n g e r s in m i s i n t e r p r e t i n g A f r i c a n f r a c ta l s , h o w c a n w e p u t t h e m t o g o o d use? S o c ia l t h e o r i s t s fr o m m a n y d i f f e r e n t d i s c i p l i n e s h a v e u se d tw o m a t h e m a t i c a l c o n c e p t s w e h a v e d is c u s s e d , r e c u r s i o n a n d t h e a n a l o g - d i g i t a l d ic h o t o m y , in c o n s t r u c t i n g t h e i r ideologies. M a n y th e o r ie s o f c o m m u n i c a t i o n h a v e a s s u m e d t h a t t h e r e is s o m e k i n d o f u n iv e r s a l e t h i c a l o r s o c ia l d i f f e r e n c e b e t w e e n u s i n g a n a l o g s ig n a ls a n d u s i n g d i g i ta l s y m b o ls . O t h e r t h e o r i e s h a v e m a i n t a i n e d t h a t re c u rsio n h a s so m e k in d o f u n iv ersal e th ic a l o r so cial valu e. B o th are u lti­ m a t e l y fa ilu re s in t h e s e n s e t h a t e t h i c s a n d v a l u e s d o n o t lie w i t h i n m a t h e ­ m a t i c a l d i s t i n c t i o n s . Yet t h e y a re a ls o o n t h e r i g h t t r a c k in t h a t s u c h a s s o c ia ti o n s c a n b e locally fo rm ed — it is ju s t t h a t d iffere n t lo c a t io n s w ill re s u lt in d if fe re n t social m e a n i n g s . S u c h lo c a lly s p ecifie d s o c i a l a t t a c h m e n t s t o f r a c t a l s c a n b e u se fu l for u n d e r s t a n d i n g c u l t u r a l p o l i t i c s in A f r i c a a n d b e y o n d . T h e p o litic s o f t h e a n a lo g -d ig ita l d i s t i n c t i o n J e a n J a c q u e s R o u s s e a u is o f t e n c r e d i t e d as a f o u n d e r o f “ o r g a n i c r o m a n t i c i s m , ” t h e t h e o r y t h a t t h e N a t u r a l is i n h e r e n t l y b e t t e r t h a n t h e A r ti f ic i a l. W h e t h e r or n o t t h i s is d e s e r v e d , J a c q u e s D e r r i d a ( 19 7 4 ) ta k e s h i m t o ta s k fo r p r o p o s i n g t h a t a natu ral/artificial difference c a n be fo u n d b e tw e e n d iffe ren t languages. J e a n Jacqu es T h e politics o f A fr ic a n fractals R o u s s e a u p r o p o s e d t h a t t h e “n a t u r a l crys o f a n i m a l s , ” m u s ic , a n d “ a c c e n t u a t i o n ” ( t h a t is, p i t c h i n t o n a t i o n in t h e h u m a n v o i c e ) a r e all a s i m i l a r ty p e o f c o m ­ m u n i c a t i o n . I n t h i s I w o u ld t e n d t o a g re e , s i n c e i n s t r u m e n t a l m u s ic a n d h u m a n p i t c h i n t o n a t i o n a re f o r . t h e m o s t p a r t a n a l o g r e p r e s e n t a t i o n s a n d s i n c e h e was pro bably th in k in g o f a n a lo g ex am p les o f a n im a l c o m m u n ic a tio n (a lth o u g h m a n y a n im a ls , fo r e x a m p l e v e r v e t m o n k e y s , use d ig i ta l c o m m u n i c a t i o n as w ell). R o u s s e a u c o n t r a s t e d t h i s t o “a r t i c u l a t i o n ” i n t h e h u m a n v o i c e , by w h i c h h e m e a n t t h e li n g u i s t i c ( a n d h e n c e d i g i t a l ) p a r t s o f s p e e c h . B u t i n s t e a d o f s e e i n g t h e d i s t i n c t i o n as tw o d i f f e r e n t types o f r e p r e s e n t a t i o n , o n e a n a l o g a n d th e o t h e r d i g i ta l , R o u s s e a u c l a i m e d t h a t a n a l o g s i g n a l s w e r e n o t a fo rm o f r e p r e s e n t a t i o n a t all. In his view, dig ital versus a n a lo g was r e p r e s e n t a t i o n versus T h e R eal. M usic, a n im a l cries, an d e m o tio n a l i n t o n a ti o n w ere s o m e h o w m o re n a tu r a l and a u t h e n t i c . W o r s e y e t, h e i n f l a t e d t h i s i n t o a c u l t u r a l d i f f e r e n c e , m a i n t a i n i n g t h a t w h i l e E u r o p e a n l a n g u a g e s w e re la r g e ly b a s e d o n ( d i g i t a l ) a r t i c u l a t i o n , t h e l a n g u a g e o f t h e n o b e l s a v a g e w as c l o s e r t o n a t u r e . O n e m i g h t h o p e t h a t D e r r i d a w o u ld c o r r e c t t h e m a t t e r a n d p o i n t o u t t h a t a n a l o g s ig n a ls a re ju s t as m u c h a r e p r e s e n t a t i o n — ju s t as m u c h fakes, j u s t as easy t o lie o r te l l t r u t h w i t h , a n d j u s t as a r t if i c ia l— as d i g i t a l s y m b o ls are. B u t h e t o o fa iled t o p r o d u c e a b a l a n c e d p o r t r a i t . D e r r i d a d i d in s ist t h a t all h u m a n lin g u is ­ tics is f u n d a m e n t a l l y dig ital ( q u ite tr u e ) , b u t h e d id n o t b o t h e r to say a w o rd a b o u t o t h e r i n o d e s o f v o c a l r e p r e s e n t a t i o n . T h i s e r r o r is d u e t o D e r r i d a ’s c o n c e r n o v e r che a u t h o r i t a r i a n id e o lo g y t h a t o r g a n i c r o m a n t i c i s m c a n p r o d u c e . For e x a m p l e , h is t o r y is full o f d i c t a t o r s w h o c l a i m e d t h a t t h e i r e t h n i c g r o u p w as t h e re a l o r n a t u r a l o n e , a n d t h a t o t h e r s w e re a r t i f i c i a l p o l l u t a n t s in t h e i r E d e n . R o u s s e a u h i m s e l f d id n o t h a v e s u c h fa s c istic t e n d e n c i e s , b u t D e r r i d a is r i g h t in p o i n t i n g o u t t h a t o r g a n i c rc m a n ti-c is r.v c a r. alv/ay s b e u s e d in t h a t w a y , n o m a t t e r w h o it is c o m i n g fro m . * O n e n e e d n o t p a n i c so m u c h , h o w e v e r , a n d b a n i s h a n a l o g sig­ n a ls fr o m e x i s t e n c e ; it is e n o u g h t o g iv e t h e m t h e s a m e e p i s t e m o l o g i c a l s t a t u s as d i g i ta l s y m b o l s — n o m o r e a n d n o less. 1 h a v e f o u n d th is e g a l i t a r i a n v ie w o f t h e a n a l o g / d ig i ta l d i s t i n c t i o n v ery d if­ ficult to p ro m o te ; it seem s t h a t e v e r y o n e h a s t h e i r o w n fav o rite view. W h e n 1 sp o k e to c h a o s t h e o r i s t R a l p h A b r a h a m , for e x a m p l e , h e e x p l a i n e d t h a t a n a l o g system s w ere in h i s view t h e r e a l m o f spirit, t h e v i b r a t i o n s o f A t m a n . P o s t m o d e r n th e o r y m a v e n J a m e s C liffo rd , t o t h e c o n tr a r y , i n s is t e d t h a t o n ly d ig i ta l r e p r e s e n t a t i o n is c a p a b l e o f t h e flexible r e a r r a n g e m e n t s t h a t c o n s t i t u t e h u m a n t h o u g h t . T h i s sa m e b a t t l e h a s b e e n p la y e d o u t in t h e h i s t o r y o f A f r i c a n c u l t u r a l s tu d ie s . D u r ­ ing th e ) 960s, realism was in vogue, a n d w h a t c o u ld h a v e b e e n a w o n d e rfu l e x p l o ­ r a t i o n o f t h e a n a l o g r e p r e s e n t a t i o n t e c h n i q u e s in A f r i c a n c u l t u r e w as o f t e n r e d u c e d t o r o m a n t i c p o r t r a i t s o f t h e “ r e a l ” a n d “n a t u r a l , ” w h ile A f r i c a n sy m b o l 193 /mpficacions 194 s y s t e m s s u f fe re d f r o m n e g l e c t . D u r i n g t h e l a t e 1 9 7 0 s , t h i s b e g a n t o r e v e r s e itself—r w i t h t h e a d v e n t o f p o s t m o d e r n i s m , A f r i c a n c u l t u r a l p o r t r a i t s b e c a m e increasin gly focused o n d iscou rse a n d sy m b o l systems, e v e n a t t h e e x p e n s e o f ig n o r­ in g a n a l o g r e p r e s e n t a t i o n s . I t is i m p o r t a n t , h o w e v e r , t o see h o w th e s e r e s t r i c t i o n s h a v e b e e n c o n t e s t e d , p a r t ic u l a r ly in b la c k i n t e l l e c t u a l c o m m u n i t i e s . H o o k s ( 1 9 9 1 , 2 9 ) s u m m a r i z e s h e r o w n r e a c t i o n to r o m a n t i c o r g a n i c i s m : “T h i s d is c o u r s e c r e a t e d t h e id e a o f ‘p r i m ­ i t i v e ’ a n d p r o m o t e d t h e n o t i o n o f a n ‘a u t h e n t i c ’ e x p e r i e n c e , s e e i n g as ‘n a t u r a l ’ t h o s e ex p re s sio n s o f b la c k life w h i c h c o n f o r m e d t o a p re - e x is t in g p a t t e r n o r s t e r e o ­ ty p e .” R o se (1993.) describ es th e h is to ry o f rap m u sic, also arising in t h e m id -1 9 7 0 s, as n o t j u s t a r e s i s t a n c e to o r g a n i c r o m a n t i c i s m , b u t as a t e c h n o c u l t u r a l r e b e l l i o n t h a t m a k e s D e r r i d a lo o k like G u t e n b e r g . C o r n e l W e s t , H o u s t o n B ak er, H o r t e n s e S p il le r s , a n d H a z e l C a r b y h a v e m a d e i n t e r v e n t i o n s in A f r i c a n A m e r i c a n i n t e l ­ l e c t u a l d is c o u rs e in s i m i l a r w ay s, as h a v e w o rk s o f b l a c k s c i e n c e f i c t i o n s u c h as G e o r g e S c h y l e r ’s Black N o M o r e , R a l p h E l l i s o n ’s invisible M a n , 2 T o n i C a d e B a m b a r a ’s T/ie S a lt E aters, S a m u e l R . D e l a n y ’s D h alg ren , a n d O c t a v i a B u t l e r ’s Xenogenesis trilogy. A n e g a l i t a r i a n v ie w o f t h e n a tu r a l/ a r ti f i c ia l d i c h o t o m y c a n b e s e e n in b l a c k i n t e l l e c t u a l h i s t o r y r u n n i n g f r o m G e o r g e W a s h i n g t o n C a r v e r ' s c o n c e p t o f “ G o d ’s K i n g d o m o f t h e S y n t h e t i c ” t o M u d i m b e ’s “ I n v e n t i o n o f A f r i c a . ” -* In d e e d , C a r v e r a n d M u d i m b e ’s c o n c e p t s .a r e q u it e sim ilar; it is n o t M u d i m b e ’s c o n ­ t e n t i o n t h a t A f r i c a n u n i t y la c k s a s p i r i t u a l b o n d , b u t r a t h e r a c e l e b r a t i o n o f t h e s p i r it o f i n v e n t i o n , w h i c h r e q u i r e s r e s i s t a n c e t o t h e E u r o p e a n c l a i m t h a t s p i r i t c a n ex is t o n ly in c a te g o rie s o f t h e n a t u r a l . A f r i c a n a n i m i s m is m a r k e d b y a n e x t r a ­ o r d i n a r y a c c e p t a n c e o f t h e relig io u s s i g n i f i c a n c e o f artifice,'* fr o m gris-gris t o t h e m o j o h a n d , a n d its t e c h n i q u e s fo r p a s s i n g i n f o r m a t i o n t h r o u g h t h e p h y s i c a l d y n a m i c s o f s o u n d a n d m o v e m e n t s h o w t h a t t h i s faith in t h e p o w e r o f a n a l o g r e p ­ r e s e n t a t i o n is n o t m i s p l a c e d . T h e p o litics of re c u rs io n W h i l e D e r r i d a was t r a s h i n g o r g a n i c r o m a n t i c i s m , M i c h e l F o u c a u l t w a s a t t e m p t ­ in g t o d o t h e s a m e for h u m a n i s m . H is h is to r i c a l s t u d ie s d e m o n s t r a t e t h a t h u m a n ­ ist g o als o f r e c u r s i o n — t o b e s e l f - g o v e r n e d , s e l f - c o n t r o l l i n g i n d i v i d u a l s — a r e n o t i n n o c e n t ; b u t r a t h e r d e v e l o p h i s t o r i c a l l y in c o m b i n a t i o n w i t h v a r i o u s t e c h ­ n i q u e s o f s o c i a l c o n t r o l . I n a n e ra w h e r e " s e l f - m a n a g e m e n t ” u s u a l ly m e a n s t h a t t h e c o r p o r a t i o n y o u w o r k for h a s d e v e l o p e d i m p r o v e d t e c h n i q u e s fo r selfe x p l o i t a t i o n , it is n o t h a r d t o see w h a t F o u c a u l t is g e t t i n g a t . A s in t h e c a s e o f D e r r i d a ’s w a r n i n g s a g a i n s t c l a i m s t h a t a n a l o g r e p r e s e n t a t i o n w ill a u t o m a t i c a l l y le a d t o m o r e e t h i c a l li v in g , F o u c a u l t w a r n s a g a i n s t s e e i n g r e c u r s i o n as a m o r a l The politics o f A fr ic a n fractals f o r m u l a . 5 W h i l e A f r i c a n a n a l o g s y s te m s raise t h e p r o b l e m o f s o m e o n e m a k i n g cla im s a b o u t w h a t is m o r e real o r m o r e n a t u r a l , A f r i c a n r e c u r s io n — espe cially t h e r e c u r s iv e a r c h i t e c t u r e o f A f r i c a n s e t t l e m e n t s — raise s t h e p r o b l e m o f h u m a n i s t c la im s . T o see h o w th i s c a n be a p r o b l e m , c o n s i d e r t h e f o l lo w in g tw o case s tu d ie s o f A f r i c a n a r c h i t e c t u r e . C a p l a n ( 1 9 8 1 ) s t u d i e d t h e r e l a t i o n b e t w e e n 'h o u s i n g a n d w o m e n ’s a u t o n o m y in T a n z a n i a . S h e d e s c r i b e d h o w t h e flex ibility o f h o u s i n g a ll o w e d w o m e n to c r e a t e n e w h o m e s if t h e y w a n t e d a d i v o r c e , o r t o e x t e n d o ld h o m e s if th e y w a n t e d t o s h ift t h e fa m il y s t r u c t u r e . A s in m a n y A f r i c a n s e t t l e ­ m e n ts, this self-organized h o u s in g c r e a t e d a self-sim ilar s tru c tu r e — fractals— w h i c h a ll o w e d g r e a t e r s o c ia l s e l f - c o n t r o l fo r w o m e n . W h e n s o c i a li s m b r o u g h t m o d ­ e r n i z a t i o n p ro g r a m s , t h i s a u t o n o m y w as t h r e a t e n e d by t h e “ i m p r o v e d ” h o u s i n g design, w h i c h s o m e tim e s re s em b le d c o n c r e t e a rm y b arracks.^ H e r e o n e w o u ld c o n ­ c l u d e t h a t f r a c ta l is b e tte r. S t o i l e r ( 1 9 8 4 ) d e s c r ib e d a S o n g h a i t o w n in w h i c h a c a s t e s y s te m e n s u r e d t h a t t h e b e s t la n d w as v o l u n t a r i l y g i v e n o v e r t o t h e h i g h e s t c a s t e m e m b e r s . It was n o t a m a t t e r o f forcing p e o p le a g a in s t t h e i r will, b u t simply u n q u e s t i o n e d c o m ­ m o n s e n s e t h a t o n e s h o u l d w a n t t o b e l o c a t e d in t h e i r p r o p e r p la c e . T h i s fractal, s e lf-o rg an iz ed a r c h i t e c t u r e w as a fo r m o f s e l f - e x p l o i t a t i o n . E v e n tu a l ly sev eral m e m b e r s o f t h e c o m m u n i t y d e c i d e d to b r e a k o u t o f th i s o p p r e s s iv e s t r u c t u r e by b u i l d i n g h o u s e s a l o n g t h e n e w h i g h w a y . T h u s l i b e r a t i o n in th i s c a s e m e a n t leaving th e fractal geometry, a n d lin in g u p in s tra ig h t E u clid ean fo r m a tio n — exactly th e o p p o s ite o f th e T a n z a n ia n villa ge s t u d ie d by C a p l a n . S to l le r ’s w o rk rficely illus­ tr a t e s M i c h e l F o u c a u l t ’s w a r n i n g a g a i n s t s i m p l i s t i c h u m a n i s t fo r m u la s : selfd e t e r m i n a t i o n is n o t n e c e ss a rily l i b e r a t i n g ; ir c a n se r v e t o s u p p o r t so c ia l c o n t r o l " r a t h e r -t h a n resist it. M e i t h e r - f r a c t a l - n c r E u c l i d e a n g e o m e t r i e s h a v e a n y i n h e r ­ e n t e t h i c a l c o n t e n t ; s u c h m e a n i n g s arise fr o m t h e p e o p l e w h o use t h e m . C o l o n i a l i s m a n d a r c h i t e c t u r a l fr a c ta ls R e n e D escartes was n o t m u c h o f a h u m a n i s t ; ' i n his view self-organized a r c h i te c t u r e is j u n k . H e m a k e s th is c le a r in h i s f a m o u s D iscourse o n Metfiodolog)1; (TJhere is less p e rfectio n in w orks m a d e o f several pieces an d in w o rks m ade by th e h a n d s o f several m a s te rs t h a n in t h o s e works o n w h ic h b u t o n e m a s te r h as w orked. T h u s o n e sees chat b u il d in g s u n d e r t a k e n a n d c o m p l e te d by a single a r c h i te c t are c o m m o n l y m o r e b eau tifu l a n d b e tt e r o rd e re d t h a n th o s e t h a t several a rc h ite c ts h a v e tried to p a t c h up. . . . T h u s 1 im ag ined t h a t peo ple w h o , h a v in g o n c e b e e n h a l f savages a n d h a v in g b e e n civilized o n ly gradually, have made their laws only to th e e x te n t th a t th e in c on ven ien ce caused by crimes 195 Implications a n d q u arre ls forced t h e m to d o so, w ould n o t b e as w ell o rd e r e d as th o s e w h o, from th e very beg in nin g o f their c o m in g together, h a v e followed th e fundam ental p recep ts o f so m e p r u d e n t legislator. . (1 6 7 3 , 12) F o r D e s c a r t e s , “se l f - o r g a n iz e d ” is s y n o n y m o u s w i t h sa v a g e s , t h e im p e r f e c ­ tio n o f b o th m aterial a n d social stru ctu re. L ack o f c o m p le te E u c lid e a n reg u lar­ ity m e a n s r a n d o m n e s s : fo r “s t r e e t s c r o o k e d a n d u n e v e n , o n e will say t h a t it is c h a n c e m o r e t h a n t h e will o f s o m e m e n u s i n g t h e i r re a s o n c h a t h a s a r r a n g e d t h e m th u s ” (p. 12). T h e lack o f C a r te s ia n c o o r d i n a te s in m a n y A f r i c a n s e t tl e m e n t s w ould th u s e v id e n c e t h e i r n e e d for t h e g u id a n c e o f c o l o n i a l reas o n . A s H u l l ( 19 7 6 ) no tes, h u g e c e n t e r s o f u r b a n life in A f r i c a w e r e i n d e e d d i s r e g a r d e d b y E u r o p e a n s as “u n s t r u c t u r e d b u s h c o m m u n i t i e s ” o n j u s t t h e s e p r i n c i p l e s . W h i l e T i m b u k t u was g r a n t e d c i t y h o o d d u e to its grid p a t t e r n o f s t r e e t s , t h e Y o r u b a c it ie s o f e q u a l p o p ­ u l a t i o n size a n d e c o n o m i c , t e c h n i c a l , a n d l a b o r s p e c i a l i z a t i o n h a v e b e e n d i s r e ­ g a r d e d as m e r e l y g i a n t v illa g e s d u e t o t h e i r la c k o f C a r t e s i a n reg u la rity .^ T h u s f r a c t a l a r c h i t e c t u r e w as u s e d as c o l o n i a l p r o o f o f p r i m i t i v i s m . T h i s d e b a t e o v e r t h e u r b a n s t a t u s o f n o n - E u c i i d e a n s e t t l e m e n t s c o n t i n u e s in t h e p o s t c o l o n i a l era (see S c h w a b 1 9 6 5 ; L lo y d 1 9 7 3 ). T h e o c c a s i o n a l C a r t e s i a n l i n e a r i t y in A f r i c a n a r c h i t e c t u r e t h r e w a h i t c h i n t o t h i s c o l o n i a l j u s t i f i c a t i o n , i n 1871 t h e G e r m a n g e o l o g i s t C a r l M a u c h “d i s ­ c o v e r e d ” t h e r u i n s o f G r e a t Z i m b a b w e . S t u n n e d by t h e e v i d e n c e o f p re c ise s t o n e c u t t i n g o n a m a s s i v e s c a le , h e p r o p o s e d t h a t t h e b u i l d i n g s w e r e n o t o f A f r i c a n d e s i g n , b u t w e r e i n s t e a d d u e to t h e Q u e e n o f S h e b a ' s v i s i t t o S o l o m o n . T h e R h o d e s i a n g o v e r n m e n t u se d t h i s e x p l a n a t i o n a s a p a r t o f its p r o p a g a n d a a g a in s t B la c k r u l e ( M a c i n t o s h a n d M a c i n t o s h , 1 9 8 9 ) . A c t u a l l y , t h e y h a d m u c h less to f e a r in t h e t r u t h : t h e s t o n e w as n o t c u t , b u t it n a t u r a l l y b r o k e i n t o l i n e a r s h e e t s ( a f t e r h e a t i n g ) d u e t o its g e o lo g i c p r o p e r t i e s . M o r e o v e r , m o s t o f t h e o u t ­ sid e w a lls w e r e o r i g i n a l l y c o v e r e d w i t h s m o o t h clay, c r e a t i n g a n o n l i n e a r s e t o f s c a l i n g s h a p e s ( w h i c h C o n n a h [ 1 9 8 7 ] r e f e r s t o as " r a n d o m c u r v e d f o r m s " ) . T h i s is n o t t o d i m i n i s h t h e r e m a r k a b l e t e c h n o l o g i c a l sk ill o f t h e c o n s t r u c t i o n , h u t t o p o i n t o u t t h a t o n e c u l t u r e ’s s i g n for “a r t if i c ia l " c a n b e a n o t h e r ’s sign for " n a t u r a l . ” E u c l i d e a n v ersus f r a c ta l d o e s n o t n e c e s s a r il y m e a n artificial versus n a t ­ ural; t h a t , t o o , is c u l t u r a l l y i n f l u e n c e d . D u rin g th e d e v e lo p m e n t of co lo n ia l cities, th e c h a o s o f A frican a rc h ite c ­ t u r e w a s u se d as b o t h s y m b o l a n d s y m p t o m o f E u r o p e a n fears o v e r s o c ia l c h a o s . P e n n a n t (1 9 8 3 ) provid es a n e x a m p l e o f th is c o n c e r n a b o u t p r o p e r s e t tl e m e n t g e o m ­ e tr y i n his e x a m i n a t i o n o f c o l o n i a l d e v e l o p m e n t in M a la w i: " T h e lan g u a g e o f th is 1930s policy discourse is significant. M e d i c a l e x p e r t s w r o te o f ‘in v e s tig a tio n s ’ s h o w ­ in g ' u n q u e s t i o n a b l y ’ a n d o f ‘a b u n d a n t p r o o f . ’ . • • L a y E u r o p e a n s s h o w e d ‘c o n ­ c e r n , ’ ‘a l a r m , ’ a n d ‘h o r r o r . ’ A f r i c a n s , w i t h t h e i r ‘p r i m i t i v e h a b i t s , ’ o f ‘p r o m is c u o u s The politics o f A/riccm fractals d e f e c a t i o n ’ f o r m e d a ‘f l o a t i n g ’ o r ‘s c a t t e r e d ’ p o p u l a t i o n in n e e d o f ‘c o n t r o l ’ a n d ' s u p e r v i s i o n ' in a ‘p r o p e r l y l a i d - o u t v il la g e o r l o c a t i o n . ’ ” I n t h e a b o v e c a s e w h e r e “p r i m i t i v e ” m i x e s - w i t h m o d e r n , t h e f r a c t a l t r a d i ­ t i o n was a t h r e a t . B u t k e p t in w h a t c o l o n i a l i s t s t h o u g h t o f as its n a t u r a l ro le , it c o u ld m a k e fractal s e t t l e m e n t s a p p e a r to b e n e f i t th e c o lo n i a l e n te r p ris e . T h e n o v ­ elist K a r e n B l i x e n ( l s a k D i n e s e n ) , i n O u t o f A fr ic a ( 1 9 8 8 ) , d e s c r ib e d h e r a t t e m p t s to lay o u t grids for A f r i c a n w o r k e r s ’ h o u s e s o n h e r r a n c h . T h e y refu sed t o fo llo w t h e s e l i n e a r i n s t r u c t i o n s a n d fit t h e i r h o u s e s in p a t t e r n s m a t c h i n g t h e ir r e g u la r c o n fig u ra tio n o f t h e la n d . T h a t s u c h e c o lo g i c a l fit c o u ld b e q u i t e efficient was n o t, h o w e v e r, lost o n t h e c o lo n ists. “T h e s q u a t t e r s ’ la n d was m o r e in te n s e ly a liv e t h a n t h e re s t o f t h e fa rm , a n d w a s c h a n g i n g w i t h t h e s e a s o n s ” (p . 9 ). A r c h i t e c t u r a l f r a c ta ls c o u l d b e p a r t o f c o l o n i a l r o m a n t i c i s m as lo n g as t h e y e n s u r e d a s u p p ly o f s e l f - s u p p o r t i n g w o rk e rs . E v e n in th e case o f so cial c o n tr o l, in d ig e n o u s fractals co uld be utilized. British c o l o n i a l po lic y , fo r e x a m p l e , a t first f a i le d in cases w h e r e t h e r e w as a d e c e n t r a l ­ ized n e t w o r k r a t h e r t h a n a la r g e h i e r a r c h y . T h i s w as a p p r o a c h e d in t h e c ase o f t h e I b o w i t h a sy s te m o f “ i n d i r e c t r u l e ” b a s e d o n “w a r r e n t c h ie f s " ( I s ic h e i 1 9 7 6 ). T h e Ib o a u t o n o m y o f s e l f - o r g a n iz a t io n w as t u r n e d a g a in s t t h e m ; in a s e n s e it was g ra s s -ro o ts c o l o n i a l i s m . T h e a r c h i t e c t u r a l e q u i v a l e n t o f t h i s s y s te m c a n b e seen, i n a m a n u a l for c o l o n i a l - e r a h o u s i n g d e s i g n s f r o m t h e A g e n c y for I n t e r n a t i o n a l D e v e lo p m e n t ( H in c h c l if f 1946, 3 1 ). H e r e t h e Ib o s’ fractal s e t t l e m e n t p a tt e r n (radial houses a r o u n d a c e n t e r in e a c h v illa g e , r a d i a l v illages a r o u n d t h e s e t t l e m e n t c e n ­ t e r ) is t i d i e d u p t o s u i t E u r o p e a n c o n c e p t i o n s o f s y m m e t r y w h i l e r e t a i n i n g th e o v e ra ll i n d i g e n o u s f r a c ta l s t r u c t u r e . F r a c t a l s a n d r a c ia l r e d is tr ic tin g In th e i n t r o d u c t i o n t o his s e m in a l F ra cta l G e o m e tr y o f N a tu r e , B e n o i t M a n d e l b r o t e x a m i n e s s o m e o f t h e d i s p a r a g i n g c o m m e n t s t h a t w e r e m a d e a b o u t t h e e a rly fractal fo rm s o f G e o r g C a n t o r , H e l g e v o n K o c h , a n d o th e r s . R e j e c t e d as “b iz a r r e ” a n d " t o r t u r o u s , ” t h e s e “d r a g o n s ” w e r e c o n s i g n e d co t h e o d d i t i e s s e c t i o n a t t h e e n d o f t h e few m a t h t e x t s t h a t w o u l d e v e n c o n s i d e r t h e m . S t r i k i n g l y s i m i l a r la n g u a g e h a s b e e n u s e d t o r e j e c t t h e o u t l i n e s o f v o t i n g d i s t r i c t s t h a t w e r e a lt e r e d t o i n c l u d e la rg e r A f r i c a n A m e r i c a n p o p u l a t i o n s , a n d t h e s e d o i n d e e d a p p e a r t o b e f r a c ta l s (fig. 1 2 .1 ) .8 W e r e t h e c o u r t s as m i s t a k e n l y h a s t y t o d i s r e ­ gard f r a c ta l s as m a t h e m a t i c i a n s w e re ? T h e E u c l i d e a n s h a p e o f v o t i n g d i s t r i c t s is n o t a n a r b i t r a r y s a m p l i n g — th is Jj? could o n ly b e d o n e by r a n d o m l y s e l e c t i n g v o t e r s fr o m e v e r y w h e r e in t h e s ta te . A c c o r d i n g to t h e 199 3 S u p r e m e C o u r t r u l i n g in Shaur v. R en o , it is m e a n t to 198 Implications Q eo rg ia c o n g r e s s io n a l d i s t r i c t x i trr 1 9 9 2 S i m i l a r i t y o f i r r e g u l a r r e d i s t r i c c i n g p a r c e r n c a n b e s e e n a t m u l t i p l e s c a le s . {O r ig in a l maps c o u r te s y o f die I | C a r l V i n s o n I n s t i t u t e , p r o [/ h ie h ig h lig h t s b y t h e a u t h o r . ) d e s i g n a t e a g e o g r a p h i c lo c a l e in w h i c h " s h a r e d i n t e r e s t s ” i n f o r m t h e v o te . T h e ;j o b j e c t i o n to c r e a t i n g a d i s t r i c t in w h i c h c o n t o u r s a re “p r e d o m i n a n t l y m otivated" b y r a c e is t h a t it c r e a t e s a b ia s in t h e s a m p l i n g o f t h e g e o g r a p h i c l o c a t i o n . T h is | w o u ld c e r t a i n l y b e t h e c a s e if w e w e r e to t a k e a r a n d o m s a m p l i n g , s e p a r a te d vot- .jf* ers by e t h n i c i t y , a n d t h e n d e s i g n a t e d th o s e e t h n i c g r o u p s as t h e v o t i n g districrs. f„ H o w e v e r , if s o m e e t h n i c g ro u p s a re d i s t r i b u t e d in E u c l i d e a n s e t t l e m e n t patterns, L* a n d o t h e r s in f r a c t a l s e t t l e m e n t p a t t e r n s , t h e n w h y c o n s i d e r E u c l i d e a n district > s h a p e s t o b e u n b i a s e d , a n d f r a c t a l d i s t r i c t s h a p e s t o b e b i a s e d ? I d o n o t know if« A f r i c a n A m e r i c a n h o u s i n g d i s t r i b u t i o n s a r e m o r e f r a c t a l t h a n o t h e r s — niuT e v e n if th e y are, I w o u ld n o t n e c e s s a r il y a s s u m e a c u l t u r a l c o n n e c t i o n to African,' f r a c ta ls — b u t t h e fa c t t h a t we n o w k n o w o f s o c i e t i e s in w h i c h f r a c ta l s e t t l e m e n t p a t t e r n s a re b e a u t i f u l f u s i o n s o f f o r m a n d f u n c t i o n s u g g e s ts t h a t w e m i g h t r e c o n s i d e r t h e i r p o t e n t i a l ro le in A m e r i c a n p o li ti c s . A f r i c a n fr a c ta ls f r o m c u l t u r a l v is io n a rie s F ra c t a ls a n d c h a o s t h e o r y h a v e b e e n i n c r e a s i n g l y m e n t i o n e d in che h u m a n i t i e s as e i t h e r a to o l o r a n o b j e c t o f c u l t u r a l a n a ly s is , b u t t o o o f t e n t h e a p p r o a c h o f t h e s e s t u d i e s h a s le ft t h e im p r e s s i o n o f m a t h e m a t i c a l i n k b lo ts a l l o w i n g w rite rs t o see w h a t e v e r t h e y p le a s e . L y o ta rd ( 1 9 8 4 ) sa w f r a c ta l g e o m e t r y as c o n t r i b u t ­ in g t o a “p o s t m o d e r n c o n d i t i o n ” w h o s e c o n t r a d i c t o r y n a t u r e w o u l d d i s r u p t a u t h o r i t a r i a n c e r t a in t y ; a m o r e c a u t i o u s v e r s i o n o f t h i s th e s is is floated in D eleuze a n d G u a t t a r i ( 1 9 8 7 ) . A t le a s t tw o a u t h o r s ( S t e e n b u r g 19 91; A r g y ro s 1 9 9 1 ) h a v e a rg u e d t h a t fractals a n d o t h e r b r a n c h e s o f c h a o s th e o r y h a v e c re a te d a d ir e c t c h a l l e n g e t o p o s t m o d e r n i s m , i n t e g r a t i n g t h e d i s r u p t i o n s it c r e a t e d . P o r u s h ( 1 9 9 1 ) a n d o t h e r s insist t h a t " d e t e r m i n i s t i c c h a o s " is a t t e m p t i n g t o s u b s t i t u t e a f e e l i n g o f free will for fa ta lity .^ S o b c h a c k ( 1 9 9 0 ) s u g g e s ts t h a t it im p li e s “a n e m b r a c e o f ir r e s p o n s ib il it y in a w o r l d a l r e a d y b e y o n d c o n t r o l . ” W h e n S o b c h a c k c it e s P e i t g e n a n d F r e e m a n in h e r c o n d e m n a t i o n o f c h a o s t h e o r y as a d e n i a l o f " t h e sp e c ific ity o f h u m a n e m b o d i m e n t a n d h i s t o r i c a l s i t u a t i o n , ” I c a n ’t h e l p b u t t h i n k o f P e i t g e n ’s fractal g e o m e t r y c o u rs e a t t h e U n i v e r s i t y o f C a l i f o r n i a at S a n t a C ruz, w h e re h e c o m m e n te d o n G e r m a n m a t h e m a t ic i a n s w h o a lte re d th e ir c a r e e r s t o o p p o s e N a z i a n t i - S e m i t i s m o r s u p p o r t p e a c e effo rts ; o r o f F r e e m a n ’s ( 1 9 8 1 ) u se o f M a r t i n L u t h e r K in g in h is d is c u s s i o n o f c h a o s in n e u ro p h y s io lo g y . H o w c a n w e c r i t i q u e t h e w o rk o f c h a o s t h e o r i s t s as la c k i n g h i s t o r i c a l sp e c ific ity a n d e m b o d i m e n t if w e ig n o r e t h e i r o w n h i s t o r i e s a n d b o d ie s ? H a y le s ’s C h a o s B ound ( 19 9 0 ) t o o k a m o r e s u b t le a p p r o a c h . L ike P o ru s h a n d o t h e r s , H a y l e s ’s li te r a ry m e t h o d a ll o w s h e r t o g lid e far t o o easily b e t w e e n u n ­ r e l a te d ideas; by t h e ti m e s h e h a s to s s e d t o g e t h e r q u a n t u m th e o r y , e n t r o p y , a n d G o d e l ’s t h e o r e m w i t h d e c o n s t r u c t i o n a n d “h o l i s m , ” o n e c a n o n ly c o n c l u d e t h a t a n y c o m p l i c a t e d idea c a n b e a m e t a p h o r for a n y o t h e r c o m p l i c a t e d id ea. B u t h e r d e ra i le d a n a ly s is o f li te r a ry w ork s, s h o w i n g d e e p p a r a l le l s b e t w e e n se lf-re fle x iv e w ritin g a n d s e l f-r e fe re n tia l m a t h e m a t i c s , su g g es ts t h a t w h e n g r o u n d e d in specific l o c a t io n s t h e fu s io n o f f r a c ta l g e o m e t r y a n d c u l t u r a l i n t e r p r e t a t i o n c a n h e p r o ­ fo u n d ly r e w a r d i n g . P a u l G i lr o y make's e x p li c it use o f f r a c ta l s in his p o r t r a i t o f t h e d iv e r s ity a n d d y n a m i c is m w i t h w h i c h b o t h t r a d i t i o n a l A f r i c a a n d t h e A f r i c a n d i a s p o r a h a v e o rg a n iz e d t h e i r c r o s s - c u l t u r a l flow s. T h e r e c u r s i v e c o n s t r u c t i o n o f h is B la c k A t l a n t i c c a n b e s e e n , fo r e x a m p l e , in t h i s q u o t e fr o m J a m e s B ro w n o n a v is it to Implications h e a r F e la K u t i in N i g e r i a : “[B]y t h i s t i m e h e w a s d e v e l o p i n g A f r o - b e a t o u t o f A f r i c a n m u s ic anc! f u n k . H is b a n d h a d a s t r o n g r h y t h m ; I t h i n k C l y d e p i c k e d u p o n it in h is d r u m m i n g , a n d B o o tsy d u g it t o o . S o m e o f t h e id e a s m y'bancl w as g e t­ t i n g fr o m t h a t b a n d h a d c o m e fro m m e i n t h e first p l a c e , b u t t h a t w as o k a y w i t h m e . I t m a d e t h e m u s ic t h a t m u c h s t r o n g e r ” ( 1 9 9 3 , 199)G i l r o y c i t e s t h e i m p a c t o f t h e V i r g i n i a J u b i l e e S i n g e r s o n t o u r in S o u t h A f r i c a in 1 8 9 0 , t h e r e t u r n o f sla v e s f r o m B razil t o N i g e r i a , t h e R a s t a f a r i c u l t u r e in Z i m b a b w e , a n d o t h e r e x a m p l e s o f “m u t a t i o n s p r o d u c e d d u r i n g its c o n t i n g e n t l o o p s a n d f r a c ta l tr a j e c t o r i e s ." P e r h a p s h i s m o s t r a d i c a l m o v e is a c l a i m for d ias p o r i c m i x i n g w i t h J e w is h c u l t u r e — W .E .B . D u B o is p a s s i n g fo r a Je w t o m a i n ­ t a i n safety in E a s te r n E u ro p e , t h e u se o f t h e E x o d u s t h e m e in M a r t i n L u t h e r K ing, J r., a n d M a r c u s G a r v e y , a n d E. W . B l y d e n ’s c h i l d h o o d in a J e w is h c o m m u ­ n i t y . 1^1 T h e f r a c ta l i m a g e r y w o r k s i n m a n y d i f f e r e n t w ay s fo r G i l r o y — fr o m t h e tu r b u le n t m e t a p h o r o f h y b rid ity to th e c o n c r e t e d e s c r ip ti o n o f s h ip s ’ p a th s a n d t r a v e l e r s ' r o u t e s ( o r “r o o t s / r o u t e s ” as h e p u t s i t ) . 11 W h i l e m u s ic is, w i t h ­ o u t d o u b t , G i l r o y ’s s t r o n g e s t e x a m p l e , h e d o e s s l ip i n t o t h e p r o b l e m a t i c la b e ls o f r e p r e s e n t a t i o n a l v e r s u s “n o n r e p r e s e n t a t i o n a l ” r a t h e r t h a n d i g i t a l v e r s u s a n a l o g , 12 b u t h e m a k e s it c l e a r t h a t t h e m u s i c r e v e r b e r a t i n g a c ro s s h i s B la c k A t l a n t i c is n e i t h e r p u r e n o r n a t u r a l . W h i l e G i l r o y is p r i m a r i l y f o c u s e d o n f r a c t a l s as s p a t i a l r e p r e s e n t a t i o n s o f b l u r r e d b o u n d a r i e s , h e a ls o b ri e fly m e n t i o n s t h e i r p o t e n t i a l for “ a s t r i k i n g im a g e o f t h e s c o p e o f a g e n c y w i t h i n r e s t r i c t e d c o n d i t i o n s ” ( 1 9 9 3 , 2 3 7 0 2 8 ) ; t h a t is, t h e a b i l i t y f o r g e o m e t r i c e x p a n s i o n w i t h i n b o u n d e d s p a c e b e c o m e s a n a n a l ­ ogy for o p p o s i t i o n a l p o litic a l e x p a n s i o n in h u m a n b o n d a g e . T h e m e t a p h o r is c a r ­ r i e d to a m o r e e x a c t i n g r e l a t i o n s h i p i n G a r y V a n W y k s s t u d y o f S o t h o - T s w a n a m u r a l s u n d e r t h e a p a r t h e i d s y s te m o f S o u t h A f r i c a . V a n W y k ( 1 9 9 3 ) f o u n d t h a t t h e litem a , o r t h e h o u s e p a i n t i n g p a t t e r n s o f t h e S o t h o - T s w a n a w o m e n , u tiliz e a l t e r n a t i o n s o f ir r e g u l a r i t y a n d r e g u l a r i t y a t s e v e r a l s c a le s , s o m e t i m e s re s u lti n g in a r e s e m b l a n c e t o f r a c ta l p a t t e r n s . N o t i n g t h a t t h e s c a l i n g is a s s o c i a t e d w i t h t h e g e o m e t r i c s t r u c t u r e o f flo w e rs, a n d f lo w e r s w i t h t h e r e g e n e r a t i v e p o w e r of w o m e n ( b o t h s p i r i t u a l l y a n d in s o c ia l s t r u g g l e s ) , V a n W y k ’s e t h n o g r a p h y c o n ­ c l u d e s t h a t t h e m u r a l s e x p r e s s e d p o l i t i c a l o p p o s i t i o n to a p a r t h e i d h y p r o v i d i n g a v is u a l a n a l o g in w h i c h “ a w o m a n c a n b e s e c r e t i v e w h i l e a t t h e s a m e t i m e h i d ­ in g n o t h i n g ” ( D e l e u z e a n d G u a t t a r i 1 9 8 7 , 2 8 9 —2 9 0 ) . A l t h o u g h t h e w o rd “ fr a c ta l " is n o w h e r e m e n t i o n e d in h is t e x t , A n t o n S h a m m a s ’s n o v e l A ra b esq u es ( 1 9 8 8 ) is a n e x e m p l a r o f n o n l i n e a r in s ig h t a n d re c u r­ s iv e c u l t u r a l c o m m e n t a r y . H e a v e r ( 1 9 8 7 ) a n a ly z e s t h e n o v e l t h r o u g h t h e N o r t h A f r i c a n a r t i s t i c fo rm o f t h e “a r a b e s q u e , ” a n d s h o w s h o w S h a m m a s h a s used th is f r a c ta l t o s u s t a i n t h e c y c lic t i m e a n d m u l t i p l e i d e n t i t i e s r e q u i r e d to a r t i c u l a t e a FIGURE I 2.2 R e l i g i o n s in s t i t u t i o n s in t h e m a p o f C a iro , 1 8 9 8 202 Jm/)Iicauons p o li ti c a l flexibility c ru c ia l t o th e su rv iv al o f c u lt u r a l diversity. “A s a n 'Israeli A r a b , ’ S h a m m a s is a m e m b e r o f a m i n o r i t y g r o u p — b u t as a C h r i s t i a n , h e falls o u t s i d e t h e Is la m i c m a i n s t r e a m o f t h e m i n o r i t y . . .• . O n t h e o t h e r h a n d , 'fie w r it e s in H e b r e w , t h e la n g u a g e o f t h e d o m i n a n t l y J e w is h c u l t u r e , w h i c h is it s e l f a m i n o r ­ ity w i t h i n t h e p r e d o m i n a n t l y A r a b M i d d l e E a s t ” (p . 4 9 ) . S u c h r e c u r s i v e n e s t i n g is e m p h a s i z e d t h r o u g h o u t t h e t e m p o r a l flow , n a r r a t i v e s t r u c t u r e , a n d c o n c e p t u a l d y n a m i c s o f t h e n o v e l . H e a v e r su g g es ts t h a t t h e “ n o n m i m e t ic g e o m e tric a l a b stra c tio n s o f th e a ra b e s q u e ” are a spatial m o d el f o r S h a m m a s . H e n o t e s t h a t in p a r t t h e s e c y c li c r e e n t r i e s a c t t o n e g a t e o n e a n o t h e r ; u n d e r m i n i n g , fo r e x a m p l e , t h e fr u itle s s a r g u m e n t o f “1 w a s h e r e f i r s t .” B u t n e g a t i o n is n o t t h e o n l y m e a n i n g b e h i n d t h e a r a b e s q u e , as H e a v e r p o i n t s o u t in a p a s s a g e t h a t tie s I s l a m i c s o c i a l s t r u c t u r e t o a n a l o g r e p r e s e n t a t i o n , r e c u r s i o n , a n d t h e s c a l i n g p r o p e r t i e s o f fra ctals. T h e a ra b e s q u e does n o t serv e o n ly a n e g a tiv e , c r itic a l f u n c t i o n ; it also bears a p o sitiv e , u t o p i a n message. It acts as a n a n a lo g u e , in t h e a re a o f visual arts, to t h e p o s itio n o f Islamic “c o n t r a c t u a l i s m ” in t h e so c ia l s p h e r e . . . . In c o n tr a s t t o w e ste r n c o rp o r a tiv is m , w ith its p re f e re n c e for h i e r a r c h i c a l s tru c tu r e s in w h i c h a lim ite d n u m b e r o f c o n c lu s io n s are d r a w n from a lim ite d n u m b e r of premises (o n th e m odel o f geometry), th e cyclical rh y th m s o f th e arabesque could w ell be said to c h a r a c t e r i z e a n “ i n d e f i n i t e l y e x p a n d a b l e ” s t r u c t u r e . T h e arabesque pro vides a fram ew ork w ith in w h ic h it b e c o m e s possible to reduce the a p p aren tly “c h a o tic variety o f life’s reality” to m an a g e a b le p ro po rtio ns, yet w i t h ­ o u t “arb itra rily s e t ti n g b o u n d s to it.” ( H e a v e r 1987, 61) C l e a r l y , w h e n H e a v e r re f e rs t o t h e l i m i t i n g d a n g e r s o f a " m o d e l o f g e o m ­ e t r y ” h e is t h i n k i n g o f E u c l i d e a n s t r u c t u r e s ; i t - i s t h e f r a c t a l geo m et-r.y-of-the a r a b e s q u e w h i c h c o n v e y s t h e h o p e f u l m e s s a g e o f S h a m m a s . I n c h a p t e r 2 we e x a m i n e d t h e a r a b e s q u e b r a n c h e s o f s t r e e ts t h a t a p p e a r in a m a p o f C a i r o , E gypt. I n a n o t h e r s e c t i o n o f t h i s m a p (fig. 1 2 . 2 ) , a w i d e d i v e r s i t y o f r e l i g i o u s i n s t i ­ tu tio n s flo w er at th e en d s o f th e se b ra n c h e s, a tte s tin g to th e p o sitiv e p o t e n ­ t i a l o f f r a c t a l s in c u l t u r a l p o l i t i c s . CHAPTER J3 — Fractals -------------------------------------in the----------------------------------------------—European---------------------------------- ---------—history of--------------------------------------------mathematics------------------------------------------ A n t h r o p o l o g i s t s h a v e r e c e n t ly t a k e n a n in c r e a s i n g i n t e r e s t in t h e c u l t u r a l a n a l y ­ sis o f E u r o - A m e r i c a n s o c i e ti e s . I n p a r t t h i s is a r e a c t i o n to t h e m a n y d e c a d e s o f focus o n i n d i g e n o u s so cieties, as if t h e i r b e h a v i o r re q u ire d e x p l a n a t i o n w h ile t h a t o f E u r o p e a n s w as s e l f - e v i d e n t . A t first t h i s “ re fle x iv e e t h n o g r a p h y ” s o u n d e d like a n in g e n i o u s way t o t u r n t a b l e s o n s o m e v e ry t r o u b l i n g a s p e c t s o f a n t h r o p o l o g ­ ical a u t h o r i t y , b u t it t o o h a s d r a w b a c k s . O c c a s i o n a l l y o n e s u s p e c ts a h i d d e n s ig h o f re l ie f fro m a n t h r o p o l o g i s t s w h o d e c i d e t h e y c a n p l a c e t h e m s e l v e s o n t h e c u t ­ tin g ed g e by “stu d y in g th e i r o w n t r ib e ” (ju st as c y b e r e t h n o g r a p h y s o m e tim e s seem s s u s p ic io u s ly c o n v e n i e n t ) . N e v e r t h e l e s s , t h e r e is a n i m p o r t a n t p l a c e f o r a n t h r o ­ po log ical stu dies o f E u ro - A m e r ic a n s . It w o u ld b e a n u n b a l a n c e d p o r t r a it if we w ere to see A f r i c a n fra c ta ls in n e e d o f c u l t u r a l an a ly s is, a n d W e s t e r n fractals as m e re ly self-evident m ath em atics. A c u l t u r a l h i s t o r y o f E u r o p e a n fr a c ta ls A n c i e n t G r e e k p h i l o s o p h y is o f t e n r e m e m b e r e d fo r P l a t o ’s r a t i o n a l r e a l m o f u n c h a n g i n g , s t a t i c fo rm s. B u t in t h e h i s t o r y o f m a t h e m a t i c s , it is i m p o r t a n t to c o n s i d e r o t h e r i n t e l l e c t u a l c u r r e n t s in t h a t so cie ty, in p a r t i c u l a r t h e p a r a d o x e s o f t h e p h i l o s o p h e r Z e n o o f E le a a n d d i e d is c o v e r y o f i r r a t i o n a l n u m b e r s by t h e P y th a g o r e a n s . 203 Implications 204 A c c o rd in g to a n c ie n t h isto rian s, P y thagoras o f S a m o s g a th e re d k n o w led g e in E g y p t a n d B a b y l o n in t h e s i x t h c e n t u r y b .c .e . a n d e sta b lish e d a s e c r e t s o c i­ e t y in M a g n a G r a e c i a { w h a t is n o w s o u t h e a s t e r n I t a l y ) . H i s d is c ip les,...in c!u d in g o n e o f t h e first r e c o r d e d w o m e n m a t h e m a t i c i a n s , T h e a n a ? s w o r e a n o a t h t o m a i n t a i n s t r i c t d i e t a r y r e g u l a t i o n s , se c r e c y , a n d a re l ig i o u s f a i t h i n n u m b e r s . T h e P y th a g o re a n co sm o lo g y w as a h a r m o n io u s u n ity based o n w h o le n u m b e r s ( i , 2, 3 . . .) a n d t h e i r r a t i o s ( f r a c t i o n s s u c h as 2 /3 , 5 /2 , e t c . ) . F r o m t h e m o t i o n o f h e a v e n l y b o d ie s to t h e laws o f m u s ic , t h e y f o u n d in c r e a s i n g e v i d e n c e fo r t h e i r a r i t h m e t i c r e l i g i o n . B u t a t s o m e p o i n t — a n d m u c h i n k h a s b e e n s p i l l e d in t h e d a t e d e b a t e — c a m e t h e d i s c o v e r y o f w h a t t h e y t e r m e d alogos, t h e “ i r r a t i o n a l ” n u m b e r s ( a n a m e t h a t w e h a v e k e p t t o t h i s d a y ) . U n l i k e w h o l e n u m b e r r a t io s , w h i c h e i t h e r t e r m i n a t e ( 5 / 2 = 2 . 5 0 0 0 0 . . .) o r r e p e a t ( 1 3 / 1 1 = 1 . 1 8 1 8 1 8 ... .), i r r a t i o n a l n u m b e r s , s u c h a s t h e s q u a r e r o o t o f t w o ( 1 . 4 1 4 2 1 3 5 6 . . •), c o n t i n u e to c h a n g e fo rev er. T h e y c a n n o t b e e x p r e s s e d as t h e r a t i o o f t w o fi n it e i n t e g e r s ; as g e o m e t r i c m a g n i t u d e s t h e y a r e “ i n c o m m e n s u r a b l e li n e s .” T h e m o s t p la u s ib l e o r i g i n for t h e P y t h a g o r e a n k n o w l e d g e o f i r r a t i o n a l s is i n a n a t t e m p t t o d e t e r ­ m i n e t h e d i a g o n a l o f a p e n t a g o n . If y o u w is h to d e t e r m i n e t h e r a t i o o f d i a g o ­ n a l t o sides fo r a r e g u l a r h e x a g o n , it is q u i t e easy, b e c a u s e all d i a g o n a l s i n t e r s e c t in t h e c e n te r . B u t t h e d i a g o n a l s o f a p e n t a g o n ju s t form a s m a ll e r p e n t a g o n . S i n c e t h e s a m e o p e r a t i o n c a n b e r e p e a t e d a g a i n a n d a g a i n , a n i r r a t i o n a l n u m b e r is e x p o s e d . 1 T h i s “ ir ra tio n a lity " in th e h e a r t o f t h e i r s p iritu al p r a c t ic e w as t o o m u c h , a n d m em b ers o f t h e g ro u p ag re e d n o t to rev eal th is s e c re t o n p a in o f d e a th . Z e n o o f E le a (fl. c a . 4 5 0 b .c.e.), a d iscip le o f P a rm e n id e s , p ro v id e d a •series o f p a r a d o x e s t h a t a ls o c o n f l i c t e d w i t h t h e n u m e r i c a l f a i t h o f t h e day. H is m o s t f a m o u s e x a m p l e is a r a c e b e t w e e n A c h ilE e s,_ th e f l e e t e s t o f r u n n e r s , a n d a t o r t o i s e . A l l o w i n g t h e t o r t o i s e a s p o r t i n g c h a n c e , A c h i l l e s g iv e s it a c o n s i d e r ­ a b l e le a d ( l e t ’s sa y jo o f e e t ) . B u t by t h e t i m e h e c a u g h t u p t o t h e p l a c e w h e r e t h e t o r t o i s e b e g a n , it h a d a l r e a d y a d v a n c e d 10 fe e t. By t h e t i m e h e g a i n e d t h a t d i s t a n c e , t h e t o r t o i s e h a s c r e p t f o r w a r d o n e fo o t. Z e n o c o n c l u d e d t h a t a l t h o u g h e x p e r i e n c e p r o v e s o t h e r w i s e , lo g ic a lly t h e t o r t o i s e s h o u l d w i n t h e r a c e . B a c k in 450 b.c.e., t h e s e p a r a d o x e s o f i n f i n i t y ( a n d i n f i n i t y ’s flip s id e , t h e i n f i n i t e s i m a l ) w e r e u n n e r v i n g , e v e n s h o c k i n g to p h i l o s o p h e r s w h o d e p e n d e d o n r a t i o n a l i t y as t h e g a t e w a y t o re l ig i o u s p e r f e c t i o n . I n P la t o 's p h i l o s o p h i c c o sm o lo g y , s p iritu a l p e r f e c t i o n w as s e e n as t h e h i g h e r le v e l o f t r a n s c e n d e n t stasis, a n d il lu s io n a n d i g n o r a n c e w e r e t h e r e s u l t o f life in o u r l o w e r r e a l m of. c h a n g i n g d y n a m i c s ( “f l u x , ” w h i c h in a n c i e n t G r e e k also m e a n s “d i a r r h e a ” ). S e v e r a l o f P l a t o ’s s t u d e n t s a t t e m p t e d t o i m p r o v e t h e m a t c h b e t w e e n t h e c h a r a c t e r i s t i c s o f m a t h e m a t i c s a n d t h e r e q u i r e m e n t s o f che s t a t i c realm . E u d o x u s p ro p o s e d to e lim in a te irra tio n a ls hy r e d e f in in g “ r a tio ,” an d Fractals in the European history o f mathematics 205 X en o c ra c e s in t r o d u c e d a d o c t r i n e o f in d iv is ib le s t o o p p o s e Z e n o ’s p ara d o x es . A r i s ­ t o t l e , n o t i n g t h a t in f in ity + in f in ity = in fin ity , s u g g e s te d t h a t t h i s “s e l f - a n n i h i ­ l a t i n g ” c h a r a c t e r i s t i c c o u l d b e e l i m i n a t e d b y r e s t r i c t i n g r e f e r e n c e t o in f in ity as a l i m i t to b e a p p r o a c h e d , r a t h e r t h a n as a t h i n g itself, a p r o p e r o b j e c t o f m a t h e ­ m a t i c a l 'inquiry.2 ' T h e P l a t o n i c re f o r m w as q u i t e s u c c e s sfu l, a n d as a r e s u l t m a t h e m a t i c i a n s in t h e f o l l o w i n g c e n t u r i e s p a id li t t i e a t t e n t i o n t o t h e k i n d s o f r e c u r s i o n t h a t led to Z e n o ’s t r o u b l i n g i n f in i te regress. O n e e a r l y e x c e p t i o n w as t h a t o f L e o n a r d o F i b o n a c c i in t h e t w e l f t h c e n tu r y . H e i n t r o d u c e d t h e first re c u r s iv e series s h o w n to b e o f use in m o d e l in g th e n a t u r a l w orld. I n c h a p t e r 7 w e saw t h a t th e F ib o n a c c i series a p p e a r s t o h a v e b e e n u tiliz e d in t h e t e m p l e a r c h i t e c t u r e a n d w e i g h t b a l­ a n c e s o f a n c i e n t E g y p t. T h e r e m a y a c t u a l l y b e a c o n n e c t i o n b e t w e e n t h e tw o. W h i l e li t t l e b i o g r a p h i c a l m a t e r i a l is a v a i l a b l e , G i e s a n d G i e s ( 1 9 6 9 ) a n d o t h e r s o u r c e s h a v e p u t t o g e t h e r a g o o d a c c o u n t o f w h a t life w as p r o b a b l y lik e for y o u n g L e o n a r d o o f P isa . F o ll o w i n g s c h o o l i n g in P isa , in w h i c h a r i t h m e t i c was larg ely b a s e d o n t h e L a t i n w r i t i n g s o f B o e t h i u s ( c i r c a 5 0 0 c .e .), L e o n a r d o ’s fath er s e n t for h im from th e N o r t h A fric a n city o f Bugia (B ou gie). T h e r e h e learned t h e I n d i a n p l a c e - v a l u e n o t a t i o n ( p r o b a b l y t h r o u g h A r a b i c so u r c e s ). H e was i n s p ir e d by t h i s i n n o v a t i o n a n d t r a v e l e d a l o n g t h e M e d i t e r r a n e a n t o C o n s t a n ­ t i n o p l e , E g y p t, S y ria, S icily, a n d P r o v e n c e , c o l l e c t i n g m a t h e m a t i c a l k n o w l e d g e fr o m b o t h s c h o l a r s a n d o r d i n a r y m e r c h a n t s . T h e r e s u l t i n g t e x t , Liber A b a c i ( B o o k o f t h e A b a c u s ) , h a s a s t r o n g Is lam ic in f lu e n c e . L e v e y ( 1 9 6 6 ) , for e x a m p l e , s h o w s t h a t m a n y o f a b u K a m i l ’s s i x t y - n i n e p r o b l e m s c a n b e f o u n d in L e o n a r d o ’s t e x t . B u t t h e F i b o n a c c i series, i n t r o d u c e d u n o b t r u s i v e l y as t h e s o l u t i o n to a p r o b l e m in r a b b i t p o p u l a t i o n g ro w th , d o e s n o t h a v e a k n o w n Is la m i c c o u n t e r p a r t . P e r h a p s it is s i m p l y a n i n d e p e n d e n t i n v e n ­ t i o n , b u t if t h e w e i g h t b a la n c e 's y s te m " w a s in use a t ' t h a t t i m e , L e o n a r d o c o u ld h a v e easily p i c k e d it u p fr o m a m e r c h a n t d u r i n g h is t r a v e ls in E gypt. A n d it is p o ss ib le t h a t t h r o u g h its re lig io u s use in a n c i e n t E g y p t t h e se rie s h a d r e t a i n e d so m e sig n ific a n c e as a n it e m o f s a c r e d o r m y s ti c a l k n o w l e d g e a n d was t h u s t r a n s ­ m i t t e d t h r o u g h s c h o l a r ly c o n t a c t . G ie s a n d G i e s (7 9 6 9 , 6 1 ) n o t e t h a t L e o n a r d o ’s p r a c t i c e o f r e d u c i n g all fra c­ tio n s to ] in che n u m e r a t o r “w e n t b a c k t o a n c i e n t E g y p t, a n d p e r h a p s d e r i v e d from t h e fa c t t h a t fr a c tio n s w e re r e g a r d e d less as n u m b e r s in t h e i r o w n r i g h t t h a n as sig ns o f d i v i s i o n .” B o y er (1 9 6 8 , 2 8 1 ) suggests t h a t t h e Liber A b a c i p r o b l e m w ith re c u rsiv e n e s t i n g o f s e v e n s ( “ S e v e n o ld w o m e n w e n t t o R o m e , e a c h w o m a n h a d se v e n m u le s; e a c h m u l e c a r r ie d s e v e n s a c k s . . . ’’) o r i g i n a t e d in its a n c i e n t E g y p t­ ia n c o u n t e r p a r t ( R h i n d M a t h e m a t i c a l P a p y ru s p r o b l e m # 7 9 ) . A n d F ib o n a c c i d oes provide a n a rra tiv e s ta te m e n t of th e recu rsiv e c o n s tru c tio n ,^ h ig h lig h tin g th e Implications 206 s a m e s e l f - g e n e r a t i n g a s p e c t o f t h e series t h a t w o u ld b e e m p h a s iz e d b y t h e a n c i e n t E g y p t i a n b e l i e f s y s te m . I f tliis i n f l u e n c e , ( w h e t h e r m e r e l y c o n t e x t u a l o r d i r e c t ) d o e s in f a c t e x i s t , it s h o u l d n o t d e t r a c t f r o m t h e g e n i u s o f L e o n a r d o ’s w o rk . H i s ‘’g e n e r a l s o l u t i o n fo r f i n d i n g “c o n g r u e n t n u m b e r s ” f o r s q u a r e s h a s b e e n h a i l e d as “ t h e f i n e s t p i e c e o f r e a s o n i n g in n u m b e r t h e o r y o f w h i c h w e h a v e a n y r e c o r d b e f o r e t h e t i m e o f F e r m a t . ” B u t w h e n it c o m e s t o t h e u s e o f t h e F i b o n a c c i s e r ie s in t h e c o n t e m ­ p o r a r y h i s t o r y o f m a t h e m a t i c s (cf. B r o o k e 1 9 6 4 ) , t h e r e is a c t u a l l y n o e v i d e n c e o f a d i r e c t c o n t r i b u t i o n f r o m F i b o n a c c i h i m s e l f . By a ll a c c o u n t s , G e r m a n a s t r o n o m e r J o h a n n e s K e p le r r e d i s c o v e r e d t h e series i n d e p e n d e n t l y in 1611, a n d it w as o n l y in t h e m i d - i 8 o o s , w i t h t h e f o r m a l p u b l i c a t i o n o f L ib e r A b a c i , t h a t F r e n c h m a t h e m a t i c i a n E d o u a r d L u c a s f o u n d t h e P is a n h i s t o r i c a l p r e d e c e s s o r a n d n a m e d it ac c o rd in g ly . T h i s fa c t h a s r e c e i v e d litt le a t t e n t i o n , a n d m o s t t e x t s p r e s ­ e n t F i b o n a c c i ’s d i s c o v e r y as if i t w e r e i n a d i r e c t i n t e l l e c t u a l l i n e o f d e s c e n t ra th e r th a n an h o n o ra r y title g iv e n to a w ell-d e se rv in g b u t d is c o n n e c te d a n t e ­ c e d e n t. F ib o n a c c i h im self seem ed u n h e s ita n t a b o u t th e m u ltic u ltu ra l c o n tr i­ b u t i o n s t o h i s w o r k ; t h e f i r s t s e n t e n c e o f L iber A b a c i s t a t e s , “T h e n i n e I n d i a n fi g u r e s a r e . . . N o d o u b t h e w o u ld h a v e b e e n q u ite c o n t e n t a ttr ib u tin g th e s e r ie s t o o r i g i n a t o r s o f a n y h e r i t a g e . F i b o n a c c i ’s s e r ie s was s i m p l y u n b o u n d e d g r o w t h ; t h e r e w a s n o i n t r o d u c ­ t i o n o f t h e i n f i n i t e e x c e p t in w ay s t h a t A r i s t o t l e w o u ld h a v e a p p r o v e d . T h e s e v ­ e n t e e n t h c e n tu r y b r o u g h t a t t e n t i o n to th e c o n c e p t o f t h e " in f i n ite s im a l” { r e v i v e d f r o m its G r e e k b a n i s h m e n t in K e p l e r ’s S te r e o m e tr ia [ 1 6 1 5 D , a n d c o n ­ v e r g e n c e t o a l i m i t . a s i n f i n i t y is a p p r o a c h e d ( e .g ., t h e a l g o r i t h m s fo r g e n e r a t ­ in g p i ) ; b u t i n f i n i t y w o u l d s till e x i s t o n l y as a n e v e r - r e a c h e d o r i e n t a t i o n r a t h e r t h a n a l e g i t i m a t e o b j e c t o f study- T h e A r i s t o t e l i a n v o i c e c o u l d s till b e h e a r d in 1831, w h e n m a t h e m a t i c i a n C a r ! F ried ric h G au ss (1 7 7 7 - 1 8 5 5 ) c a u t i o n e d his friend S c h u m a c h e r a g a i n s t in f in i ty : " I m u s t p r o t e s t m o s t v e h e m e n t l y a g a i n s t y o u r use o f t h e in f in ite as s o m e t h i n g c o n s u m m a t e d , as th is is n e v e r p e r m i t t e d in m a t h e ­ m a t ic s . T h e i n f i n i t e is b u t a. f a ^ o n d e p a r l e r , m e a n i n g a l i m i t t o w h i c h c e r t a i n r a t i o s m a y a p p r o a c h as c lo s e ly as d e s i r e d w h e n o t h e r s a re p e r m i t t e d t o in c r e a s e i n d e f i n i t e l y . ” B u t G a u s s ’s d i s t i n c t i o n w a s s h o r t - l i v e d . A s w e s a w in c h a p t e r 1, t h e w o r k o f G e o r g C a n t o r , w h i c h h a d p r o d u c e d t h e first f r a c t a l , t h e C a n t o r set, e n d e d t h e A r i s t o t e l i a n v ie w o n i n f in i ty . L ik e F i b o n a c c i , C a n t o r t o o m a y h a v e h a d s o m e n o n - E u r o p e a n i n f l u e n c e i n h i s w o rk . T h e C a n t o r s e t (fig. i- 3 -ia ) w a s h is v i s u a l i z a t i o n o f t r a n s f i n i t e n u m b e r t h e o r y . I t s h o w s t h e i n t e r v a l o f ze ro t o o n e o n t h e re a l n u m b e r li n e , a n d i n d i ­ c a t e s t h a t t h e n u m b e r o f p o i n t s is n o t d e n u m e r a h l e — t h a t is, g r e a t e r t h a n i n f i n ­ ity. B u t a t t h e t i m e , p u r e m a t h e m a t i c s w a s o n ly o n e o f C a n t o r ’s c o n c e r n s . H is Fractals in the European Ktstor^y 0 / mathematics real fa s c in a t io n w as in t h e th e o l o g ic a l im p lic a tio n s ; t h e in c r e a s in g classes o f in f in ­ ity h e d is c o v e re d s e e m e d to p o i n t to w a rd a religio us t r a n s c e n d e n t a l . C a n t o r ’s b io g ­ r a p h e r s differ g reatly o n t h e c u l t u r a l s i g n if i c a n c e o f t h i s p o i n t . E. T. B ell felt t h a t C a n t o r ’s J e w is h e t h n i c o r i g i n r u l e d h is life, a n d h e m a d e s e v e r a l r e m a r k s a b o u t t h e i n h e r i t a n c e o f p e r s o n a l i t y t r a i t s — p a r t i c u l a r l y d i s t u r b i n g in l i g h t o f h is II II II II II II 1111 1111 FIGURE T h e II II 1111 1111 I3 .I C a n to r set (a) T h e first f r a c t a l , c r e a t e d b y G e o r g C a n t o r i n 1 8 7 7 . ( b ) T h i s d e s i g n is f o u n d o n t h e t o p o f co lu m n s in t h e t e m p l e s o f a n c i e n t E g y p t . G e o r g C a n t o r ’s f t o s i c r u c i a n b e l i e f s a n d h i s c o u s i n M o r t i z C a n t o r, a n e x p e r t o n che g e o m e t r y o f E g y p t i a n a rt , m a y h a v e p u t h i m in c o n t a c t w i t h t h i s E g y p t i a n design. ( b, f r o m F u n r ie r 1 8 2 1 .) I m p li c a t io n s r e m a r k s o n C a n t o r ’s a r c h r i va l , t h e J e w i s h m a t h e m a t i c i a n L e o p o l d K r o n e c k e r : “T h e r e is n o mo r e v i c i o u s a c a d e m i c h a t r e d t h a n t h a t o f o n e J e w for a n o t h e r w h e n t he y disagree o n purel y sc i enti fic ma t t e r s . W h e n t w o in t e l l e c t u a l J ews fall o u t t he y di s a g re e al l ove r , t h r o w r e s e r v e t o t h e dogs, a n d d o e v e r y t h i n g in t h e i r p o w e r t o c u t o n e a n o t h e r ’s t h r o a t o r s t a b o n e a n o t h e r i n t h e b a c k ” ( Be l l 1 9 3 9 , 5 6 2 - 5 6 3 ) . A n o t h e r C a n t o r b i o g r a p h e r , J. W. D a u b e n , says t h a t s i n c e C a n t o r ’s m o t h e r was R o m a n - C a t h o l i c “s h e was by d e f i n i t i o n n o n -J e w i s h , t h u s it fol lows t h a t G e o r g C a n t o r was n o t Jewi s h, c o n t r a r y t o t h e v i e w w h i c h h a s p r e v a i l e d in p r i n t for m a n y y e a r s ” ( D a u b e n 1 9 7 9 ) . N a z i s c h o l a r s s o l v e d t h e i r w o r r i e s by s p r e a d i n g a s t o r y t h a t C a n t o r wa s f o u n d a b a n d o n e d o n a s h i p b o u n d for S t . P e t e r s b e r g ( G r a t t a n G u i n e s s 1971, 352). A c t u a l l y C a n t o r ’s J e w i s h i d e n t i t y wa s q u i t e c o m p l e x . H i s fami l y h a d i n d e e d c o n v e r t e d t o C h r i s t i a n i t y , b u t h e was we l l a w a r e o f h i s h e r i t a g e . H e r e f e r r e d t o hi s g r a n d m o t h e r as “ t h e I s r a e l i t e ” a n d w r o t e a re l i g i ou s t r a c t t h a t a t t e m p t e d t o s h o w t h a t t h e r e wa s n o v i r g i n b i r t h , a n d t h a t t h e r eal f a t h e r o f J e s us C h r i s t wa s Jo s e p h o f A r im a th e a . C a n t o r e v e ntua lly jo ined the Rosicrucians, w h o s e mysti­ c a l/ s c i en t i f i c a p p r o a c h t o a s u p p o s e d E g y p t i a n o r i g i n f o r all r e l i g i o n s p r o b a b l y a p p e a l e d n o t o n l y t o his i n t e l l e c t u a l i n t e r e s t s , b u t al s o t o h i s s y n c r e t i c e t h n i c ­ ity. C a n t o r c h o s e a H e b r e w l e t t e r as h i s n e w s y mb o l : t h e a l e p h , b e g i n n i n g o f t h e a l p h a b e t , was u s e d t o r e p r e s e n t t h e b e g i n n i n g o f t h e n o n d e n u m e r n b l e sets. W h i l e His b i o g r a p h e r s a r g u e d J e w o r n o t - J e w, off o r o n , z e r o o r o n e , C a n t o r h i m ­ self p r o v e d t h a t t h e c o n t i n u u m f r o m zero t o o n e c a n n o t b e d e l i m i t e d b y a n y s u b ­ d i v i s i o n process, n o m a t t e r h o w l o n g its a r g u m e n t s . G i v e n C a n t o r ’s R o s i c t u c i a n t h e o l o g y a n d t h e p r o x i m i t y o f h i s c o u s i n M o r i t z C a n t o r — a t t h a t t i m e a l e a d i n g e x p e r t in t h e g e o m e t r y o f E g y p t i a n a r t ' ( C a n t o r 1 8 8 0 ) — it m a y b e t h a t G e o r g C a n t o r s a w t h e a n c i e n t E g y p t i a n r e p r e ­ s e n t a t i o n o f t h e l o t u s c r e a t i o n m y t h (fig. 1 3 . 1 b ) , a n d d e r i v e d i n s p i r a t i o n f r o m t hi s A f r i c a n f r act al for t h e C a n t o r set. W e m a y n e v e r k n o w for c e r t a i n , b u t t h e g e o m e t r i c r e s e m b l a n c e is q u i t e s t r o n g . A s n o t e d in c h a p t e r 1, C a n t o r ’s m a t h e m a t i c s was c o n s i d e r e d u t t e r l y i m p r a c ­ ti cal; it w a s n o t u n t i l B e n o i t M a n d e l b r o t t h a t f r a c t a l g e o m e t r y b e c a m e usef ul t o science. M a n d e l b r o t reports t h a t his in s p ira tio n c a m e from a study o f l o n g - te rm r i ve r f l u c t u a t i o n s by B r i t i s h c i v i l s e r v a n t H . E. H u r s t . H u r s t e x a m i n e d t h e fl ood v a r i a t i o n s o v e r s ev e r al c e n t u r i e s a n d c o n c l u d e d t h a t it c o u l d b e c h a r a c t e r i z e d in t e r m s o f a s c a l i n g e x p o n e n t . La t e r , M a n d e l b r o t real ized t h a t t h i s w a s t h e s a m e s cal i ng m a t h e m a t i c s t h a t t h e “r e m a r k a b l e c u r v e s " o f C a n t o r a n d o t h e r s d esc r i be d, But w h e r e d i d H u r s t find a rel iabl e s o u r c e for sever al c e n t u r i e s o f flood d a t a ? H u r s t l i ved in Eg y p t for 6 2 years a n d wa s a b l e t o d e m o n s t r a t e l o n g - t e r m s c a l i n g in N i l e flood r ecor ds b ec a u s e o f t h e a c c u r a t e “ n i l o m e t e r ” r e a d i ng s g o i n g b a c k fifteen c e n - F r a c t a ls in t h e E u r o p e a n h is t o r y o f m a t h e m a t ic s curies. A t t e m p t s t o find p a t t e r n s in t h e flo od s a re q u i t e a n c i e n t in t h e N i l e v a l ­ ley; in s o m e w ays, H u r s t a n d M a n d e l b r o t w e r e s i m p l y t h e l a t e s t — a n d m o s t s u c ­ cessfu l— p a r t i c i p a n t s in c h a t s e a r c h . R e c u rs io n a n d sex: a c r o s s - c u ltu r a l c o m p a riso n T h r o u g h o u t t h e e x p l o r a t i o n o f A f r i c a n fra ctals, we failed to find a n y o n e c u l t u r a l f e a t u r e t h a t w a s p e r s i s t e n t l y a s s o c i a t e d w i t h t h e s e fo r m s . T h e y r a n g e d from p r a c t i c a l c o n s t r u c t i o n t e c h n i q u e s t o a b s t r a c t t h e o l o g i c a l ic o n s , f r o m w i n d ­ screen s to k in s h ip structures, f ro m e s t h e ti c p a tt e r n s to d i v i n a t i o n te c h n iq u e s. T h e r e is n o s i n g u la r “r e a s o n ” w h y A f r i c a n s use f r a c ta l s , a n y m o r e t h a n a s i n g u l a r r e a ­ s o n w h y A m e r i c a n s like r o c k m u s ic . S u c h e n o r m o u s c u l t u r a l p r a c t i c e s j u s t c o v e r t o o m u c h s o c ia l t e r r a i n . A t b e s t w e c a n m a k e a l o w e r - d i m e n s i o n a l p r o j e c t i o n o f such h ig h - d im e n s io n a l d y n am ics, th e silh o u ecte th a t ap p ears g iv en o n e p a r tic ­ u la r a x is o f i l l u m i n a t i o n . T h i s s e c t i o n w ill fo c u s o n t h e r e l a t i o n b e t w e e n r e c u r ­ s i o n in m a t h e m a t i c s a n d s e x u a l i t y i n c u l t u r e . S e x is c o n v e n i e n t in t h a t o t h e r research ers h a v e d e v e lo p e d A fric a n -E u ro p e a n co m p a riso n s , a n d th a t sexual r e p r o d u c t i o n is o f t e n c o n n e c t e d t o r e c u r s i v e c o n c e p t s . T a y lo r ( 1 9 9 0 ) d e s c r ib e s s e x u a l i t y in R w a n d a as b a s e d o n t h e c o n c e p t o f a “f r a c ta l p e r s o n ” in w h i c h s o c i e t y is p e r c e i v e d “n o t in t e r m s o f m o n a d i c i n d i v i d ­ u a ls b u t i n t e r m s o f . . . s t r u c t u r e s o f m e a n i n g w h o s e p a t t e r n s r e p e a t t h e m s e l v e s in s lig h tly v a r y i n g fo rm s like t h e c o n t o u r s o f a fra c ta l t o p o g r a p h y ” (p . 1 0 2 5 ) . H is analysis o n e x p re s s io n s o f th is s o c i a li ty in te r m s o f t h e c i r c u l a t i o n o f fluids is used to e x a m i n e th e failure o f p ro g ram s to e n c o u ra g e c o n d o m use. C a r o l y n M a r t i n S h a w ( 1 9 8 9 , 1 9 9 5 ) a n a ly z e s K ik u y u s e x u a l i t y in r e l a t e d w ays a n d p ro v i d e s a n i l l u m i ­ n a ti n g c o n tr a s t to E u ro p e a n sexuality. U s in g F o u c a u lt’s c ri ti q u e o f h u m a n i s m , S haw , c h a l l e n g e s t h e u su al p o r t r a i t o f E u r o p e a n s e x u a l r e p r e s s i o n a n d A f r i c a n s e x u a l lic e n se . S h e d e m o n s t r a t e s t h a t in b o t h cases, t h e so c ia l s y s te m c o n t r o l s s e x u a l b e h a v io r , b u t w h ile t h e E u r o p e a n lo cu s o f c o n t r o l is in t h e p r i v a t i z a t i o n o f p l e a ­ s u r e , t h e K i k u y u s ’s s e x u a l r e g u l a t i o n o c c u r s t h r o u g h p u b l i c e x p r e s s io n s o f p l e a ­ su re a n d “s o c i a li ty o f i n d i v i d u a l c o n s c i e n c e . ” F o r e x a m p l e , s h e h i g h l i g h t s t h e p r a c t ic e o f ngwe/co, in w h i c h te e n a g e r s w r a p t h e m s e l v e s w i t h a few l e a t h e r strips, oil t h e i r b o d ie s , a n d e n g a g e in a p u b l i c d is p la y o f s e x u a l b e h a v i o r . F r o m a E u r o ­ p e a n p o i n t o f v ie w t h i s s o u n d s lik e a n u n r e g u l a t e d orgy, b u t S h a w fo u n d t h a t t h e practic e was a m e t h o d o f p r e v e n t in g te e n a g e p re g n a n c ie s a n d c h a n n e l i n g t h e t e e n s ’ s e x u a l d e s i r e i n t o so c ia lly a p p r o v e d f o r m s .4 W h e n w e l o o k a t m a n y A f r i c a n f r a c t a l s w e c a n see a n e m p h a s i s o n s e x u ­ a lity in t e r m s o f r e p r o d u c t i o n . T h e s e lf- s im i la r it y o f t h e B a m a n a c h i w a r a a n t e ­ lo p e h e a d d r e s s a n d m e r u n k u n f e r t i l i t y p u p p e t , t h e s e l f - g e n e r a t i n g D o g o n I 210 I m p lic a tio n s c o s m o lo g y , t h e c y c l i c k i n s h i p i c o n o g r a p h y o f t h e M i t s o g h o , F a n g , a n d B a lu b a , t h e i t e r a t i o n s o f b i r t h i n g in t h e N a n k a n i a r c h i t e c t u r e , a n d m a n y o t h e r c ases o f r e c u r s i o n a r e c lo s e ly t i e d t o s e x u a l r e p r o d u c t i o n . T h u s o n e c o n t r i b u t i n g fa c to r t o t h e A f r i c a n m a t h e m a t i c a l e m p h a s i s o n r e c u r s i o n c o u l d b e tills A f r i c a n c o n str u c tio n o f sexuality th r o u g h p o sitiv e pu b lic d o m a in ex p ressio n s. T h e E u r o p e a n c o u n t e r p a r t o f S h a w ’s th e o r y w o u ld p r e d i c t t h e o p p o s i te , a n d i n d e e d w e find t h a t t h e b a n i s h i n g o f i n f in i te reg ress in t h e P l a t o n i c r e f o r m was c lo s e ly t i e d t o a k i n d o f s e x u a l p r o h i b i t i o n . In P l a t o ’s S y m p o s iu m , S o c r a t e s e x p l a i n s t h a t t h e r e is a h i e r a r c h y o f r e p r o d u c t i o n . L o v e b e t w e e n a m a n a n d a w o m a n w ill o n l y re su lt in a flesh c h i l d , a c r e a t u r e o f flux w h o w ill e v e n t u a l l y die, a t b e s t p r o d u c i n g m o r e flux. L o v e b e t w e e n a m a n a n d a bo y ( l o v e r a n d b e l o v e d ) is h i g h e r , b e c a u s e it c a n re s u lt in r a i s i n g t h e b o y t o a h i g h e r p l a n e — t h a t o f a p h il o s o p h e r. A n d a p h i l o s o p h e r c a n h a v e a “b r a i n c h il d ,” a p e r f e c t id e a t h a t n e v e r c h a n g e s o r dies. T h e P l a t o n i c ideal o f s t a t i c , e t e r n a l p e r f e c t i o n c o n f l ic t s w i t h t h e e v e r - c h a n g i n g d y n a m i c o f s e x u a l r e p r o d u c t i o n . T h e G r e e k p r e f e r e n c e for t h e s t a ­ t i c s h a p e o f t h e A r c h i m e d e a n s p ira l s u g g e s ts t h i s P l a t o n i c i d e a l , ju s t as t h e g r o w i n g s h a p e o f t h e l o g a r i t h m i c s p ira l su g g es ts t h e A f r i c a n e m p h a s i s o n f e r t i l ­ ity a n d r e p r o d u c t i o n . O f c o u r s e , t h i s is a g ro ss g e n e r a l i z a t i o n ; t h e r e a re , for e x a m p l e , p l e n t y ' o f A r c h i m e d e a n spirals in A f r i c a n design s. C o n v e r s e l y , E u r o p e a n m a t h e m a t i c i a n J a c o b o B e r n o u l l i w a s u t t e r l y d e d i c a t e d t o t h e l o g a r i t h m i c s p ira l a n d sp e c ifie d t h a t o n e w o u l d b e e n g r a v e d o n h i s t o m b s t o n e . B u t t h e s t o n e c u t ­ te r d id n o t go a g a in s t t h e g ra i n o f h is c u lt u r e ; B e r n o u ll i’s g ra v e is still m a r k e d w ith a n A r c h m e d e a n s p i r a l (fig. 13-2). I t w o u ld be d a n g e r o u s to su g g e s t t h a t t h e r e is a n e t h i c a l d i f f e r e n c e a t s t a k e h e r e , as s o m a n y o r g a n i c r o m a n t i c i s t s h a v e m a i n t a i n e d . A g a i n , t h e r e is n o h i s t o r i c a l e v i d e n c e for a c o n s i s t e n t r e l a t i o n s h i p b e t w e e n m a t h e m a t i c a l d i s t i n c - FIGURE 13 .2 B e r n o u lli’s to m b s to n e Although Bernoulli asked f o r a logarithmic spiral to be inscribed on his tombstone, the engraver was apparently only familiar with che linear spiral. (From Mnor 1987, courtesy Birk/iriuser Verfag AG, Basel, Sifiiterland.) F r a c t a l s in th e E u r o p e a n h is t o r y o f m a t h e m a t ic s d o n s a n d th e e th ic s o f th e i r users. S o m e strictly linear, logical th in k e rs like B e rtran d R u s s e l l a n d N o a m C h o m s k y h a v e b e e n f a m o u s for t h e i r p r o g r e s s i v e e t h i c a l s t a n d p o i n t s , j u s t as s o m e h o l i s t i c o r g a n i c i s t s h a v e b e e n p r o n e t o fascism . A n d o f course vice versa. W h a t d o e s c o u n t for e th ic s is h o w peo ple a re able to use m a t h e ­ m atics' in t h e p a r t i c u l a r e v e n t s a n d ideaS t h a t s u r r o u n d e d t h e i r life. W i t h t h a t in m i n d , l e t ’s lo o k a t t h r e e o f t h e i n n o v a t o r s w h o b r o u g h t r e c u r s i o n i n t o E u r o ­ p ean co m p u tatio n al m athem atics. T h e story o f A d a L o v e la c e is w e ll k n o w n in c o m p u t i n g s c i e n c e history. H e r f a m e s t e m s fr o m h e r w r i t i n g s in 1 8 4 3 o n fh e m a t h e m a t i c a l p o s s i b i l i t i e s o f C h a r l e s B a b b a g e ’s p r o p o s e d “a n a l y t i c a l e n g i n e ”— a p l a n fo r a m e c h a n i c a l d i g i ­ tal c o m p u t e r . L o v e la c e is o f t e n p r o m o t e d as a r e c o v e r e d fe m in i s t a n c e s t o r , a p o s i­ ti o n t h a t t e n d s to o v e r e s t i m a t e h e r a c h i e v e m e n t s a n d o b sc u r e h e r o w n t h i n k i n g . A g a in s t th e s e red u ctiv e p o rtra its, S t e i n ( 1 9 8 5 ) h as w rit te n a d etailed, critical e x a m ­ i n a t i o n o f L o v e la c e t h a t re v e a ls a m u c h m o r e i n t e r e s t i n g a n d c o m p l e x sto ry t h a n th e p o p u larizations h a v e allo w ed . L o v e l a c e ’s m o t h e r w as a lw a y s w o r r i e d c h a t s h e m i g h t h a v e i n h e r i t e d t h e n o t o r i o u s s e x u a l p r o c l i v i t i e s o f h e r f a t h e r , L o r d B y ro n . H e r c h i l d h o o d r e v o l v e d a r o u n d s t r i c t l y p r e s c r i b e d e d u c a t i o n a l a c t i v i t i e s , a n d a t ti m e s s h e w as f o r c e d to lie perfectly still, w ith bags o v e r h e r h a n d s to w ard o ff any “w ildness.’’.T his repressed u p b rin g in g e v e n tu a lly in sp ired re b e llio n in th e fo rm of a n a tte m p te d e lo p e ­ m e n t , b u t t h e fa iled affair left h e r h u m i l i a t e d a n d r e p e n t a n t . S h e w r o te to a f a m ­ ily fr ie n d , W i l l i a m K in g , r e q u e s t i n g m a t h e m a t i c a l i n s t r u c t i o n as a c u r e for h e r sin fu l im p u lse s. K in g a g re e d , s e n d i n g h e r b o t h m a t h e m a t i c a l a n d relig io u s tex ts. B u t d e s p i t e h e r d e c l a r a t i o n s to a p p l y h e r m a t h e m a t i c a l i m a g i n a t i o n “t o t h e g r e a t e r glory o f G o d , ” s h e t u r n e d a w a y f r o m t h e m o r a l i z i n g o f K in g t o t h e m o r e g la m o r o u s s o c ia l c o m p a n y o f B a b b a g e a n d h i s f a m o u s “ t h i n k i n g m a c h i n e s . ” B a b b a g e ’s m o t i v a t i o n s w e r e far r e m o v e d fro m K i n g ’s relig io u s i n t e l l e c t u a ltsm . H e w as p r i m a r i ly c o n c e r n e d w i t h e c o n o m i c a n d s c i e n ti fic p ro gress. T h i s s w i t c h fro m K in g to B a b b a g e w as a n a c t o f i n d e p e n d e n c e , a n d L o v e l a c e b e g a n t o tu r n h e r i m a g in a t io n lo ose. W h i l e p u rs u in g a m u c h m o r e i n t e n s e area o f m a t h e ­ m a t i c a l stud y, h e r re lig io u s t h i n k i n g a ls o t o o k a n e x p a n d e d tu r n . S h e b e g a n to de sc rib e h e rs e lf a n d h e r w o rk in te r m s o f m a g ic a l im agery: t h e m e c h a n is m s o f s y m ­ bol m a n i p u l a t i o n w e re “m a t h e m a t i c a l s p r i t e s , ” a n d s h e a d v is e d B a b b a g e to a llo w h i m s e l f t o be “u n r e s i s t i n g l y b e w i t c h e d ” by “t h e H i g h P rie s te s s o f B a g g a g e ’s E n g in e.” S t e i n a ls o n o t e s t h a t it was a c t u a l l y B a b b a g e w h o first d r e w u p t h e " t a b l e o f s t e p s ” c o n s t i t u t i n g t h e first c o m p u t e r p r o g r a m s . B a b b a g e w as h a v i n g d i f f i ­ c u lty o b t a i n i n g f u n d i n g for his w o r k , h o w e v e r , a n d re aliz ed t h a t L o v e l a c e ’s so cial p o s i t i o n a n d n o t o r i e t y — b o t h as t h e d a u g h t e r o f B y r o n as w ell as a “L a d y o f 212 Im p l ic a tio n s M a t h e m a t i c s ”— c o u l d b e p u t to h i s a d v a n t a g e . T h e r e p u t a t i o n o f L o v e l a c e as t h e o r i g i n a t o r o f p r o g r a m m i n g s t e m s f r o m t h i s p u b l i c r e l a t i o n s p lo y o f B a b b a g e . T h e r e w as, h o w e v e r , o n e t a b l e f o r w h i c h A d a w a s w h o l l y r e s p o n s i b l e : t h e r e c u r s i v e g e n e r a t i o n o f a s e q u e n c e k n o w n as t h e B e r n o u l l b n u m b e r s . M o o r e ( L977) sta te s t h a t th is ta b le used recu rsiv e p ro g ra m m in g . H u sk e y a n d H u sk ey ( 1 9 8 4 ) , a p p a r e n t l y r e f e r r i n g t o t h i s c l a i m , s u g g e s t t h a t t h i s is a c o n f u s i o n w i t h L o v e l a c e ’s d e s c r i p t i o n o f m a t h e m a t i c a l “ r e c u r r e n c e g r o u p s ” a n d n o t e t h a t t h e t e r m “ r e c u r s i v e p r o g r a m m i n g " g e n e r a l l y r e f e r s t o a p r o c e d u r e t h a t c a ll s i t s e l f ' (i.e ., s e l f - r e f e r e n c e ) — im p o s s ib le fo r L o v e l a c e s i n c e h e r c o d e h a d n o p ro c e d u r e s . B u t t h e y a ls o n o t e t h a t L o v e l a c e i n t r o d u c e d a n e w c o d e n o t a t i o n t o d e s c r i b e w h a t s h e r e f e r r e d t o as “a c y c l e o f a c y c l e , ” w h i c h w o u l d b e e q u i v a l e n t t o t h e r e c u r s i v e s t r u c t u r e o f n e s t e d i t e r a t i o n i n use to d a y . S ig n i f i c a n tl y , t h i s i t e r a t i v e r e c u r s i o n w a s t h e o n e p r o g r a m fo r w h i c h B a b ­ b ag e c la i m e d c re d it: “W e d is cu ssed v ario u s illu s tr a t io n s t h a t m i g h t b e in t r o d u c e d : I s u g g e s te d s e v e r a l, b u t t h e s e l e c t i o n w a s e n t i r e l y h e r o w n . S o a ls o w as t h e a l g e ­ braic w o rk in g o u t o f th e d ifferen t pro b lem s, e x c e p t, in d e e d , t h a t re la tin g to th e n u m b e r s o f B e r n o u i l l i [sic], w h i c h I h a d o ff e re d to d o t o s a v e L a d y L o v e l a c e t h e t r o u b l e ” ( q u o t e d in S t e i n 1 9 8 5 , 8 9 ) . T h e a p p r o p r i a t i o n m ay h a v e b e e n a n t i c i p a t e d b y L o v e la c e : S t e i n n o t e s t h a t i n t h e l e t t e r s c o n c e r n i n g t h i s p r o g r a m , L o v e l a c e is a t y p i c a l l y v a g u e — s h e h a d a lw a y s b e e n o v e r d e p e n d e n t o n B a b b a g e fo r m a t h e m a t i c a l s p e c i f i c s — a n d s p e c ­ u l a t e s t h a t t h e v a g u e n e s s w as a d e l i b e r a t e m o v e t o p r o t e c t h e r i t e r a t i v e i n n o ­ v a t i o n . M a n y f e m i n i s t s h a v e w r i t t e n a b o u t m a l e e n v y o f w o m e n ’s r e p r o d u c t i v e c a p a c i t y , a n d t h e r e m i g h t w e ll b e a p a r a l l e l in B a b b a g e ’s a p p r o p r i a t i o n o f L o v e l a c e ’s r e c u r s i v e a c h i e v e m e n t . B u t t h e o r g a n i c i s t v e r s i o n s o f s u c h a n a ly s e s p o rtra y t h e c o n f l i c t in te r m s o f w o m e n b e in g m o r e n a t u r a l o r e m b o d i e d , a n d m e n b e i n g m o r e a r t i f i c i a l o r a b s t r a c t . In t h i s s t o r y o f m a l e w o m b e n v y a n d t h e p r o ­ t e c t i v e m o t h e r , it is t h e d i g i t a l a b s t r a c t i o n o f r e c u r s i o n , n o t c o n c r e t e e m b o d i ­ m e n t , o v e r w h i c h t h e s t r u g g le is fo u g h t . T h e b i r t h i n g m e t a p h o r w a s m e n t i o n e d by L o v e l a c e h e r s e lf ; t h e f i n i s h e d p r o g r a m m i n g s t u d y w a s “h e r first c h i l d . " C o n ­ tra ry t o P l a t o , s e x u a l r e p r o d u c t i o n is n o t in o p p o s i t i o n t o t h e a b s t r a c t r e a l m o f m a t h e m a t i c s ; L o v e l a c e u s e d h e r m a t h e m a t i c s t o r e b e l a g a i n s t a t t e m p t s to li m i t h e r to a repressive fe m in in ity a n d used th is a rtific ia l s e x u a lity — a b e w itc h in g h i g h priestess, je a l o u s l y g u a r d i n g h e r p r o g r a m m i n g p r o g e n y — t o d e v e l o p t h e first c o m p u ta tio n a l recu rsio n . I n t h e d i s c u s s i o n o f t h e m a t h e m a t i c a l t h e o r y o f c o m p u t a b i l i t y in c h a p ­ t e r 10 , we n o t e d t h a t t h e s e t o f “p r i m i t i v e r e c u r s i v e f u n c t i o n s , " d e v e l o p e d by R o zsa P e t e r , h a d t h e g r e a t e s t c o m p u t i n g p o w e r s h o r t o f a T u r i n g m a c h i n e . U n l i k e L o v e l a c e , P e t e r ’s c a p a b i l i t y as a m a t h e m a t i c i a n is u n c o n t e s t e d ; in fac t, F r a c t a l s in th e E u r o p e a n h is t o r y o f m a t h e m a t ic s s h e is w id e ly r e g a r d e d as “che m o t h e r o f r e c u r s i v e f u n c t i o n t h e o r y ” ( M o r r i s a n d H a rk le r o a d 1990). B u t sh e, too, im p lie d t h a t p arallels ex isted b e tw e e n h er g e n d e r i d e n t i t y a n d m a t h e m a t i c s ; m a i n t a i n i n g t h a t w o m e n c o u ld p r o v i d e a s p e ­ c ia l i n s i g h t t h a t m e n c o u l d n o t ( A n d r e k j a 1 9 7 4 , 1 7 3 ) . S i n c e w e k n o w t h a t , as a m a t h e m a t i c i a n , s h e w o u ld n o t b e t h i n k i n g o f t h i s s p ec ial in s ig h t as b e i n g m o re c o n c r e t e o r less lo gical, it m a y b e t h a t P e t e r also m a d e c o n n e c t i o n s b e t w e e n s e x ­ ual r e p r o d u c t i o n a n d r e c u r s i o n . F o llo w in g P e t e r ’s class o f p rim itiv e recu rsiv e fu n c tio n s , o n e reach es th e upper lim it o f re c u rsiv e p o w e r in t h e T u r i n g m a c h i n e . A l a n T u r i n g ’s c o n t r i b u t i o n s w ere n o t o n l y in t h e m a t h e m a t i c a l a b s t r a c t i o n s o f c o m p u t i n g , b u t in its a p p l i c a t i o n t o artific ia l i n t e l l i g e n c e as w ell. In h is c la ssic p a p e r t i t l e d “C o m p u t i n g M a c h i n ­ ery a n d I n t e l l i g e n c e , ” h e p r o p o s e d w h a t is n o w c a l l e d t h e T u r i n g te s t. A t first, m o s t d e f i n i t i o n s o f m a c h i n e i n t e l l i g e n c e w e r e b a s e d o n a p a r t i c u l a r ta s k or b e h a v i o r (e.g., c h e s s p la y i n g ) . B u t as t h e field o f a r t ifi c ia l i n t e l l i g e n c e ( A I ) has d e v e l o p e d , t h e s e h a v e s h o w n t o b e i n c r e a s i n g l y i n a d e q u a t e , a n d t h e T u r i n g test is w id e ly r e g a r d e d as t h e m o s t r e l ia b l e d e f i n i t i o n fo r A l ( i n fact, y early T u r in g tes ts a re n o w h e l d , w i t h n o m a c h i n e w i n n e r s t h u s far). T u r i n g b e g i n s by d e s c r i b i n g a g a m e in w h i c h a m a n a n d a w o m a n are b e h i n d a d o o r a n d a n s w e r q u e s t i o n s fr o m a n i n t e r r o g a t o r by w r i t t e n replies. T h e i n t e r r o g a t o r m u s t d e t e r m i n e w h o is t h e m a n a n d w h o is t h e w o m a n ; b o t h m u s t try to d e c e i v e h i m in t h e i r a n s w e rs . T u r i n g t h e n s u g g e s te d r e p l a c i n g o n e p e r s o n w i t h a n A I m a c h i n e ; t h e T u r i n g te s t h o l d s t h a t if t h e i n t e r r o g a t o r c a n n o t d i s ­ t i n g u i s h p e r s o n f r o m m a c h i n e , t h e n o n e h a s c r e a t e d t r u e m a c h i n e i n t e ll ig e n c e . T u r i n g ’s b i o g r a p h e r , A n d r e w H o d g e s , su g g es ts t h a t t h i s “i m i t a t i o n g a m e ” was i n s p ir e d by T u r i n g ’s o w n life: s t r u g g l i n g to d e f i n e h is i d e n t i t y as a h o m o s e x u a l in a h o m o p h o b i c so ciety . B o t h t h e T u r i n g m a c h i n e ’s a b il it y to i m i t a t e o t h e r m a c h i n e s a n d t h i s g a m e o f c o g n i t i v e i m i t a t i o n e c h o t h e so c ia l e x p e r i e n c e o f gays w h o liv e in a c o m m u n i t y w h e r e t h e y m u s t p r e t e n d t o b e s o m e o n e t h e y a re n o t . A n d to s o m e e x t e n t , t h e e n d le s s s e l f - r e f e r e n c e o f m e t a m a t h e m a t i c s was T u r i n g ’s h i d i n g p l a c e fro m t h e a n t i g a y w o r l d s u r r o u n d i n g h i m . B u t t h e s e x u a l g u e ss in g g am e o n w h ic h th e T u rin g te st was b a se d w o rk e d a g a in s t su c h n o rm a tiv e g e n d e r r e s t r i c t i o n s : it s u g g e s te d g e n d e r as s o m e t h i n g m o r e flu id , less fi x e d — a featu re w h ic h th e v irtu al c o m m u n itie s o n th e I n te r n e t h a v e sta rte d to d e m o n ­ s t ra te (cf. S t o n e 1995; T u rk le 1 99 5). D o u g la s H o f s t a d t e r ( 1 9 8 5 , 1 3 6 -3 6 7 ) , a m o d ­ ern m a s te r o f recu rsio n , h as also w r i t t e n a b o u t t h e p o te n t ia l for a m o r e fluid g e n d e r i d e n t i t y in d ig i ta l d y n a m i c s . M a t h e m a t i c s h a d a d o u b l e m e a n i n g for T u r i n g . It w as b o t h a n e m o t i o n a l sh ie ld , a c lo s e d w o rld o f e n d le s s in t e r i o r s e lf-r e fe re n c e , as well as a n o p e n i n g in to c o n s c i o u s n e s s a n d c o m m u n i t y . I n t h e e n d , t h i s d e s i r e fo r o p e n i n g k i l l e d T u r in g : im p lic a t i o n s d u rin g a robbery in v e s tig a tio n h e a d m itte d h is h o m o s e x u a lity to p o lic e d e te c ­ ti v e s a n d w as a r r e s t e d a n d f o r c e d t o s u b m i t t o h o r m o n e t r e a t m e n t s . T h i s e v e n ­ t u a l l y d r o v e h i m to s u i c id e . I t w as a tr a g ic f a i r y - t a le e n d i n g : h e k i l l e d ‘‘h i m s e l f by e a t i n g a n a p p l e d i p p e d in p o i s o n . H o d g e s w r it e s a b o u t t h i s d e a t h in t e r m s o f t h e d o u b l e m e a n i n g t h a t m a t h e m a t i c s h a d in h is life. “L o n e l y c o n s c i o u s n e s s o f selfc o n s c i o u s n e s s w as a t t h e c e n t e r o f h i s id eas. B u t t h a t s e l f - c o n s c i o u s n e s s w e n t b e y o n d G o d e l i a n self-referenc e, a b s t r a c t m i n d t u r n i n g u p o n its a b s t r a c t self. T h e r e w as in h i s life a m a t h e m a t i c a l ' s e r p e n t , b i t i n g its o w n ta il forever,-■•hut th e r e - w a s • a n o t h e r o n e t h a t h a d b i d h i m e a t fro m t h e t r e e o f k n o w l e d g e . ” I n A f r i c a t h e s e tw o s e r p e n t s are o n e ; s e x u a l r e p r o d u c t i o n ex is ts in t h e s a m e p u b l i c r e a l m as s o c i a l i n t e r c o u r s e . T h a t is o n e p o s s ib le r e a s o n w h y w e see r e c u r ­ s i o n — t h e s n a k e t h a t b i t e s its o w n ta i l— so p r o m i n e n t l y e m p h a s i z e d in A f r i c a n fractals, a n d a p o ss ib le e x p l a n a t i o n for w h y t h e s e p i o n e e r s o f r e c u r s i o n in E u r o p e h a p p e n e d to b e p e o p l e w h o t o o k is sue w i t h s e x u a l r e p r e s s i o n . T h a t ' s n o t t o say t h e r e is a d e t e r m i n i s t i c l i n k b e t w e e n t h e tw o . I n a n a l o g f e e d b a c k th e o r y , for e x a m p l e , w e see b o t h a n t i - a u t h o r i t a r i a n f e m in i s ts , li k e N o r b e r t W i e n e r ( H e i m s 1 9 8 4 ) , as w ell as a u t h o r i t a r i a n p r u d e s lik e H o w a r d O d u m ( T a y lo r 1 9 8 8 ). M a t h e ­ m a t i c s is n o t a m e r e r e f l e c t i o n o f p e r s o n a l in t e r e s t s , n o r is it a n a b s t r a c t i o n t h a t is e n t i r e l y d i v o r c e d fr o m o u r lives. W e m a k e m e a n i n g for o u r s e l v e s o u t o f w h a t ­ e v e r m e t a p h o r s — t e c h n i c a l o r o t h e r w i s e — w e fin d u sefu l; c o n v e r s e l y , p e r s o n a l m e a n i n g s c a n o f t e n in s p ir e n e w t e c h n i c a l ideas. W h i l e r e c u r s io n is p r o m i n e n t in A f r i c a n fractals, it h a s b e e n less so in E u r o ­ p e a n f r a c t a l g e o m e t r y . 6 I n t h e h i s t o r i c a l a p p e n d i x t o T h e F r a c ta l G e o m e tr y o f N a tu r e , M a n d e l b r o t p ro v id e s an e r u d i te h is to r y o f m a t h e m a t i c a l d e v e l o p m e n t s t h a t l e d t o h i s w o r k ; r e c u r s i o n is n e v e r m e n t i o n e d . E v e n w h e n r e c u r s io n , d o e s c o m e u p in t h e f r a c t a l g e o m e t r y l i t e r a t u r e , t h e t r e a t m e n t is ty p i c a l l y i n f o r m a l o r cursory. For e x a m p le , S a u p e (1 9 8 8 , 7 2) m erely n o te s t h a t "in so m e cases th e p r o c e d u r e c a n b e f o r m u l a t e d as a r e c u r s i o n . ” ^ S i m i l a r l y , t h e f r a c t a l t i m e se r ie s p r o d u c e d by d e t e r m i n i s t i c c h a o s is ra r e ly r e g a r d e d as t h e p r o d u c t o f f e e d b a c k lo o p s , a n d in o n e o f t h e few s t u d i e s t h a t is f o c u s e d o n t h i s r e l a t i o n s h i p , M e e s ( 1 9 8 4 , 1 0 1 ) m e r e ly s t a t e s t h a t “c h a o s is c e r t a i n l y p o s s ib le in f e e d b a c k s y s te m s .” O n t h e c o n t r a r y , it is n o t t h a t c h a o s is p o s s ib le w i t h fe e d b a c k ,- b u t t h a t c h a o s is i m p o s s i b l e w i t h o u t it. It w o u ld b e i n a c c u r a t e t o say t h a t E u r o p e a n m a t h e m a t i c s h a s d is r e g a r d e d r e c u r s i o n in g e n e r a l , a n d p e r h a p s r h e o b s e r v a t i o n I a m m a k i n g is s i m p l y d u e to d is c ip l in a r y s p e c i a li z a ti o n ; t h e r e is n o r e a s o n w h y s o m e o n e s t u d y i n g a p p l i c a t i o n s o f g r a p h i c s to a n a ly s is a n d m e n s u r a l t h e o r y s h o u l d n e c e s s a r il y b e t h i n k i n g a b o u t T u r i n g m a c h i n e s o r re c u r s iv e f u n c t i o n s . B u t it is p r e c i s e l y t h i s l a c k o f n e c e s s ity in m a t h e m a t i c s t h a t is so e a sily f o r g o t t e n in a d i s c i p l i n e w h e r e c e r t a i n t y goes F r a c t a l s in th e E u r o p e a n f u s t o r ) o f m a th e m a t ic s b e y o n d t h a t o f a n y e m p i r i c a l s c i e n c e im a g i n a b l e . M a t h e m a t i c s is b o t h a n i n v e i v t i o n a n d a d is co v ery . W e d i s c o v e r t h e c o n s t r a i n t s i n h e r e n t in t h e fa b ric o f sp a c e a n d t i m e , c o n s t r a i n t s t h a t a r e t h e s t u f f o f w h i c h o u r u n i v e r s e is c o m p o s e d . B u t m a t h e m a t i c s d o e s n o t sto p th e r e . T h e c o n s t r a i n t s a re n o t ju s t n e g a tio n s , b u t r a t h e r t h e b u i l d i n g b lo c k s w i t h w h i c h f u r t h e r m a t h e m a t i c s is c o n s t r u c t e d . A n d lik e an y c o n s t r u c t i o n , t h e r e a re c h o i c e s t o b e m a d e , d e c i s i o n s a b o u t h o w t h e s e b u i l d i n g b l o c k s a r e to b e c o n n e c t e d , i n t e r r o g a t e d , a n d d e p l o y e d in f u r t h e r d iscov ery . T h i s is w h e r e t h e h u m a n side o f m a t h e m a t i c s e n t e r s t h e p i c t u r e , e s p e c ia lly t h a t m o s t h u m a n o f e n d e a v o r s , c u l t u r e . C o n v e r s e l y , c u l t u r e is n o t m e r e w h i m , a p u r e l y s u b j e c t i v e m a t t e r o f c h o o s i n g f a v o r e d s o c ia l p r a c t ic e s . T h i s is w h e r e th e m a t h e m a t i c a l sid e o f h u m a n i t y e n t e r s t h e p i c t u r e , fo r we a re o n l y free t o c o n ­ s t r u c t c u l t u r e w i t h i n t h e c o n s t r a i n t s o f t h e u n i v e r s e in w h i c h w e live. N e i t h e r m a t h e m a t i c s n o r c u l t u r e s h o u l d b e v ie w e d as firmly fixed o n t h e u n i v e r s a l/ lo c a l d i v i d e ; t h e r e a re d iv i s i o n s w i t h i n d i v i s i o n s n e v e r e n d i n g . CHAPTER -Futures— for— A frican -fractals- M o s t a n t h r o p o l o g i s t s h a v e l o n g a b a n d o n e d t h e t e n d e n c y t o c r e a t e a fr o z e n “a n c i e n t t r a d i t i o n ” i n d e f i n i n g i n d i g e n o u s s o c i e ty ; c h a n g e a n d s y n t h e s i s a r e n o w i nt eg r a l p a r t s o f t h e c u l t u r a l p o r t r a i t . S o , t oo, w i t h A f r i c a n fr actal s; t h e y ar e n e c e s s a r i l y as m u c h o f t h e f u t u r e as t h e y a r e o f t h e p a s t . F r a c t a l s i n A f r i c a n c o n te m p o r a r y a rts T h e r e are m a n y works of m o d e r n A fri c a n professional art w h i c h in c o r p o ra t e a s p e c t s o f fr actal s, s p a n n i n g a w i d e r a n g e o f c u l t u r a l v i e w p o i n t s . A t t h e N a t i o n a l M u s e u m i n Y a o u n d e , C a m e r o o n , o n e c a n see o r g a n i c r o m a n t i c i s m i n N y a m e ’s p a i n t i n g s o f l o g a r i t h m i c sp i r a l s m o r p h i n g i n t o p e o p l e . T h e d o u b l e - s i d e d p o s t ­ m o d e r n m e t a l s c u l p t u r e s o f L e g b a i n B e n i n , b y a r t i s t s s u c h as K o u a s s , s h o w a c y b o r g 1 trickster w h ose tr a d itio n a l bifurcating abilities are ready for th e b in a r y c o d e s of n e w te c h n o l o g i e s . In East Africa, p a i n t e r G e b r e Kristos D e s ta p r o d u c e s n o n l i n e a r s c a l i n g c i r c l e s h e d e s c r i b e s as p u r e a b s t r a c t i o n i s m ( M o u n t 1 97 3 , 1 1 8 ) . A f r i c a n f r ac t a l s c o n t i n u e t o e v o l v e . B e s i d e s b e i n g p r e s e n t i n p r o ­ f e s s i o n a l s t u d i o art , f r ac t a l s h a v e a l s o a p p e a r e d in l a r g e - s c a l e p u b l i c a r t wo r k s , s u c h as o n t h e f a c a d e o f t h e U n i v e r s i t y o f D a k a r l i b ra r y (fig. 1 4 . 1) . T h i s s c a l i n g d e s i g n , in w h i c h t h e a l t e r n a t i o n o f p a i n t e d r e c t a n g l e s at t h e sm all scale 216 F u t u r e s f o r A f r i c a n fr a c t a ls 217 FIGU RE 14.1 T h e lib r a ry o f th e U n i v e r s i t y o f D a k a r This design makes use of both self-similarity (the vertical alternation of painted rectangles looks like the alternation of buildings) and nonlinear scaling (the rectangle width decreases rapidly as you go toward the center). m a t c h e s t h e a l t e r n a t i o n o f t h e b u i l d i n g w alls a t t h e large scale, is r e m i n i s c e n t o f c e r t a i n A f r i c a n fabrics. O n e o f t h e m o s t a c t i v e a r e a s o f t o d a y ’s A f r i c a n a r t c o m e s n o t fr o m p r o ­ f e s s i o n a l s t u d i o s , b u t r a t h e r f r o m t h e u n d i s t i n g u i s h e d s e l le rs o f t o u r i s t a rt . T o u r i s t a r t was fo r m a lly d i s r e g a r d e d in t h e p r o f e s s i o n a l a r t w o r ld , b u t c u l t u r a l s t u d i e s h a v e i n c r e a s i n g l y s h o w n c h a t t h i s is a d u b i o u s p o s i t i o n . F irst, n e i t h e r t h e “ t r a d i t i o n a l a r t i s t ” c r e a t i n g ro y a l w o r k s fo r a k i n g , n o r a r t s t u d e n t s t r y i n g t o p l e a s e t h e i r i n s t r u c t o r s , n o r e v e n p r o f e s s i o n a l s t u d i o a r t is t s w h o m u s t a ls o b e c o n c e r n e d w i t h sales a r e c o m p l e t e l y fr e e t o c r e a t e w h a t e v e r t h e y w i s h , so t h e r e is n o r e a s o n t o s i n g le o u t t h e c r e a t o r s o f t o u r i s t a r t for b e i n g c o n s t r a i n e d . S e c o n d , o p p o r t u n i t i e s f o r p r o f e s s i o n a l s t u d i o a r t i s t s a r e few, a n d t h e t o u r i s t m a r k e t c r e a t e s a large n u m b e r o f e c o n o m i c o p p o r t u n i t i e s ; it s e e m s s u s p i c i o u s t o d i s r e g a r d t h i s v i b r a n t a c t i v i t y in f a v o r o f a t i n y e li te . A n d fin ally , as C u l l e r s ( 1 9 8 1 ) n o t e s , to u r i s m is n o t t h e o p p o s i t e o f a u t h e n t i c c u l t u r e , r a t h e r t o u r i s m c rea tes a u t h e n t i c i t y . C u l l e r s ’s o b s e r v a t i o n was r e p e a t e d to m e by M a x ( h e d i d n o t w a n t h is last n a m e to be u se d ), a S e n e g a le s e a rt is t in D a k a r w h o sells t o t h e to u r is t tr a d e . M a x c o m p l a i n e d t h a t h is m o s t c r e a t i v e w o r k — t h e d e s i g n s w h i c h c a m e t o h i m in d r e a m s — was diffic ult t o sell b e c a u s e o f t h e t o u r i s t c o n c e p t i o n o f t r a d i t i o n a n d a u t h e n t i c i t y . L ik e m a n y c r e a t o r s o f t o u r i s t arc in D a k a r, h e p r o d u c e s i m i t a t i o n s o f t h e k o r a , t h e S e n e g a l e s e s t r i n g e d i n s t r u m e n t t h a t f e a t u r e s a s in g le f r e t r u n ­ n i n g d o w n t h e c e n t e r a n d a h a n d g rip o n b o t h sides. F igu re 14.2 s h o w s t h e u su al k o r a m o d e l , a l o n g w i t h M a x ’s i n n o v a t i o n , t h e re c u r s iv e k o r a . T h e re c u r s iv e k o r a m a k e s use o f e a c h h a n d g rip a s t h e f r e t o f tw o s m a ll e r k o ras . 1 a s k e d M a x • 2 l8 /mplicali'ons if h e h a d e v e r c o n s i d e r e d c o n t i n u i n g to s m a l l e r sc a le s, a n d h e s a i d t h a t h e h a d o n c e clone so, b u t t h a t it was im p o s s ib le t o sell s u c h i n n o v a t i v e w o rk ; to u r is t s d id n o t w a n t a n y t h i n g t h a t s m a c k e d o f o rig in a l it y . F r a c ta ls in A f r i c a n c o n te m p o r a r y a r c h i t e c t u r e M a n y in d ig e n o u s A fric a n designs h a v e b e e n in c o r p o ra te d i n t o m o d e r n a rc h i. t e c t u r a ! p r o j e c t s in A .frica, a n d s o m e o f t h e s e h a v e b e e n f r a c t a l s . F o r e x a m p l e , t h e S i e r p i n s k t - l i k e i t e r a t i v e t r i a n g l e s f r o m M a u r i t a n i a w e r e u s e d in a n i n s t i t u ­ t i o n a l b u i l d i n g in S e n e g a l , a n d t h e c ir c le s o f c ir c le s in t h e a r c h i t e c t u r e o f W e s t A f r i c a n v il la g e s b e c a m e t h e b a sis o f a d e s i g n fo r a b u i l d i n g c o m p l e x in d o w n ­ t o w n B a m a k o , t h e c a p i t a l o f M a l i (fig. 1 4 .3 ). O n e o f t h e m o s t p o t e n t v i s io n s o f a n A f r i c a n f r a c ta l f u t u r e h a s c o m e fro m t h e a r c h i t e c t u r a l s t u d i e s o f D r. D a v i d H u g h e s a t K e n t S t a t e U n i v e r s i t y in O h i o . W o r k i n g as a F u lb r ig h t s c h o l a r in sev eral A f r i c a n c o u n t r i e s , H u g h e s ( 1 9 9 4 ) p u t t o g e t h e r a p o r t r a i t o f w h a t h e t e r m e d “A f r o c e n t r i c a r c h i t e c t u r e , " w h i c h e m b o d i e s s e v e r a l a s p e c t s o f t h e f r a c t a l m o d e l . F irst, H u g h e s c o m b i n e d a c h a r ­ a c te riz a tio n of th e self-o rg an izin g p ro p e rtie s o f A fric a n b u ild in g d esig n (a n " o r g a n i c a r c h i t e c t u r e " w h i c h “g r o w s fr o m its s i t e ” ) w i t h its s e l f - s i m i l a r p r o p ­ e r t i e s ( w h a t h e t e r m e d “t h e o u t s i d e / i n s i d e r e l a t i o n s h i p , ” a m u t u a l s h a p i n g o f u n its, clu sters o f u n its, a n d c o m m u n a l spaces fo rm ed by th e s u r r o u n d in g c lu s ­ te r s). S e c o n d , h e e x p l i c i t l y r e j e c t e d p r i m i t i v i s t o r n a t u r a l i z i n g p o r t r a i t s . W h i l e F i u i t r e s f o r A f r i c a n f r a c t a ls 239 n o t i n g its e n v i r o n m e n t a l h a r m o n y , H u g h e s a ls o e m p h a s i z e d t h a t A f r i c a n a r c h i t e c t u r e is a lw a y s a n i n t e n t i o n a l a c t o f d e s i g n a n d s e m i o t i c s , n o t m e r e l y a n u n c o n s c i o u s a d a p t a t i o n t o t h e e c o s y s t e m . I n h is f r a m e w o r k , '‘t r a d i t i o n ” i n c l u d e s t h e t r a d i t i o n o f i n n o v a t i o n , o r as G a t e ' 5- ( i p 8 8 ) p u t s it, t h e A f r i c a n t h e m e ot “r e p e t i t i o n w i t h r e v i s i o n . ” FIGURE I 4 . 3 I n d i g e n o u s fra cta ls in m o d e rn a rch itectu re (a) Here a traditional Mauritanian fractal design is used in a modern building in the Casamance, Senegal, ( b ) The DPC building in Burkina Faso, using traditional scaling cylinders with contem ­ porary construction techniques. Architects such as Issiaka Isaac Drabo have made many large-scale buildings based on this syncretic approach. 220 Jmpiications G i v e n t hi s c o m b i n a t i o n o f s e l f- o r g a n i z e d s t r u c t u r e a n d i n t e n t i o n a l d e s i gn , it is n o t sur prising t h a t H u g h e s ’s wo r k led h i m t o a be a ut i f ul e x a m p l e o f t h e p o t e n ­ ti al f r ac t a l f ut ur e . Fi g u r e 14.4 s h o w s a d e s i g n by A l e x N y a n g u l a , o n e o f U y g h e s ’s s t u d e n t s at t h e C o p p e r b e l t U n i v e r s i t y in Z a m b i a ( H u g h e s 1 9 9 4 , ‘’1 6 5 - 1 6 6 ) . T h i s ar c h i t e c t u r e provi des a powerful syncr eti c fusion of in d i g e n o u s a n d m o d e r n for ms. T h e figure t r a c e d b y t h e w a l k w a y s h o w n i n t h e g r o u n d p l a n is a classic f ig u r e 14.4 D e s i g n for K itiv e C o m m u n ity C linic (a) Kitwe C o m m u n i t y Clinic in Zambi a; design by David Hu g h e s and Alex Nyangula. (b) Kitwe C o m m u n i t y Clinic ground plan. CPhotos courtesy David Hughes.) FIGURE 1 4 . 5 F r a c ta l.ite r a tio n s o f N y a n g a l a ’s c o m m u n i t y c l i n i c d e s i g n Fractal based on Nyangula’s architectural design. T he ‘‘active lines” of the generation process have been removed, as have any self-intersecting hexagons. 222 I m p l ic a tio n s e x a m p l e o f ch e f r a c t a l b r a n c h i n g p a t t e r n r e f e r r e d t o a s a C a y l e y t r e e (s e e S c h r o e d e r 1 9 9 1 , 8 7 - 8 8 ; P e i t g e n e t al. 1 9 9 1 , 1 9 - 2 0 ) , a n d c a n b e e x t e n d e d fro m t h e tw o i t e r a t i o n s g i v e n by N y a n g u l a t o in finity. A d d i n g t h e hexag on sT C a s y n ­ c r e t i s m b e t w e e n t h e c y l i n d e r o f Z a m b i a n in d i g e n o u s a r c h i t e c t u r e a n d t h e r e c ­ ti li n e a r form s o f m o d e r n m a t e ri a ls ) v io la te s t h e C a y le y r e q u i r e m e n t t h a t t h e g ra p h is s e l f - a v o i d i n g ( t h a t is, t h a t t h e b r a n c h e s d o n o t i n t e r s e c t ) . S i n c e I w a s i n t e r ­ e ste d in e x p l o r i n g th e fra c ta l s t r u c t u r e by t a k i n g N y a n g u l a ’s d e s i g n t o h i g h e r i t e r ­ a tio n s , I m a d e tw o a d ju s tm e n ts for th is p ro b l e m . O n e is su gg ested by t h e a p p r o a c h e l e v a t i o n s k e t c h ( H u g h e s 1 9 9 4 , 1 6 7 ) , w h e r e it is c l e a r t h a t t h e c e n t r a l xtnit is sligh tly larger t h a n t h e o th e r s . T h i s m e a n s t h a t s e l f - i n te r s e c ti o n will be fo restalled to h i g h e r i t e r a t i o n s . 2 T h e o t h e r is sim p ly t h e e l i m i n a t i o n o f u n i t s w h e n e v e r th e y o v e r l a p . W i t h th e s e tw o q u a li f i c a ti o n s , N y a n g u l a ’s d e s i g n m a k e s for a n in f in i te l y e x p a n d a b l e ( y e t b o u n d e d ) a r c h i t e c t u r e , a s s h o w n i n figure 1 4 .5 . S u c h f l e x ib i l­ ity c o u l d c o n t r i b u t e to t h e e ffo rts to e n c o u r a g e a m o r e p a r t i c i p a t o r y a p p r o a c h to A fric a n a rc h ite c tu ra l d e sig n (F a th y 1973; O z k a n 1997). If w e t a k e a n a e r i a l v i e w o f t h e m o d e r n E u r o p e a n s e t t l e m e n t o f P a ris , F r a n c e , w e w o u l d see l i n e a r c o n c e n t r i c c i r c l e s s u r r o u n d i n g i t s c e n t e r o f s o c i a l p o w e r. T h e d i f f e r e n c e b e t w e e n t h i s l i n e a r , r a d i a ll y s y m m e t r i c s e r ie s o f c i r c l e s a n d A f r i c a ’s n o n l i n e a r , d e c e n t r a l i z e d a r c h i t e c t u r e is p e r h a p s s u b t l e , b u t i m p o r ­ t a n t . T h e t e r m " A f r o c e n t r i c " is m i s l e a d i n g in t h a t “c e n t r i c " is m u c h m o r e t h e g e o m e try o f P aris th a n of L o g o n e -B irn i, M o k o u le k , L ab b ezan g a, a n d th e o th e r A f r i c a n a r c h i t e c t u r e s w e h a v e e x p l o r e d . H u g h e s ’s c a l l f o r a “m u l t i d i m e n s i o n a l A f r o c e n t r i s m ” is b o t h a n a f f i r m a t i o n o f “ A f r o ” a n d a c h a l l e n g e t o “ c e n t r i s m ” ; it is a c a ll for c u l t u r a l p o r t r a i t s t h a t d o n o t r e d u c e to a s i n g l e o n e - d i m e n s i o n a l c e n te r b u t ra th e r co m b in e th e b o u n d a rie s of trad itio n w ith th e in fin ite e x p a n ­ sion o f in n o v a tio n . A f r i c a n fr a c ta ls in m a t h e d u c a t i o n S e v e ra l re se a rc h e rs h a v e i n d e p e n d e n tly e x p lo re d fractal a s p e c ts o f A f r ic a n m a th e m a tic s . C h o n a t G e tz of th e U n iv e rsity of th e W itw a te rs ra n d h a s c reated I t e r a t e d F u n c t i o n S y s t e m s i m u l a t i o n s o f Z u lu b a s k e t w e a v in g . J o h n S im s, m a t h e ­ m a t i c i a n a n d a r t i s t a t t h e R i n g l i n g S c h o o l o f D e s ig n in F lo r id a , h a s d e v e l o p e d fra c ta l p a t t e r n s based o n B aku ba .rafia c l o t h ( a n d in s p ired by his A f r i c a n h e r i ta g e ) . I n c h a p t e r 5 w e e n c o u n t e r e d t h e l u s o n a a n a ly s is o f P a u l u s G e r d s , a p r o f e s s o r at U n t v e r s i d a d e P e d a g o g ic a o f M a p u t o , M o z a m b i q u e , w h o s e p ro lific w r i t i n g s h a v e r e c e n t l y r a n g e d fr o m t h e e t h n o m a t h e m a t i c s o f w o m e n 's a r t in s o u t h e r n A f r i c a ( G e r d e s 1 9 9 8 a ) to t h e use o f M o z a m b i q u e b a s k e t w e a v i n g g e o m e t r y in m o d e l ­ in g f u l l e r e n e m o l e c u l e s ( G e r d e s 1 9 9 8 b ). F u t u r e s f o r A f r i c a n fr a c t a ls W h i l e t h e r e a re c l e a r l y b e n e f i t s t o u t i l i z i n g i n d i g e n o u s k n o w l e d g e for d e v e l o p m e n t a n d e d u c a t i o n in A f r i c a , A f r i c a n frac tals m i g h t also b e o f use in th e U n i t e d S t a t e s . D e s p it e t h e low m a t h e m a t i c s p a r t i c i p a t i o n o f A f r i c a n A m e r i c a n s t u d e n t s as a n e t h n i c g r o u p , it h a s b e e n d e m o n s t r a t e d t h a t c h a n g e s in t h e l e a r n ­ in g e n v i r o n m e n t c a n i m p r o v e t h e i r m a t h e m a t i c s p ro f ic ie n c y to le v e ls e q u a l to t h e m a jo rity p o p u l a t i o n . E v i d e n c e suggests t h a t a l t h o u g h d ir e c t i n s t i t u t i o n a l b a r ­ riers in e c o n o m i c a l l y d i s a d v a n t a g e d s c h o o l s , s u c h as t h e e m p h a s i s o f v o c a t i o n a l o v e r a c a d e m i c s u b j e c ts ( D a v i s 1 9 8 6 ) a n d l a c k o f c o m p u t e r a c c e s s ( A n d e r s o n , W e l c h , a n d H a rri s , 1 9 8 4 ) c a n a c c o u n t for s o m e o f th i s d if f e r e n c e , m o r e s u b t le c u rr ic u la r c h a n g e s c a n play a key ro le in r e t e n t i o n a n d a c h i e v e m e n t . For e x a m p le , B a ra tz e t a l. ( 1 9 8 5 ) f o u n d t h a t A f r i c a n A m e r i c a n s t u d e n t s a re m o r e li k e ly to use c o m p u t e r s f o r r o u t i n e d rill; h e n c e , t h e p r o b l e m is n o t s i m p l y t h e a v a i l a b i l ­ ity o f c o m p u t e r s , b u t a ls o t h e i r s t y le o f u t i l i z a t i o n . T h e N a t i o n a l A s s e s s m e n t o f E d u c a t i o n a l P ro g r e s s ( 1 9 8 3 ) s t u d y o f m a t h p e r f o r m a n c e in s e v e n t e e n - y e a r - o l d A fric a n A m e ric a n s re p o rte d th e g re a te st d e fic ie n c ie s a t t h e a p p lic a tio n s level, a n d s e v e r a l r e s e a r c h e r s ( U s i s k i n 198 5; D a v i s 1 9 8 9 ; M a l c o m 1 9 8 3 ) h a v e r e c o m ­ m e n d e d r e v i s i o n o f c o u r s e s to e m p h a s i z e m o r e i n t e r d i s c i p l i n a r y a n d “re a lw o r l d ” m a t h e m a t i c s i n s t r u c t i o n as w e ll as “ a c t i o n - o r i e n t e d ” p ed a g o g y . C o m p u t e r - b a s e d l e a r n i n g h a s d e m o n s t r a t e d t h e c a p a b i l i t y fo r b o t h i n t e r a c t i v e a n d i n t e r d i s c i p l i n a r y m a t h e m a t i c s i n s t r u c t i o n (K e it e l a n d R u r h v e n 1 9 9 3 ) , a n d S t i f f e t al. ( 1 9 9 3 ) s p e c i f ic a lly p o i n t t o c o m p u t e r - b a s e d le a r n in g , as a p r o m i s i n g fo r u m fo r b r i n g i n g t h e s e c h a n g e s t o A f r i c a n A m e r i c a n s t u d e n t s . T h e s e n e e d s c o u l d be d i r e c t l y a d d r e s s e d by a p p l y i n g A f r i c a n f r a c ta ls t o t h e c la s s r o o m . I n a d d i t i o n to c h a n g e s in s t r u c t u r a l a s p e c t s o f m a t h e m a t i c s t e a c h i n g , s e v e r a l r e s e a r c h e r s a n d i n s t r u c t o r s h a v e i n i t i a t e d c u l t u r a l l y e n r i c h e d c u r r ic u l a . T h e r a t i o n a l e for t h i s a p p r o a c h c o m e s f r o m a v a r i e t y o f p e r s p e c t i v e s (e.g., V y g o tsk ian le a rn in g th e o ry ). P o w ell (1 9 9 0 ) n o te s th a t p e rv a s iv e m a in s tre a m stereo ty p es o f scien tists a n d m a t h e m a tic ia n s co n flic t w ith A fric a n A m e ric a n c u ltu ral o rie n ta tio n . S im ila r c o n flic ts b e tw e e n A fric a n A m e ric a n id e n tity and m a t h e m a t i c s e d u c a t i o n in t e r m s o f s e l f - p e r c e p t i o n , c o u r s e s e l e c t i o n , a n d c a r e e r g u i d a n c e h a v e b e e n n o t e d (cf. H a l l a n d P o s t m a n - K a m m e r 1 9 8 7 ; B o y e r 1 9 8 3 ). B u t we s h o u l d n o t a ss u m e t h a t th is c o n s t i t u t e s a p r o b l e m o f “se lf-e s te e m .” T h e r e l a t i o n b e t w e e n c u l t u r a l i d e n t i t y a n d l e a r n i n g is q u i t e c o m p l e x ; it w o u ld be n a i v e to su g g e s t t h a t t o d a y ’s A f r i c a n A m e r i c a n s t u d e n t s h a v e t h e s a m e r e l a ­ t i o n to i d e a s a b o u t t h e i r a n c e s t r y as d i d s t u d e n t s in p r e v i o u s d e c a d e s , a n d in n o case h as th e re ever b e e n a sim p le “m im ic ry ” of A fric a n cu ltu re. R ath er, e th n o g r a p h ic research (H e b d ig e . 1987; M e r c e r 1988; R ose 1994) show s th a t A f r i c a n A m e r i c a n y o u t h a c ti v e ly c o n s t r u c t i d e n t i t y u sin g a w id e v a r i e ty o f c u l ­ t u r a l sig n ifie rs . 224 ] m j) lic a tio n s F o r th is re as o n , a p p li c a ti o n s o f A f r i c a n fractals will h a v e to .stress d esig n tools a n d g u id e d dis c o v ery , a n d a v o id p assiv e p r e s e n t a t i o n . W h i l e “ i n t e r a c t i v e ” h a s b e c o m e a c a t c h w o r d in m u l ti m e d i a , m a n y o f th e s e system s m e r e ly use t h e c o m p u t e r like a slide p ro je c to r, w i t h s t u d e n ts pressing d if fe re n t b u t t o n s t o see v a r i o u s im ages. M u l t i m e d i a in t h i s fo r m h a s a d i s t i n c tl y “c a n n e d ” feel t o it. T h e d e s i g n a p p r o a c h , in c o n t r a s t , offers s t u d e n t s to o ls for c o n s t r u c t i n g p a t t e r n s o f t h e i r o w n c r e a t i o n . T h a n k s t o m a n y p a r t i c i p a n t s — in p a rtic u la r, p r o g r a m m e r s T Q B erg a n d J a r o n S a m p s o n , a n d m i n o r i ty m a t h - e d u c a t i o n sp ecialist G l o r i a G i l m e r — we h a v e s t a rte d d e v e l o p m e n t o f a n A f r i c a n fra c ta ls s o f tw a r e m a t h lab. T h e la b b e g in s w i t h s i m u ­ la t io n s o f t r a d i t i o n a l A f r i c a n p a t t e r n s a n d s h o w s s t u d e n t s h o w t h e m a t h e m a t i c a l s t r u c t u r e b e h i n d t h e s e d e s i g n s offers t h e m to o ls t o c r e a t e t h e i r o w n . A g a i n , i t is i m p o r t a n t to stress t h a t A f r i c a n A m e r i c a n s t u d e n t s a r e n o t e x p e c t e d t o b e i n t e r e s t e d in t h e m a t e r i a l o u t o f a s i m p l e i d e n t i t y r e f l e c t i o n , a n y m o r e t h a n t h e y w o u ld n e c e s s a rily be i n t e r e s t e d in w e a r i n g D n s h ik i s h i r t s a n d A fro s . R a t h e r , it is t h e o p p o r t u n i t y t o c r e a t e n e w c o n f i g u r a t i o n s a n d s y n t h e s e s t h a t C o m b in e tr a d it io n a n d i n n o v a t i o n t h a t are significant. A t t h e J u n e 199 6 m e e t ­ in g o f t h e C o l u m b u s U r b a n Y o u t h C o n f e r e n c e , w e e x p l o r e d t h e s e c o n n e c t i o n s w i t h a class o f e i g h t e e n 1 2 - y e a r - o ld A f r i c a n A m e r i c a n s t u d e n t s . T h e first class m e e t i n g i n t r o d u c i n g t r a d i t i o n a l a r c h i t e c t u r e w as a n e a r d is a s te r ; d e s p i t e m u l t i m e d i a a n d m a n i p u l a t i v e s , it a p p e a r e d t h a t t h e p r i m i e i v i s t a s s o c i a t i o n s w ith “ m u d h u t s ” w e r e a s t r o n g d e t e r r e n t . T h e f o l l o w i n g s e s s io n , u s i n g t h e G h a n a i a n log s p i r a l— c e l l u l a r a u t o m a t a — o w a ri r e l a t i o n s , w as q u i t e su c c e s s fu l, p r o b a b l y b e c a u s e t h e c o m b i n a t i o n o f t r a d i t i o n a l re lig io u s k n o w l e d g e a n d m a t h e m a t i c a l g r a p h i c s s e n t a m o r e c l e a r a n t i p r i m i t i v i s t m e s s a g e d B u t in a d e s i g n e x e r c i s e w h e re th e s tu d e n ts b eg an w ith c o m p u te r g ra p h ic s sim u la tio n s o f th e G h a n a ia n l o g a r i t h m i c s p i r a l p a t t e r n s , t h e y s h o w e d l i t t l e i n t e r e s t in p r o d u c i n g f u r t h e r im itations o f t h e A f r i c a n d e s i g n s . R a t h e r , t h e s t u d e n t s q u i c k l y c a u g h t o n to v is u a l c o r r e l a t e s o f t h e e q u a t i o n p a r a m e t e r s a n d b e g a n a fr e e -fo r- a ll c o m p e t i t i o n t o see w h o c o u l d m a k e t h e m o s t b izarre p a t t e r n s . T h e i r i n t e r e s t a p p e a r e d to be s p a r k e d by t h e A f r i c a n c o n n e c t i o n s , h u t q u i c k l y w e n t b e y o n d t h e m . P erh ap s m ore im p o rta n t th a n m itig a tin g a d ire c t co n flic t b e tw e e n e th n ic id e n t it y a n d m a t h e m a t i c s , u sing A f r i c a n fractals in t h e c la s s ro o m m i g h t h e l p guard a g a in st a n o v e re m p h a sis o n b io lo g ical d e te r m in is m , w h ic h h a s b e e n fou n d ad versely to affect m a t h e m a t i c s le a r n in g . G e a r y ( 1 9 9 4 ) rev ie w s cro s s-c u ltu ra l s t u d ­ ies t h a t i n d i c a t e t h a t w h i l e c h i l d r e n , t e a c h e r s a n d p a r e n t s in C h i n a a n d j a p a n t e n d t o v ie w d if fic u lty w i t h m a t h e m a t i c s as a p r o b l e m o f t i m e a n d e f f o r t, t h e i r A m e r i c a n c o u n t e r p a r t s a t t r i b u t e d i f f e r e n c e s in m a t h e m a t i c s p e r f o r m a n c e to i n n a t e a b il it y ( w h i c h c a n t h e n b e c o m e a self-fu lfillin g p r o p h e c y ) . F o r A f r i c a n A m e r i c a n s , b i o l o g i c a l d e t e r m i n i s m h a s b e e n c l o s e ly a s s o c i a t e d w i t h m y t h i c F utures fo r A fr ic a n fractals s t e r e o t y p e s a b o u t “p r i m i t i v e p e o p l e ” (e.g., t h e f a b l e t h a t A f r i c a n s c o u n t “o n e , t wo, t h r e e , m a n y ”). By s h o w i n g t h e p r e s e n c e o f c o m p l e x m a t h e m a t i c a l c o n c e p t s in A f r i c a n c u l t u r e , w e c a n m e n d s o m e o f t h a t d a m a g e . S i n c e r e d u c t i v e m y t h s o f b i o l o g i c a l d e t e r m i n i s m a r e d e t r i m e n t a l t o m a t h e m a t i c s l e a r n i n g for s t u d e n t s o f all e t h n i c b a c k g r o u n d s , all s t u d e n t s c o u l d p o t e n t i a l l y b e n e f i t f r o m t h i s m a t e r i a l . Finally, w e s h o u l d n o t e t h a t t h e i n c r e a s i n g use o f m u l t i c u l t u r a l c u r r i c u l u m ma t e r i a l s in t h e arts a n d h u m a n i t i e s h a v e n o t b e e n m a t c h e d in t h e sci ences. T h i s c o u l d s e n d a m e s s a g e t o m i n o r i t y s t u d e n t s t h a t t h e i r h e r i t a g e is o n l y p e r t i n e n t t o t h e a r t s a n d h u m a n i t i e s , a n d t h a t t h e s c i e n c e s a r e r e a l l y for p e o p l e f r o m o t h e r e t h n i c groups. In a d d i t i o n , s o m e text s s u c h as Multicultural Mathematics ( N e l ­ s o n 1 9 9 3 ) h a v e e m p h a s i z e d o n l y C h i n e s e , H i n d u , a n d M u s l i m e x a m p l e s , so t h a t e v e n in c ase s w h e r e m u l t i c u l t u r a l i s m is used, A f r i c a n m a t h ma y be left o u t ( s ee K at z 1 992 for a s i mi l a r c r i t i q u e ) . A n d o f t h e f e w t e x t s t h a t d o use A f r i c a n m a t h , a lm o s t all e x a mp l es are re s t ri ct ed t o p r i ma r y sc h o o l level. A g a i n , thi s restri c­ t i o n m i g h t u n i n t e n t i o n a l l y i m p l y p r i m i t i v i s m (i.e., t h a t m a t h e m a t i c a l c o n c e p t s f r o m A f r i c a n c u l t u r e a r e o n l y c h i l d l i k e ) . F o r thi s r e a s o n , o u r l a b ’s i n c l u s i o n of a d v a n c e d t o p i c s s u c h as f r ac t a l g e o m e t r y , c e l l u l a r a u t o m a t a , a n d c q m p l e x i t y a r e w o r t h t h e e x tr a effort t o t i e i n t o a s e c o n d a r y s c h o o l c u r r i c u l u m ( w i t h o u t o v e r ­ l o o k i n g t h e use o f s t a n d a r d t o p i c s s u c h as l o g a r i t h m i c s ca l i n g , g e o m e t r i c c o n ­ struction, and trigonometry), W h i l e t h e mu l t i m e d i a l a b ’s mo s t significant p o t e n t i a l for i m p r o v i ng e d u c a t i o n is in m a t h e m a t i c s , we s h o u l d n o t i gnor e A f r i c a n St u d i es . A f r i c a n art, for e x a mp l e , is i ncr easi ngly used in s e c o n d a r y s c h o o l s across t h e n a t i o n , a n d use o f our lab c o u l d g r eat l y e n h a n c e s u c h courses. First, as n o t e d a b o v e , it p r o v i d e s a n a l t e r n a t i v e to d e t r i m e n t a l m i s r e p r e s e n t a t i o n s o f A f r i c a n s as “p r i m i t i v e " p e o p l e . In art h i s t o r y l es s ons , for i n s t a n c e , s t u d e n t s o f t e n l e a r n a b o u t t h e g e o m e t r i c basis for G r e e k a r c h i t e c t u r e o r R e n a i s s a n c e p a i n t i n g , w h i l e c o m m e n t a r y o n A f r i c a n w o r k s is o f t e n r es t r i ct e d t o d i s c u s s i on o f " n a t u r a l n e s s ” o r “e m o t i o n a l e x p r e s s i o n . ” S e c o n d , t h e l ab aids in i n t e g r a t i v e c u r r i c u l a d e v e l o p m e n t (see R o t h 1 9 9 4 o n difficul ti es in t h i s ar ea) . It w o u l d a l l o w m a t h t e a c h e r s w h o w o u l d li ke t o i n c l u d e e t h n o m a t h e m a t i c s c o m p o n e n t s in t h e i r t e a c h i n g t o refer t o e x a m p l e s in w h i c h s t u d e n t s are al r eady e ng a g e d , a n d w o u l d p r o v i d e art t e a c h e r s w i t h n e w t ool s for d e s i g n a n d analysis. S i m i l a r a d v a n t a g e s c o u l d b e o b t a i n e d i n o t h e r A f r i c a n S t u d i e s areas. I n f o r m a t i o n tec h n o lo g ie s a n d s u s t a i n a b l e d e v e l o p m e n t T h e use o f i n d i g e n o u s k n o w l e d g e s y s t e m s i n d e v e l o p m e n t g o e s b a c k t o c o l o ­ n i a l a p p r o p r i a t i o n s , b u t in t h e p o s t c o l o n i a l c o n t e x t t h e s e s y s t e m s h a v e t a k e n o n n e w m e a n i n g as a s i g n o f e i t h e r e p i s t e m o l o g i c a l i n d e p e n d e n c e , o r a t l e a s t 226 implicatfoni a m o r e e g a l i t a r i a n v i e w o f k n o w l e d g e s y s te m s . In c h a p t e r 10 , f o r e x a m p l e , w e sa w t h e s c a l i n g s p i r a l s o f J o l a s e t t l e m e n t a r c h i t e c t u r e t h a t a r o s e f r o m t h e i r c ir c u la r b u ild in g s; th e F re n c h re s e a rc h o rg a n iz a tio n E N D A h asT b u ilt a n i m p l u v i u m c r e a t e d by t h e c o m b i n a t i o n o f m o d e r n m a t e r i a l s a n d t h i s t r a d i ­ t i o n a l J o l a d e s i g n . A n o t h e r o f E N D A ’s ru r a l d e v e l o p m e n t p r o j e c t s t h a t i n c o r ­ p o r a t e b o t h t r a d i t i o n a l f r a c t a l a r c h i t e c t u r e a n d m o d e r n t e c h n i q u e s is s h o w n i n f i g u r e 14 .6. I n c h a p t e r 6 w e saw h o w t h e s c a l i n g p a t t e r n s o f k c n t e c l o t h w e re c r e a t e d to m a t c h t h e sc a lin g o f s a c c a d i c eye m o v e m e n t s as t h e y s c a n fro m t h e face to t h e body. T h e G h a n a i a n B r o a d c a s t i n g C o r p o r a t i o n , G h a n a ’s n a t i o n a l t e l e v i s i o n c h a n n e l , h a s c o n t i n u e d t h i s p r a c t i c e in t h e c o n t e x t o f m o d e r n i n f o r m a t i o n t e c h ­ n olo gie s, utilizing t h e scalin g p a t t e r n o f k e n t e c l o t h in t h e i r test p a t t e r n (fig. 14.7). W h e r e a s t h e t r a d i t i o n a l s c a l in g w as a p p l i e d t o t h e h u m a n v is u a l s c a n , t h i s t e c h nologized versio n m a kes use o f th e sa m e p a t t e r n for t e s ti n g t h e v id e o sca n . A sim ple a p p l i c a t i o n , b u t it s h o w s t h a t A f r i c a n f r a c t a l s a re n o t j u s t r e s t r i c t e d t o l o w - t e c h a d a p t a t i o n s ; t h e y c a n a ls o p r o v i d e s o m e useful b ri d g e s b e t w e e n t r a d i t i o n a l a n d h i g h - t e c h w orld s. I n c h a p t e r 10 w e sa w t h a t t h e r e w e r e ties b e t w e e n t h e t r a d i t i o n a l k n o w l ­ e d g e s y s te m s s u p p o r t e d b y A f r i c a n f r a c ta l s a n d t h e p r o d u c t i v e m a i n t e n a n c e o f th e s e s o c ietie s in w h a t P e r B ak w o u ld call a s t a t e o f se lf-o rg a n iz e d c ritica lity . T h i s su g g es ts t h a t m o s t o f t h e in d i g e n o u s A f r i c a n s o c i e t i e s w e r e n e i t h e r u t t e r l y a n a r ­ c h i c , n o r frozen in s t a t i c o r d e r ; r a t h e r , t h e y u ti li z e d a n a d a p t i v e f le x ib ility t h a t c o u ld b e a p p l i e d t o m o d e r n d e v e l o p m e n t . B u t d e c a d e s o f r e s e a r c h h a v e s h o w n figure 1 4 .6 M o d e rn ise d f r a c t a l v illag e This ENDA project in Burkina Faso combined the traditional fractal structure with modern construction techniques. Futures fo r A fr ic a n fracials 227 FIGURE I 4 . 7 K e n t e c lo th in th e Q h c in a ia n B r o a d c a s tin g C o r p o ra tio n t e s t p a t t e r n Kente cloth pattern is used in the upper right-hand quadrant of the large circle. t h a t a t o p - d o w n a p p r o a c h t o d e v e l o p m e n t , e v e n t h a t m a k i n g use o f i n d i g e n o u s k n o w l e d g e , is o f t e n less e f f e c t i v e t h a n a b o t t o m - u p , “grass r o o t s ” a p p r o a c h . A d o p t i n g in f o rm a tio n tec h n o lo g y t o nir al areas could prov id e t h e o p p o rtu n it y for p u t t i n g A f r i c a n f r ac t a l s t o w o r k i n s u s t a i n a b l e d e v e l o p m e n t . I n a d d i t i o n t o t h e n e e d for b o t t o m - u p a u t h o r i t y , r e s e a r c h e r s h a v e d e m o n ­ s t r a t e d t h e c r i t i c a l r o l e o f w o m e n i n A f r i c a n d e v e l o p m e n t ( e. g . , B o s e r u p 1 9 70 ; N e l s o n 1 9 8 1 ; A d e p o j u a n d O p p o n g 1 9 9 4 ) ; p a r t i c u l a r l y in t e r m s o f t h e g e n d e r e d d i v i s i o n o f l a b o r in r u r a l s o c i e t i e s ( B e n e r i a 1 9 8 2 ) . W h i l e m u c h o f t hi s a n a l y s i s h a s f o c u s e d o n t h e v u l n e r a b i l i t y o f w o m e n i n b e a r i n g t h e b r u n t o f e c o n o m i c c h a n g e , it h a s a l s o s t n r t e d h n e w a p p r e c i a t i o n f o r t h e e x t e n s i v e k n o w l e d g e s y s t e m s t h a t e x i s t e d i n p r e c o l o n i a l w o m e n ’s a c t i v i t i e s . S i n c e m a n y o f t h e s e p r a c t i c e s c o n t i n u e t o d a y ( a l b e i t i n m o d i f i e d f o r m ) , w o m e n ’s i n d i g e ­ n o u s k n o w l e d g e sy s t ems h a v e b e c o m e a n i m p o r t a n t r e s o u r c e in n e w a p p r o a c h e s to d e v e lo p m e n t. S o m e obvious challenges in cl ude e n v i r o n m e n t a l d am ag e (increasing salin­ ization, def o r es t a t i o n , a n d d e s er t i f i c a t i o n ) , e x t e r n a l e c o n o m i c pressures ( t h e m o v e to c a s h - c ro p p in g , tou ri sm , a n d m i g r a t i o n to cities; abus e of p o w e r by pr i va te c o r p o r a t i o n s ) , incr eas ed disease ( A I D S a n d o t h e r viruses), poli ti cal u nr es t ( e t h n i c conflict, u n c o n t r o l l e d m ili ta r y force, abuse of a u th o r it y ) , a n d dam a g e t o t h e s o c i o c u l t u r a l s y s t e m ( d i s r u p t i o n s o f w o m e n ’s t r a d i t i o n a l a u t h o r i t y , loss of t r a d i t i o n a l k n o w l e d g e s y s t e m s ) . W h i l e al! o f t h e s e a r e far t o o l a r g e t o be a d d r e s s e d by a n y o n e a p p r o a c h , n o n e o f t h e m c a n be v i e w e d i n i s o l a t i o n f r o m t h e o t h e r s . I n Ni g e r i a, for e x a m p l e , t h e S h e l l P e t r o l e u m D e v e l o p m e n t C o m p a n y I m p l ic a tio n s 228 b e g a n o p e r a t i o n s in O g o n i l a n d t h a t e v e n t u a l l y le d t o w i d e s p r e a d e n v i r o n ­ m e n t a l d a m a g e ; a t t e m p t s t o p r o t e s t t h r o u g h t h e p re s s a n d o t h e r c o m m u n i c a ­ t i o n e v e n t u a l l y led t o t h e e x e c u t i o n o f O g o n i w r i t e r K e n S a r o - W i w a ( S o y n i k a 1 9 9 4 ) . F r e e d o m o f t h e p re s s is n o t a s e p a 'r a t e issu e from,, p r o t e c t i o n o f t h e e n v iro n m e n t. It is righ t to decry abuse o f a u th o rity , b u t r e p la c in g o n e a u t h o r i t y w ith a n o t h e r is n o t n e c e s s a r i l y g o i n g t o p r o v i d e a l o n g - t e r m s o l u t i o n . A f r i c a n f r a c ta l s s u g g e s t t w o a l t e r n a t i v e a p p r o a c h e s . F ir s t, w h a t is n e e d e d is n o t E. F. S c h u m a k e r ’s c a ll fo r " s m a ll is b e a u t i f u l , ” b u t r a t h e r , a s e l f - o r g a n i z e d a p p r o a c h t o c h a n g e s in t h e re la tio n s b e tw e e n scale a n d th e s o c i o e n v ir o n m c n ta l sy stem s— n o t just a p p r o ­ p ria te tec h n o lo g y , b u t a p p ro p r ia te scalin g . S e c o n d , 'm o r e c ritic a l a tt e n ti o n n e e d s t o be p a i d to t h e a r t i f i c i a l / n a t u r a l d i c h o t o m y , w h i c h t e n d s t o b e t r a p p e d in e i t h e r t h e o r g a n i c i s t s ’ d e s i r e fo r u n t o u c h e d n a t u r e ( e .g ., H u g h e s 1 9 9 6 ) , o r t h e t e c h n o - o p t i m i s t ’s d e s i r e fo r r e s o u r c e e x t r a c t i o n . A n a l t e r n a t i v e t o t h e s e d a m a g i n g e x t r e m e s c a n b e f o u n d in C a l e s t o u s j u m a ’s 1 9 8 9 classic, T h e G e n e Fh m ters. R a t h e r t h a n a p r e s e r v a t i o n i s t p e r s p e c ti v e , i n w h i c h i n d i g e n o u s s o c i e t y w o u l d b e p o r t r a y e d as n a t u r a l e l e m e n t s o f a n u n c h a n g i n g e c o s y s te m , o r a t e c h n o c r a t i c p r o f ite e r in g p e r s p e c t i v e , . i n w h i c h a g ri­ c u l t u r a l d e v e l o p m e n t is m e r e ly a q u e s t i o n o f m a x i m i z i n g y ie l d s w i t h i m p o r t e d s tra in s , J u m a p ro v i d e s e v i d e n c e for in d i g e n o u s a g r i c u l t u r a l a c t i v i t y as s u s t a i n a b l e b i o t e c h n o l o g y . H is s t u d i e s s h o w a l o n g - s t a n d i n g A f r i c a n t r a d i t i o n o f n e w s e e d v a riety d e v e lo p m e n t th a t c o m b in e d eco lo g ical su s ta in a b lilty w ith in n o v a tio n a n d e x p e r i m e n t a t i o n . T h e s e p r a c t ic e s h a v e b e e n t h r e a t e n e d hy c o r p o r a t e m o n o c r o p p i n g , w h i c h c a n c a u s e soil d e p l e t i o n , o v e r - d e p e n d a n c e ora i n s e c t i c i d e s , loss o f g e n e t i c v a r i a t i o n , a n d o t h e r s o c ia l a n d e c o l o g i c a l crises, as w e ll as t h e a p p r o ­ p r i a t i o n o f t h e s e g e n e t i c re s o u r c e s by a b i o t e c h n o l o g y i n d u s t r y w i t h li t t l e i n t e r ­ est in i n d i g e n o u s le g a l rig h ts . J u i n a n o t e s t h a t t h e c h a l l e n g e n o w f a c i n g A f r i c a n a g r i c u lt u r a li s t s is n o t j u s t p r e s e r v a t i o n o f b io d iv e rsity , b u t a ls o a c c e s s to t h e legal, t e c h n i c a l , a n d f i n a n c i a l a p p a r a t u s t h a t w o u ld a ll o w t h e m t o r e a p t h e p ro fit t h a t c o u l d s u s t a i n s u c h e c o l o g i c a l l y s o u n d efforts. F r o m t h e v i e w p o i n t o f c o m p l e x i t y th e o r y , J u m n ’s c r i t i q u e s u g g e s ts t h a t w e a re t r a p p e d b e t w e e n t h e p e r i o d i c sta sis o f t h e p r e s e r v a t i o n i s t s ’ li m i t c y c le , a n d t h e w h i t e n o i s e o f t h e p r o f i t e e r i n g p o s i t i v e f e e d b a c k l o o p . A s w e saw in t h e s e m a t h e m a t i c a l m o d e l s , b o t h a r e l a c k i n g in f l e x i b l e i n t e r a c t i o n s w i t h m e m o r y ; t h e l i m i t c y c le b e i n g t o o t i e d t o it, a n d t h e w h i t e n o i s e b e i n g t o o free f r o m it. I n f o r m a t io n te c h n o l o g ie s h a v e t h e p o t e n t i a l to p r o v i d e th is m e m o ry , d o c u m e n t i n g i n d i g e n o u s k n o w l e d g e fr o m s e e d v a r i e t i e s a n d soil t y p e s t o g e n e s e q u e n c e s to e c o t o p e s . By p r o v i d i n g i n f o r m e d ru ra l a c c e s s t o i n f o r m a t i o n t e c h n o l o g i e s , A f r i c a n a g r i c u l t u r a l i s t s c a n ta k e a s t e p t o w a r d p r o t e c t i n g t h e i r g e n e t i c re s o u rc e s Futures fo r A fr ic a n fractals f r o m a p p r o p r i a t i o n a n d m o v e t o w a r d J u m a ’s a p p r o a c h , w h i c h we m i g h t call “ b i o t e c h - d i v e r s i t y ” (cf. H a r a w a y 1 9 9 7 ; S h i v a 1 9 9 7 ) . T o v i e w i n d i g e n o u s k n o w l e d g e as a s e l f - o r g a n iz i n g s y s t e m is o n e t h i n g , b u t c r e a t in g t h e same b o t t o m - u p a p p r o a c h for a synth esis o f ecological sustainabli li ty a n d t e c h n o l o g i c a l ' d e v e l o p m e n t is a m u c h gr e a t e r c h a l l e n g e . For e x a m p l e , Rus s e l Bar s h n o t e s : “T h e r e is a n i m p l i c i t a s s u m p t i o n in t h e r e s e a r c h m e t h o d o l ­ ogy used t o e l i c i t t r a d i t i o n a l p h a r m a c o l o g i c a l k n o w l e d g e t h a t t hi s i n f o r m a t i o n is r e c o r d e d a n d t r a n s m i t t e d d i g i t a l l y ( n u m b e r s a n d / o r w o r d s ) . . . [ r a t h e r t h a n ] internalizing an an a lo g m o d e l ” (19 9 7 , 3 3 - 3 4 ) . N a t i v e Seeds, a b o t a n i c a l o rg an i z at i o n d ed ic a te d to t h e c o n t i n u a t i o n of indi genous p l a n t st ock, h a s b e e n c r e a t i n g a “ c ult ural m e m o r y b a n k ” t h a t will record b o t h a n a l o g a n d d i g i t a l i n f o r m a t i o n o n N a t i v e A m e r i c a n a g ri c u lt u r e . T h e c o n ­ cept, originating from P hilippine e th n o b o t a n is t Virginia N azarea-Sandoval ( 1 9 9 6 ) , d o c u m e n t s t h e c o m b i n a t i o n o f c u l t u r a l a n d b i o l og i c a l i n f o r m a t i o n a b o u t t h e crops , seeds, f a r m i n g , a n d u t i l i z a t i o n m e t h o d s . T h e i n f o r m a t i o n , i n c l u d i n g v i d e o i n t e r v i e w s , is s t o r e d o n C D - R O M , w i t h a cc e ss c o n t r o l l e d e n t i r e l y by t h e i n d i ge no u s farmers. In t h e U . S . c o n t e x t , w h i c h is o v e r l o a d e d w i t h e l e c t r o n i c t e c h ­ nology a n d e t h n o c i d e , this a p p r o a c h ma k es sense, b u t th e A fr i c a n c o n te x t , w i t h its e n o r m o u s i n d i g e n o u s p o p u l a t i o n a n d s p a r se e l e c t r o n i c t e c h n o l o g y , will call for t e c h n i q u e s t h a t c a n h a v e a w i d e r i m p a c t , o n e t h a t i n c l ud e s d e v e l o p m e n t o f a t e c h n o l o g i c a l i n f r a s t r u c t u r e as well. If t h e r e is t o be soci al t r a n s f o r m a t i o n t h r o u g h grass-roots t e c h n o l o g i c a l i n n o ­ v a t i o n , it will r e q u i r e m u c h m o r e p a r t i c i p a t i o n t h a n a g r i c u l t u r a l s y s t e m s a l o n e . O t h e r k i n d s o f i n f o r m a t i o n t e c h n o l o g y d e v e l o p m e n t c o u l d i n c l u d e f l exibl e e c o ­ n o m i c n e t w o r k s , w h i c h a l l o w s ma l l -scale business''- > • c o l l a b o r a t e in t h e m a n u ­ f a c t u r e ot p r o d u c t s a n d se r vi c e * u . c y c o u l d n o t p r o d u c e i n d e p e n d e n t l y . T h e s e n e t w o r k s h a v e c r e a t e d s t r o n g r e v i t a l i z a t i o n i n c e r t a i n r ur al a r e a s o f E u r o p e ( Sabel a n d Piore 1990), a n d h a v e s h o w n p r o mi se in pil ot studies in t h e rural U n i t e d S t a r e s as wel t (e.g., A C E n e t in s o u t h e r n O h i o ) . T h e use o f c o m p u t e r s t o o r g a ­ nize p r o d u c t i o n a n d v e n d i n g a n d p r o v i d e d y n a m i c s e a r c h e s for t h e a p p r o p r i a t e m a r k e t n i c h e — o n e w h i c h w o u l d b e e n v i r o n m e n t a l l y a n d soci all y s u s t a i n a b l e as wel l as pr of i t a b l e— c o u l d s p r e a d t h e b e n e f i t s o f n e w i n f o r m a t i o n t e c h n o l o g i e s to r he mi crobusi ness level w i t h o u t h a v i n g t o p ut a lap t o p in every pushcart, anti microf i n a n c i n g p r o g r ams h a v e al r ea d y p r o v e d successful in m a n y T h i r d W o r l d c o u n t r i e s (Sera geld in 1997). A f r i c a n t r a d i t i o n s o f d e c e n t r a l i z e d d e c i s i o n m a k i n g c o u l d a l s o be c o m ­ b i n e d w i t h n e w i n f o r m a t i o n t e c h n o l o g i e s , c r e a t i n g n e w f or ms t h a t c o m b i n e d e m o c r a t i c r u l e w i t h c o l l e c t i v e i n f o r m a t i o n s h a r i n g . T h e i d e a o f “e l e c t r o n i c d e m o c r a c y " h a s s l owl y b e e n d e v e l o p i n g o v e r t h e I n t e r n e t ; b u t t h e ef f o r t s h a v e 229 230 I m p l ic a tio n s b e e n h a m p e r e d by t h e t e n d e n c y t o a s s u m e t h a t v i r t u a l v o t i n g m u s t b e t h e s a m e as o r d i n a r y v o ti n g . P e r h a p s t h e n e u r a l n e t sty le o f A f r i c a n d e c i s i o n m a k i n g c o u ld b e u t i l i z e d i n t h e W e s t as w e ll, w i t h v o t e r s i n d i c a t i n g p r o p o r t i o n a b s t r e n g t h s fo r v a r i o u s o p t i o n s . C o n v e r s e l y , p e r h a p s t h e r e a r e w a y s to” a p p l y c o m p u t e r m e d i a t o e n h a n c e A f r i c a n d e c i s i o n m a k i n g . O n e a p p r o a c h w o u ld b e t h e d e v e l o p ­ m e n t o f c o m m u n i t y n e t w o r k s t h r o u g h p u b l i c - a c c e s s t e r m i n a l s ( S c h u l e r 1 9 9 5 ). A n d t h e e n o r m o u s d e v e l o p m e n t in e l e c t r o n i c s e c u r i t y m e a s u r e s , c r e a t i n g sy s­ te m s t h a t sty m ie e v e n t h e m o s t s o p h i s t i c a t e d h a c k e r s ( e n c r y p t i o n c o d e s , fin g er­ p r i n t s c a n n e r s , e t c . ) , m i g h t f i n d u s e s in p r e v e n t i n g v o t e r f r a u d t h a t is so c o m m o n in u n s t a b l e p o l i t i c a l r e g i m e s . N ig e ria n A m e r ic a n c o m p u te r e n g in e e r E g o n d u O n y e je k w e h a s started e ff o rts t o a p p ly i n f o r m a t i o n t e c h n o l o g y n e t w o r k i n g in A f r i c a n d e v e l o p m e n t a l p r o j e c t s u s i n g c o m p l e x i t y t h e o r y a s a g u i d i n g p r i n c i p l e . O n e a r e a s h e c it e s is th e p r o b l e m o f la n d o w n e r s h i p (fo r e x a m p l e , see C h a r n l e y 1 9 9 6 ) . S h e n o t e s t h a t t h e c o n t i n u a l d i v i s i o n o f l a n d p r o m o t e d by t h e c o l o n i a l le g a c y o f t e n r e s u l t s in u n p r o d u c t i v e e c o n o m i e s o f s c a le , b u t t h a t g o v e r n m e n t o w n e r s h i p t e n d s t o m a k e c o n d i t i o n s w o rse hy a d d i n g m o r e h ie r a r c h y . “R e s o l v i n g t h e l a n d p r o b l e m re q u ires a n o n - h i e r a r c h i c a l m e t h o d o f o r g a n i z a t i o n , a sy s te m in w h i c h c o o p e r a t i v e b e h a v ­ i o r is r e w a r d e d a t t h e s a m e t i m e t h a t i n d i v i d u a l i n n o v a t i o n c a n fl o u ris h ; a c o m ­ b i n a t i o n o f c o o p e r a t i o n a n d c o m p e t i t i o n like we see in c e ll u la r a u t o m a t a a n d o t h e r c o m p u t a t i o n a l m o d e l s o f s e l f - o r g a n iz i n g sy s tem s . W h a t b e t t e r w ay to e n c o u r a g e t h i s t h a n t h r o u g h c o m p u t i n g a n d i n f o r m a t i o n n e t w o r k s ? ” ** N e i t h e r t h e A f r i c a n f r a c ta l s f r a m e w o r k n o r d i s s e m i n a t i o n o f i n f o r m a t i o n te c h n o l o g ie s offers p a n a c e a s . M y p o i n t is, rath er, t h a t t h e sh ift in p e rs p e c tiv e often c a l l e d for in d e v e l o p m e n t n e e c t n o t b e e i t h e r c o n s e r v a t i v e r e t u r n t o -the p a s t, n p r t h e e p i s t e m o l o g i c a l e q u i v a l e n t o f a n a l i e n i n v a s i o n . A f r i c a n f r a c t a l s o ff e r a f r a m e w o r k t h a t is b o t h r o o t e d i n i n d i g e n o u s c u l t u r e s a n d c r o s s - p o l l i n a t e s w i t h n ew hybrids. APPENDIX -Measuring--------------------------------------------- th e fractal---------------------------------------------dimension---------------------------------------------o f African---------------------------------------------settlem ent--------------------------------------------architecture-------------------------------------------- T h e r e a re s e v e r a l d i f f e r e n t w ays co e s t i m a t e t h e f r a c t a l d i m e n s i o n o f a s p a t ia l p a t t e r n . In t h e c a s e o f M o k o u l e k (fig. 2 . 4 o f c h a p t e r 2 ) we h a v e a b l a c k - a n d w h i t e a r c h i t e c t u r a l d i a g r a m , w h i c h a l l o w s us to d o a t w o - d i m e n s i o n a l v e r s i o n o f che r u l e r size v e r s u s l e n g t h p l o t s w e sa w i n c h a p t e r 1. By p l a c i n g t h e a r c h i ­ t e c t u r a l d i a g r a m o f M o k o u l e k u n d e r g rid s o f i n c r e a s i n g r e s o l u t i o n , a n d c o u n t ­ in g tire n u m b e r o f g rid c e lls t h a t c o n r a i n s o m e p a r t o f t h e d i a g r a m , we c a n p l o t t h e in c r e a s e o f a r e a w i t h d e c r e a s i n g c e ll size ( j u s t as we o b t a i n e d a p l o t o f t h e i n c r e a s i n g l e n g t h w i t h d e c r e a s i n g r u l e r size). F ig u r e a .i s h o w s t h e re s u lts , i n d i ­ c a t i n g a f r a c t a l d i m e n s i o n o f 1.67— n o t t o o far f r o m t h e i .53 f r a c t a l d i m e n s i o n t h a t is o b t a i n e d a n a l y t i c a l l y fr o m t h e c o m p u t e r s i m u l a t i o n . F o r t h e a e r i a l p h o t o o f L a b b a z a n g a (fig . 2 .5 o f c h a p t e r 2 ) w e h a v e a n im ag e in s h a d e s o f gray, a n d t h e s i m p l e g r i d - c o u n t i n g m e t h o d c a n n o t be app lie d . It is p o s s i b l e t o r e d u c e t h e g ra y s c a l e t o b l a c k a n d w h i t e , b u t a n a l t e r n a t i v e m e t h o d a ll o w s us to m a k e a m o r e d i r e c t m e a s u r e o f t h e s c a l in g p r o p e r t i e s . F i g ­ ure a . 2a s h o w s _ t h e m e t h o d for f i n d i n g t h e s c a l i n g s l o p e o f 1 / F n o is e in a o n e ­ d i m e n s i o n a l t i m e s e r ie s hy a p p l y i n g a F o u r i e r t r a n s f o r m . In fig u re a .2b w e see h o w t h i s c a n b e a p p l i e d to a t w o - d i m e n s i o n a l s p a t i a l d i s t r i b u t i o n by s w e e p ­ in g t h e s a m e s p e c t r a l d e n s i t y m e a s u r e a r o u n d in p o l a r c o o r d i n a t e s . R a t h e r t h a n tire l i n e o f o n e - d i m e n s i o n a l 1 / F n o i s e , a t w o - d i m e n s i o n a l d i s t r i b u t i o n is A p p en d ix 232 c h a r a c t e r i z e d b y a c o n e . It is d i f f i c u l t t o s h o w t h e e n t i r e c o n e , b u t w e c a n t a k e h o r i z o n t a l s l i c e s (fig. a .2b), w hich show b a z a n g a a n d its f r a c t a l s i m u l a t i o n (fig. a sim ilar c h a r a c t e r i s t i c s f o r b o t h L ab - .3) log (cell size) FIGURE A.J M e a s u r i n g t h e f r a c ta l d i m e n s i o n o f M o k o u l e k frequency O n e - d i m e n s i o n a l t i m e s e r i e s f o r 1/ F n o i s e . 1/ F n o i s e s p e c t r a l d e n s i t y from i - D Fo u rier transform . low fre q u e n c ie s a t h ig h p o w e r j-D F o u rier tra n sfo rm , w ith fre q u e n c y in p olar coo rd in ates: w id e r c ir c le = h i g h e r fre q u e n c y . T h e l i n e o f 1/ F n o i s e is r o t a t e d t o b e c o m e a c o n e . h ig h fre q u e n c ie s at low p o w e r F I G U R E A . 2 .. U s i n g a 2 ' D F o u r i e r t r a n s f o r m to d e t e c t f r a c ta l s p a tia l d is tr ib u tio n s low fre q u e n c ie s a t h ig h p o w e r h ig h fre q u e n c ie s a t low po w er a h ig h f re q u e n c ie s a t lo w p o w e r lo w fre q u e n c ie s a t h ig h p o w e r FIGURE A .3 R e s u l t s o f a 2 - V F o u r ie r t r a n s f o r m a p p l i e d to a e r ia l p h o t o o f L a b b a z a n g a (a) Spectra for aerial photo of Labhazunga (fig. 2.50 from chapter 2). (b) Spectra for fractal image (Jig. 2.5b from chapter 2). Note that the axes of symmetry in the fractal can he seen in this spectral density distribution, while none exist for that of Labbazanga. Notes' c h a p te r i i n t r o d u c t i o n to fr a cta l g e o m e t r y j. For a hexagon example, see Washburn and Crowe (1988, 237). Numerical examples can be found in Crump (1990, 39-40, 50-54, 105-106, 128-133). 2 . The number 10 was not only a basis for counting, but it also appeared in Chinese nat­ ural philosophy. In acupuncture, for example, the number 10 is created by the combi­ nation of the "five elements" (wu-yiin) and the binary yin/yang. 3. Michael Polanyi (1966) referred to this as "tacit knowledge.” c h a p te r 2 Fractals in A f n c c . n settlement architecture 1. O n triangular churches, see Norberg-Schulz (1965, 172); for che Pantheon, see ibid., 124. 2. Another passage, “path of the serpent,” is used only by royalty. It alternates left and right as it approaches the center of the palace, and thus creates a scaling zigzag pattern. The implication seems to be that even royalty must negotiate the fractal ranking, but they can traverse it in a more direct route. 3. American readers are probably most familiar with nuclear families, but in Africa the family structure typically extends to much larger networks. The English term "cousins,” for example, emphasizes the nuclear family by .lumping all these relatives together, while many African kinship systems have distinct terms for paternal parallel cousins, mater­ nal parallel cousins, paternal cross cousins, etc. 4. T he status difference between front and back is also expressed in the Ba-ila term for slave: “one who grows up at the doorway” (Smith and Dale 1968 [ 1920I vol. 1, 304). 5. This is another meaning for the term "participant simulation.” In the first meaning, briefly mentioned in the introduction, I defined it as an effort in cooperative modeling and analysts, a rechnologized version of recent attempts in collaborative ethnography by some anthropologists and their informants. In that sense it supports che humanist goals 236 N otes of self-governing autonomy. But in the Mokoulek case I am also using it in the post­ modernist sense, a participant in a virtual world. The contrasting meanings and their consequences are discussed in detail in chapter 10, where the two are brought together. 6. T h e results were published in Eglash and Broadwetl (1989), and are reproduced in the appendix. "" ch ap ter 3 F ra c ta ls in c ro ss -c u ltu ra l co m p ariso n 1. in general, anthropologists divide nonstate societies between “band” organization, which is entirely decentralized and based mainly on consensus, and “tribal” organiza­ tion, in which there is an official leader but otherwise little political hierarchy. The term "tribe” is controversial, however, since colonialists often used it to deny the existence of indigenous state societies, so it is important ro separate the technical designation front its colloquial use2. This is a complex designation in cultural studies, since the label of “traditional’'— or worse yet, "authentic”— was used by colonial authorities to exercise control over indigenous populations, and still o c c u r s in the neocolonial context to valorize the ‘‘van­ ishing native” while appropriating their cultural resources. See Minh-ha (1986), Anzaldua (1987), Clifford (1988), and Bhaltba (1990) for discussion of some of these issues. 3. Crowe and Nagy (1992), for example, have done extensive analysis of Fiji decoration, and found 1 2 o f the 17 mathematically possible two-color strip symmetries, but none of the designs they show are fractal. 4. Of course, nothing is absolutely certain when it comes to ancient history. Several researchers have suggested that the Coptic designs from Egypt were an important influence 011 the Celtic interlace patterns, and some Italian floor tiles were created by North African artisans (Argiro 1968, 22). But one could just as easily argue the influ­ ence in reverse. Given the history of trade routes and travel, we should not attempt to reduce designs to a singular origin; the goal is to see how any one society has built up its particular repertoire of designs— from whatever sources— as part of a dynamic yet culturally cp. <ific practice. chapter 4 Intention and invention in design 1. This spatial metaphor of “underlying”— truth beneath the surface— can be a delusion if we assume that there is never more than one true "essence” to be found. On the other hand, claiming that no model is more accurate a generalization than any other is equally misguided. 2. T h e postwar era marked a significant change in the role of nature as a potential model for scientific discovery, as seen in the emerging disciplines of cybernetics and bionics (Gray 1995). chapter 7 Numeric systems 1. It is unfortunate that an otherwise excellent paper comparing African and Australian ethnomathematics (Watson-Verran and Turnbull 1994) fails to make this distinction between the iterative generation of linear and nonlinear number series. 2. Readers who recall the definition of nonlinear functions as involving, at minimum, some­ thing tike may be puzzled by the idea of a nonlinear additive series. That is because most of us were first exposed to the definition of “nonlinear” in the context of continuous functions (e.g., differential equations). But discrete iteration (what is often called a “dif­ ference equation") can produce nonlinear steps with simple addition. 3. After giving a lecture on Bamana divination in the United States, I was approached by a mathematics faculty member who was quite taken by this phrase. "That’s just like us,” he exclaimed. "We get the power of mathematics only at the cost of our social defor­ mity as nerds.” N u ic s T h e s e r i e s w a s first i n t r o d u c e d a s a n e x a m p l e o f a r e c u r s i v e l y c o m p u t a b l e a p e r i o d i c s t r i n g b y A x e l T h u e ( 1 8 6 3 - 1 9 2 2 ) , u s in g rh e r e p l a c e m e n t rules tial 0. 0 -4 0 1 , 1 —» 10, w ilh an in i­ M o r s e d i s c o v e r e d it s a p p l i c a t i o n t o d e t e r m i n i s t i c c h a o s , i n w h i c h it m o d e l s i h c fra ctal tim e series p r o d u c e d by c e r ta in n o n l in e a r e q u a tio n s. S e e S c h r o c d e r ( t y y i , 2 6 4 - 2 6 8 ) o n these aspects o f the seq u en ce. 5 . O n e - d im e n s io n a l version s c a n s h o w all th e d y n a m ic s o f tw o d im e n sio n s, an d ca n e v e n b e used as a k i n d o f p a r a l l e l c o m p u t e r . C o n s i d e r , fo r e x a m p l e , a ru le t h a t in e a c h i t e r a t i o n t h e n u m b e r o f c o u n t e r s i n a c u p is r e p l a c e d b y t h e s u m o f i t s e l f a n d its l e f t neigh bor. S tartin g w ith on e: 0100000 —4 0110000 —4 0 1 2 1000 -4 0133 1 0 0 —4 0 1 4 6 4 1 0 . T h i s f o u r t h i t e r a t i o n g i v e s us c h e b i n o m i a l c o e f f i c i e n t s f o r e x p a n s i o n o f ( a + b ) “*, w h ic h e q u a ls a4 + CHAPTER 3 I. 4 a 3b + 6 a 2b2 + 4 ab^ + iA Recursion T h e s t a n d a r d t e r m i n o l o g y is s o m e w h a t a m b i g u o u s , s i n c e “ r e c u r s i o n ” is s o m e t i m e s u s e d t o r e f e r s p e c i f i c a l l y t o w h a t w e w i l l c a l l “ s e l f - r e f e r e n c e , " a n d a t o t h e r t i m e s it is u s e d i n t h e m o r e g e n e r a ! s e n s e a p p l i e d h e r e . “ I t e r a t i o n ” is u s e d in it s n o r m a l d e f i n i t i o n , a n d fo r t h e l e a s t p o w e r f u l w e w i l l u s e t h e t e r m “ c a s c a d e . ” T e c h n i c a l l y , t h e s e t h r e e t y p e s o f recu rsio n ro u g h ly co r r e sp o n d to T u r i n g m a c h in e s , p u s h - d o w n a u t o m a ta , an d finites ta te a u t o m a t a , b u t th e s e m o d e l s are a lit tle t o o a b s tr a c t to b e d ir e c t ly useful in h e l p ­ in g readers d e v e lo p a sen se o f t h e d is t i n c t io n s t h a t are o f in terest h ere. 2. S a g a y ( 1 9 8 3 ) e x p lic it ly m e n tio n s s ta rtin g w i t h th e sm all sh a p e in th e ce n te r, w h erea s t h e I p a k o E le d e row s lo o k lik e t h e y m ig h t b e b e t t e r d e s c r ib e d as a p r e e s ta b lis h e d lin ear s e q u e n ce (a lth o u g h S a g a y d o es n o t g iv e details h e re ). 3. A c t u a l l y , it is n o t w a x t h a t is u s e d i n m u c h o f A f r i c a , b u t r a t h e r a l a t e x c r e a t e d b y b o i l ­ i n g t h e s a p o f c h e E u p h o r b i a p l a n t . W i l l i a m s n o t e s t h a t it c a n p r o d u c e l o n g , d e l i c a t e th rea d s th a t are im p o ssib le for w a x . 4 . P e lto n (19 8 0 , 230) co n trasts rh e sin g u la r r a n d o m e v e n ts o f th e N a tiv e A m e r i c a n t r i c k s t e r m y t h s w i t h “ t h e le s s e p i s o d i c , m o r e n a r r a t i v e m y t h s o f L e g b a a n d O g o - Y u r u g a [ i n A f r i c a ] . ” T h e r e a s o n f o r c h e d i f f e r e n c e is p a r t l y m a t h e m a t i c a l . T h e N a t i v e A m e r i ­ c a n c o n c e p t o f u n p r e d i c t a b i l i t y is b a s e d m o r e o n c h a n c e ( s e e A s c h e r 1 9 9 1 , 8 7 - 9 4 ) , w h i l e t h e A f r i c a n c o n c e p t t e n d s co b e c l o s e r t o d e t e r m i n i s t i c c h a o s , as w e s a w in B a m a n a sand d iv in atio n . 5. C u r t i n ( 1 9 7 1 ) s h o w s t h a t c h e s l a v e t r a d e f r o m w h a t is n o w n o r t h e r n S e n e g a l d i m i n ­ i s h e d a f t e r 1 7 0 0 , a n d t h a t t h e N i g e r i a n a r e a d i d n o t b e g i n m a j o r a c t i v i t y u n t i l aft er. 1 7 3 0 . T h i s still le a ve s th e p ossib ility chat F uller c a m e fro m th e area o f p re s e n t-d a y B e n i n an d ^ G h a n a " w h i c h w o u l d 'i Y iT t 0 6 ' f a r s o u t h ' t o h a v e ' d i r e c t l y s h a r e d i n f l u e n c e s w i t h t h e B a s s a r i , b u t H o l l o w a y ( 1 9 9 0 , 1 0 ) n o t e s t h a t V i r g i n i a n s s h o w e d s o m e p r e f e r e n c e for A f r ic a n s from the S e n e g a in b ia n regio n . 1. 1 q u a lif ie d th is as “ s t a n d a r d ” b e c a u s e t h e r e h a s b e e n a g r o w i n g c o n c e r n t h a t a n t h r o ­ pologists m ay h a v e o ve re m p h a s ize d th e im p o rta n c e o f age-grade an d kin sh ip by p ro ­ j e c t i n g t h e i r o w n d e s i r e s a s w e l l a s c h e i n t e r e s t s o f t h e i r i n f o r m a n t s . S h a w ( 1 9 9 5 ) , fo r e x a m p le , s h o w s h o w L o u is L ea ky's d e s c r ip tio n o f th e e x tr e m e o b e d ie n c e o f th e K ik u y u to th e ir ag e -g ra d e syste m w a s c o l o r e d b o t h b y L e a k y's d e sire for che o rd er o f a “sm a ll E n g lish v illa g e ” th a t h e n e v e r e x p e r ie n c e d ( h a v in g g r o w n u p w ith m ission a ry p aren ts) a n d t h e K i k u y u e l d e r s ’ o w n i n t e r e s t s in r e c e i v i n g t h e i n i t i a t i o n p a y m e n t s t h a t w e r e o v e r ­ d u e to th em . ’ . In a d d i t i o n t o che a s s o c i a t i o n o f t h e v e r t i c a l w i t h t h e s p ir itu a l, F e r n a n d e z su g g e s ts th a t t h e s p a t i a l d i s t i n c t i o n d e r i v e s f r o m t h e F a n g ’s p e r i o d i c c l a n f i s s i o n / r e l o c a t i o n . T h e f r a g ­ m e n t a t i o n o f a s o c i a l g r o u p c o m e s w i t h h o r i z o n t a l m o v e m e n t a n d is s e e n a s t h e r e s u l t o f s t a g n a t i o n o r s t r i f e , w h i l e t h e e s t a b l i s h m e n t o f t h e g r o u p i n a n e w l o c a t i o n is s e e n as p o s it iv e r e g e n e r a t i o n , b u i l d i n g f r o m t h e g r o u n d up. L M a u r e r a n d R o b e r t s ( 1 9 8 5 , 2 5 ) d e s c r i b e che T a b w a b e lt, a le a th e r strip w i t h b a n d s o f b e a d s o r w i r e a s r e p r e s e n t a t i o n s o f a s i n g l e d e s c e n t l i n e . S i n c e t h e T a b w a use c h e m p a n d e Notes 238 disk to represent the expansion of all kinship groups from a singular origin, it is not unreasonable to think of the irmkaba belt as a lower-dimensional projection of the tnpancle disk. If one is willing to speculate so wildly that even [ would hesitate to do so, the aardvark's winding tunnel could be viewed^as a three-dimensional spiral pro­ jected onto the two-dimensional mpande disk, just as the belt is a'bne-dimensional projection of the mpande spiral. A similar practice, the “Poincar£ slice,” is used in non­ linear dynamics (see Abraham and Shaw .982). 9. It is important to understand that the problem is not one of “authenticity.” 1 agree with the critiques of modernist anthropology’s tendency to make one individual represen­ tative of an entire society and to focus on a false homogeneous past. In ethnomathematics we are interested in the invention of mathematical concepts; so it doesn't matter whether the source is an- entire society or a single creative individual. What does matter is the precision and accuracy of che math, and it is here that the interpretive flexibility offered by narratives presents problems. jo. Note that I wrote "has trouble with” rather than “cannot do"— in fact, a programmer could write a kind of “metaloop” of iteration that would figure out how many nestings are needed. But in doing so, the program has to be able to refer to a part of itself (its loops), so this is already a partial or limited self-tefetence. Of course we could then play the same trick, demanding that we can’t tell ahead of time how many metaloops wili be needed, and our sfnarty-pants programmer could again make a meta-metaloop, and soon. It is only when we generalize the trick itself that full self-reference-will be required. And even then, it too will meet up with undoable tasks— because that very property of not bounding the process ahead of time leaves it vulnerable to other problems. As Alan Turing proved for computing, and as Kurt Godel showed for all mathematics in general, any system that is sufficiently powerful to fully utilize self-reference will have to be incomplete in its ability to resolve all the theorems it can ask (see Hofstadter 1980). 11. The most specific connection made by Taylor is the possibility that the material attrib­ uted to Hermes-Thoth was derived from some of che Egyptian priesthood writings men­ tioned by Clement of Alexandria. 12. Stephan ides (1922, 192) suggests a more direcc sub-Saharan origin of alchemy, enter­ ing Egypt around 7 1 8 b . c . e ., following the invasions o f Ethiopia. 13. T h a t’s not to say that the Legba drum beats were random; but the drumming did indeed have an unexpected change of pacechapter 10 Complexity J, The analog/digital dichotomy in computing is often confused with other dualisms. The same terms are used by engineers to describe the continuous/discrete dichotomy, and by cognitive scientists to discuss “reasoning by analogy” versus inductive analysis, but these distinctions are irrelevant to the sense in which it is used here. Musical nores, for example, are excellent examples of analog communication, but they are entirely dis­ crete. See Eglash (1993) for details. 2. Blum et al. show that an analog Turing machine would be susceptible to tire halting problem. See Eglash (1992, 1998c) for more details on this recent history of cybernetics. We can think of the wave/particle duality in physics as another indication that the analog/digital distinction is fundamentally egalitarian. 3. We can also look at this in terms of psychopathology. A neurotic will often repeat the same phrase over and over, while a psychotic tends to be talking “word salad,” a jumble of nonsense. In both cases, their mental relation to memory is pathologi­ cally simplified: the neurotic slavishly follows memory, whilethe psychotic completely ignores it. Complex information processing requires a dynamic interaction with memory, a nontrivial recursive loop. 4. For example, say there are choices A, B, and C. A wins, bur B and C voters say, "If only I trad known A was going to win, I would have been willing to vote the other way.” N otes Tank and Hopfield (1987, 106) contrast this one-shot majority rule voting with the collective-decision-making process in neural nets: “In a collective-dectsion commit­ tee the members vore together and can express a range of opinions; the members know all about the other votes and can change their opinions. The committee gener­ ates . . . whac might be called a sense of the meeting."--5. Recall that we scaled down P to a number between 0 and 1 . That means that (1 - Pn ) will always he a fraction, which reduces P,i— in face, the larger Pn, the smaller the fraction. 6. The reason it never lands back on exactly the same spot is not because of external noise; it is rather for the same reason that the number P never repeats. Gottfried Mayer-Kress suggested that a good way to understand this is to note that the drunken driver never scops missceering, even while che sober one is overpowering him. I suspect that this com­ bination of negative feedback and positive feedback is at the heart of every case of deter­ ministic chaos, although I have yet to prove it. in Eglash (1992) I reported that che Lorenz attractor consisted of only positive feedback, but this turns our to be incorrect. In terms of dynamical systems theory (Abraham and Shaw 1982; Devaney J986), positive feedback is the counterpart to spreading in phase space, and negative feedback corresponds to folding in phase space. The phase-space combination of local spread­ ing and global folding is a common definition for chaos; the conjecture simply trans­ lates the phase-space definition into a control theory formulation. 7. I’ve oversimplified the relations here. For example, a finer distinction can be made about ‘'disorder” if we consider white-noise versus brown-noise distribution on a surface (Gardner 1978; Voss 1990). In Brownian motion, a particle moves in a random, con­ tinuous trajectory; given an infinite amount of time, such “brown noise” will approach a two-dimensional curve. In white noise, single points on che surface are selected at random, so an infinite amount of time will still only leave us with disconnected points, which is a zero-dimensional curve. Between zero and one dimension, we have objects like the Cantor set, and between one and two dimensions we have objects like the Koch curve. This is slightly different when we chink about noise as a single time-varying sig­ nal (as in acoustic noise) because the single points of the white distribution will also be connected into a continuous (but nondifferentiable) curve, now of dimension one, while brown noise as a time series will still be at dimension two. 8. Achebe himself prevents such a reading by highlighting a precoloniaf corastrophe that befalls his main character, Okonkwo. At the same time, the contrast between Okonkwo’s misery due to indigenous accident and his suicide as a result of che colonial encounter makes it clear rhaf these are entirely different orders of chaos. y. There is also a good illustration of collective fractal generation in the arts: che Mbutt bark-cloih design shown in chapter 3 is actually llie producr of multiple artists. c h a p te r ) 1 T h e o r e tic a l /rm n e iu o r k s in c u ltu r a l stu d ie s o f k n o w le d g e 1. Popper might object to the characterization of “fractal geometry minus dimensional mea­ sures,” since it sounds like an ad hoc adjustment, but the important thing is that the four attributes (scaling, recursion, infinity, and dimension) were tested in a more or less falsifiable manner. Whether or not one can still call it fractal geometry if one of the four is missing is an important question; but we need to address the possibility of a weak characterization of recursion in European fractals before making that judgment. 2. This should not necessarily he assumed to mean "closer to nature,” since it could also refer to an indigenous knowledge system chat promotes good ecological practices; but the ambiguity is problematic. 3. In fact I’m not— my master’s degree is in systems engineering, anti although I rook a few graduate seminars in mathematics for my interdisciplinary Ph.D. (thanks to die flex­ ibility of the History of Consciousness board ar the University of California at Santa Cruz), 1 wouldn’t dare call myself a mathematician in professional company, 1 have always 239 N otes 240 tried to introduce myself as an crhnomnthematician during field work, but sometimes translation problems took time to get that across. 4. Worth it not just in ethical and methodological terms; it often came to my aid in dire circumstances. On a hot road neat the Lake Chad region, I was stopped by military police who were clearly looking for a bribe. I was released only when I began to launch into a lengthy explanation of fractal geometry. Knowing the Baka counting system saved my skin when a group of teenagers in a village in southern Cameroon took me for a disrespectful tourist; unlike the gendarmes, they were delighted to find mathematics in their midst. 5. O n the role of neologisms in the work ofCesaire, see Clifford (1988). On the construction of negritude as a set of binary oppositions, see Mud ini be ( jy88). 6. For example, the octopus arose, millions of years before vertebrates but has a nervous system more sophisticated than that of some reptiles (see Eglash 1984, 161). This is a dangerous analogy, of course, because people often confuse biological and cultural evo­ lution. Here are two crucial differences. First, cultural evolution is Lamarckian— we can pass our acquired knowledge to the next generation— while biological evolution is Darwinian, with the rare lucky mutant having an advantage that is then passed on. Second, the timescales are of different orders of magnitude. Significant biological evolution requires on the order of a million years, while dramatic cultural evolution requires no more than a few thousand years. This is why human beings have such a tiny amount of genetic variation: the first modern humans, from their singular origin in Africa, quickly spread across the earth over a few thousand years. Our nearly identical genetic composition is a result of speedy Lamarckian cultural evolution adapting us into these new environments. c h a tte r 12 The politics o/ A f r i c a n f r a c t a ls 1. Derrida's promotion of arbitrary signifiers and artificiality was not the sole voice for this position. Black activists like James Boggs (1968) have also been champions of artifice. Wittig’s (1973) L e s b ia n B o d y takes a topic that was often created as the unassailable ground of feminist meaning, the authentic physical self, and dismantles this construction through textual erotics. Like Derrida, she shows that a system of arbitrary symbols is just as capable of carrying the kind of human essence often attributed to the Real or Natural. 2. Angela Davis has pointed out Ellison’s denaturalizing tropes in lectures at UCSC; her recent work continues to tease out these threads of self-assembly in black cultural iden­ tity and community. 3. My favorite illustration of analog artifice in black intellectual works occurs in chap­ ter 11 of Audre Lourd's Zflmi. Like Witfig (1973), she describes the self-assembly of a lesbian body, hut her techniques for this artificial reconstruction come through the ana­ log media of scent, vibration, and form. See Eglash (1995) for other examples. 4. Consider, for example, the mojo hnnd/dataglove comparision in Dery (1994, 210), or the following passage from Williams (1974, 40): “ ‘Simply anything can become a God,' a Yoruba informant once remarked. ‘This button (pointing to the dashboard of the car in which we were), ‘it only needs to he built up hy prayer’ (hy invocation).” 5. Similar views can be found in several other intellectual works of the time; e.g., Joreen’s (1972) critique of the women's movement, "Tyranny of Structurelessness." There ate, of course, many centralist critiques of decentralization, but Joreen's text cook a more complex angle of analysis. See Ehrlich (1979) for a critical view. Invocations of African royalty in black cultural representations are typically viewed as commen­ tary on self-esteem. While that may he true, in most cases there are hints that it. also serves to question the humanist control enacted in a political democracy that can sup­ port such deep economic subservience (see Queen Lntifa's "Queen of Royal Badness” in Smith J990). N otes 6. In face, this was how 1 got started on African fractals. It occurred to me that aerial photos might show the difference between these architectural designs as fractal ver­ sus Euclidean. Pat Caplan generously provided me with aerial photos of the area in which she worked, and the indigenous housing did indeed appear to be less Euclidean. 7. Recursive architectural structure is linguistically indicated by the Yoruba term for homestead: ot k a 0 1, or "“house within the house.” 8. The 1993 Supreme Court ruling in Sfiau/ v. Reno used the terms “bizarre” and “snakelike,” the latter echoing historian John Fiske’s 1812 characterization of a "dragon Iike” contour, a phrase changed to "salamander” and finally to “gerrymander" (after Mass­ achusetts governor Elbridge Gerry) by political cartoonist Gilbert Stuart. 9. The insistence that stochastic variation implies free will and deterministic variation implies domination is made by several authors besides Porush (e.g., Hakim Bey). I think that individuals or groups can indeed create such associations, just as they can create the opposite (e.g., that a simple bounded system can still have the liberty of infinite variation, as we will see argued by Gilroy, Van Wyk, and Heaver). The error is in assum­ ing universal meaning to what has to be local semiotics. A closer examination of the social meanings for statistics (Porter 1986) reveals that its political associations are often dependent on modernist concepts of humanist individualism, which is strongly critiqued in the Foucaultian and other postmodernist analyses championed by Porush, Hayles, Sobchack, and others. 10. Just as important is the reverse influence, e.g., Jewish jazz musician' Mezz Mezzrow pass­ ing for black while in prison so that he could play in the band. 11. Gilroy’s work in this area should be seen as part of a larger community of researchers and cultural workers (e.g., artists) who have developed a postmodern emphasis on hybrid-, ity, creolization, and other impure identities (cf. Minh-ha 1 9 8 6 ; Anzaldua 1 9 8 7 ; Bhabba 1 9 9 0 ; Sandoval 1 9 9 5 ; Haraway 1 9 9 6 ) . 12. Digital and analog are also confusing terms because digital technology is now commonly used to generate the analog waveforms of music. But it is necessary to see how these representations are layered. The electronic “on-off” .code pulses are actually noisy waveforms that must be processed with analog control circuits at the lowest level of the silicon chip; eventually they are decoded in binary form, then converted to an elec­ trical waveform that will modulate the speaker. T h e resulting acoustic waveform can be analog, digital, or— especially in the case of rap music— somewhere in between. See Eglash (1993) for details. chapter 13 Frncwls in t/ie Europemi h i s t o r y o f m a t h e m a t ic s 1. According to ancient accounts, the discovery of irrationals was in the middle of the fifth century b.c. e. Modern scholars generally agree that the proof for the incommen­ surability of the square of a diagonal with respect to its side, first mentioned explicitly in Plato's dialog T/ienems, is too abstract to have been used at this time. Von Fritz.(1944) provides a resolution for this conflict in his speculative reconstruction of Hippasus’ analy­ sis of the pentagon. See Knorr (1975) and Fowler (1987) for discussion of the origi­ nal texts relevant to this area. 2. Plato was not the only influence at the time, nor were irrationals only granted one per­ spective. Fowler (1987), for example, maintains rhat the significance of irrationals has been misunderstood and suggests that even Plato presented their proof as “a source of interesting and fruitful problems” rather than as a disturbing paradox. Nevertheless, it was the homogeneous representations of Platonic thought deployed centuries later, not irs contemporary diversity, which would matter for the intuition and practice of modern mathematicians. 3. "We add to the first number the second one, i.e., 1 and 2 , the second to the third; the third to the fourth; the fourth to the fifth . . . and it is possible to do this order for an infinite number of months” (trans. Maxey Brooke). 241 242 N o te s 4. Similar analysis was provided hy Henry Louis Gates ( 1990) and others in the censor­ ship trial of rap group 2 Live Crew, maintaining that the explicit sexual lyrics were not acultura! profanity but rather modern variations of a long-standing black tradition of public sexual commentary. * v',-« 5. Tuana (1989), for example, notes that the male homunculus theory,‘which locates the active principle of birth in sperm only, dominated European medical thinking from Aris­ totle to van Leeuwenhoek (and in some senses even to the present; see Hnrrouni 1997). Again, the African version is in strong contrast; recall from chapter 8 that the Fang believe that the homunculus or active principle is contained in the female blood (the division is more egalitarian than the European model, however, since the male Fang are said to provide a complementary protective, skeletal principle). 6. That is, prior to complexity theory, at which point advances in the application of frac­ tal geometry were made precisely because of the growing recognition of a relationship between computational recursion and self-organizing phenomena. Complexity theory is a marker distinguishing the transitional postmodernism of the 1970s from the stable postmodernism of the 1980s (Eglash 1998c). 7. The qualification is not inaccurate; the problem is that sometimes the authors of this text { T h e Science o f F r a c t a l Images) use the term “recursion" to mean iteration, and some­ times (as in this case) it means self-referential programming. This level of ambiguity would not be tolerated for any other mathematical terminology used in the text. chapter 14 Futures f o r A f r i c a n f r a c t a ls 1. For more on cyborgs, see Haraway (1996) and Gray (1995). In fact, if I had used a large enough size difference, self-intersection could have been avoided altogether, but 1 think that would not do justice to the African tradition of putting similar-sized houses together— a tradition that has its roots in egalitarian socioeconomic structure, and one to which Nyangula was no doubt sensitive. 3. But there was more to it than that. Perhaps in port because it implied a Platonic view, it made sense to the students chat religious symbolism would be mathematical, while .something as concrete as 0 mud wall was too hard to reimage. There was also the visual effect of seeing computer simulations of the African log spirals; for a generation brought up on video games and MTV, this placed it in a contemporary framework. Finally, there was something about the religious subject matter itself— the very concept of a "life force" expressed as a self-organizing system— that may have created a resonance for these students. 4. 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Sett mod 2 additive scries, 186-189 aesthetics- See esthetics affine transformation, 75 Afiocenirism, 180-181, 218, 222 age-grade, 68, 87, 121, 124, 237116 agriculture, 24, 31, 125, 227-229 Agudoawu, Kofi, 107 Akan, 77-78, 81, 104 alchemy, yy, 100, 101, J40, 141,2381112 algorithm, 38, 47, 6 1, 68, 77, 97, 113, 118, ' 3 3 . «53 —) 5 4 . 17°. ' 7 4 .206 analog, 151-154, 158-161, 164, 192-194,200, 202, 214, 229, 23811m , 2, 240113, 24ini 2 Ananse, 137 Angola, 68, 186 animism, 194 anthropology: authority in, 183—184; function­ alist, 170; mathematical, 185-187, 191; modernist, 131,238; reflexive, 95; struc­ turalist, 18 x, 188. S ue a ls o ethnography apartheid, 184, 200 apei iodicity, 108, 172 .Arabic culture, 98-99, 205 archaeology, 61, 87, 89 architecture: African, 4-8, 19-40, 87-89, 110—111, 124, 126-128, 131, 135, 148-149, 162-164,'^6.174, '95-199, 205, 210, 216-222, 224, 226; American, 3 - 5 . 3 9 . 4 9 - 5 0 . 55. ' 9 7 - ' 9 9 ; Chinese, 4; European, 3, 20, 39, 48-51, 55, 89, 174, 195-196, 225; Indian, 47-48; Mative Am. t k ; n, 39-42; South Pacific, 47 Aristotle, d, 51, 147-148, 205-206, 242 arithmetic, 86-108 arithmetic series. Sec additive series art education, 225 artificial intelligence, 213 Ascher, Marcia, 45, 47, 186, 237114 Ashanti, 137 authenticity, 74, 184, 193-194, 217, 23602, 238119, 240m authority, 31, 133, 183, 186, 203, 227—228 Babbage, Charles, 21 1-212 Badinne, Nfally, 162, 164 Ba-ila, 26-29, 5 5 . ' >o. 23 5 n 4 Bak, Per, 161,170, 226 Baka, 183, 240114 Baker, Houston, 194 Bakuba, 172-173, 222 Baluha, 130, 166, 210 Bambara, Toni Cade, 194 Bamilekc, 24-25 Banneker, Benjamin, 55,90, 182, 183 253 In d e x 254 B an tu, 62 C h a i t i n , G re g o ry , 153 Banyo, 34-36 c h a o s , 9 3 , 9 5 , 103, 108, 143, 159, 162, 168, 1 7 4 , 1 8 2 , 1 9 0 , 1 9 3 , 1 9 7 , 1 9 9 , 2 1 4 , 2 37iM b asket w eaving, 4 5 -4 6 , 222 B nssari, 1 2 1 - 1 2 2 , 2 3 7 0 5 chi w ara, 1 2 4 -1 2 5 , 127, 134, 209 B a ta m m a lib a , 121, 126, 135 C h in e se m ath em atics, 4, 4 7 -4 8 , 185, 225, 23502 B atty , M ic h a e l, 4 9 - 5 0 b ead w ork, j J3 , j 19, 166, 2 3 7 0 5 C h o k w e , 6 1 , 6 8 , 6 9 , 7 0 , 8 4 , 187 B e l l , E r i c T . , 2 0 7 —2 0 8 C h o m s k y , N o a m : c o g n itiv e th e o r y of, 2 11; h ie r a rc h y of, 1 5 6 - 1 5 8 B e m b e , j 23 B enin, 91, 1 2 4 ,1 4 1 -1 4 3 , 166, 182, 216, C h ristia n ity , 20, 4 8 , 9 0 , 127, 1 3 5 -1 3 6 , 149 cities. 237 n 5 See B etg, T Q , 224 c la s s , 81 a rch itectu re B ern o u lli, Ja c o b o , 210 C lif f o rd , J a m e s , 131, 1 8 3 , 1 9 3 , 236112, 2 4 0 0 5 B e y , H a k i m , 2 4 1n 9 co astlin es, 1 5 .1 7 b i n a r y c o d e , 9 5 , 9 8 , 1 01 c o lo n ialism , 1 9 5 -1 9 7 b in o m ia l c o efficien ts, 2 3 7 n s c o m p l e x i t y , 5, 4 5 , 6 8 , 1 4 6 , 1 5 1 - 1 7 6 , 1 8 4 , 1 8 9 , b io lo g ical d e te r m in is m , 187, 191, 2 2 4 - 2 2 5 biology, 3 , 3 4 , 8 4 , 1 0 2 - 1 0 5 , 1 0 7 - 1 0 8 , 1 24, 225, 2 2 8 ,2 3 0 c o m p u t e r : a n a l o g , 1 5 1 —1 5 5 , 1 5 8 - 1 6 1 , 1 6 4 - 1 6 6 ; 1 3 1 ,1 3 3 , 1 4 1 ,1 5 9 ,1 8 9 , 191, 2 2 7 -2 2 9 , c a l c u l a t i o n by, 7 4 , 8 9 , 9 7 , 1 5 1 ; i n d e v e l o p ­ 24006 m en t, 2 2 9 -2 3 0 ; ed u catio n , 2 2 3 -2 2 5 ; h ard ­ w a r e , 9 5 , 9 8 , 1 0 1 ; p r o g r a m s , 1 1 0 —1 1 2 , 1 3 2 , b io tech n o lo g y , 2 2 8 -2 2 9 b irth , 34, 9 0 , 109, 127, 131, 133, 168, 170, 135, 1 3 7 -1 3 8 , 188, 2 11; s im u la tio n , 3 ,1 2 , 21, 28, 31, 3 2 , 34, 38, 6 i , 208, 210, 212, 24205 B lix e n , K a r e n (lsak D in e s e n ), 197 71, 77, io r- ro 4 , 147, 172; th eo ry , 146, 1 5 6 -1 5 8 , 2 1 2 - 2 1 4 B l y d e n , E. W ., 2 0 0 C o n g o . See D e m o c ra tic R e p u b lic o f C o n g o body, 12, 6 3 - 6 5 , 7 5 -7 6 , 1 3 1 -1 3 3 , 164, 226, C onw ay, Jo h n H o rto n , 103 -1 0 4 , (70 c o o r d in a te sy stem s: C a r te s i a n , 3 - 5 , 4 2 , 8 5 , 24011m , 3 196; polar, 2 3 1 -2 3 4 ; sp h erical, 83 Boggs, Ja m e s, 2 4 0 m B ourdier, Jc n n -P a u f, 3 2 - 3 3 C o p t ic d e sig n , 2 3 6 0 4 b raid in g . See h airsty les c o rn ro w s. See h airsty les b rid ew ealth , 89 cosm ology, 4 3 - 4 4 , 4 8 , 1 3 1 - 1 3 5 , 2 0 4 , 210 B ro ad w ell, P eter, 3 1 c o u n t i n g : b a s e six , 122; b a s e t e n , 4 , 9 9 , 2 3 5 0 2 ; base tw o, 8 9 - 9 1 , 100 b ro n ze s c u lp tu re , 1 3 8 -1 3 9 C ro w e , D o n a ld , 4 7. 48 B row n, Jam es, 1 9 9 -2 0 0 b ro w n noise, 2 3 9 0 7 C row ley, A lcister, 99 B u r k i n a F a s o , 3 1 - 3 3 , 182 C ru tc h fie ld , Ja m e s, 1 5 9 -1 6 0 , 174 cy b ern etics, 2 3 6 n 2 , 2 3 8 0 2 B u tler, O c t a v i a , 194 B w am i, 52, 123 • cyborgs, 2 1 6 , 2 4 2 m B w iti, 129 D a n , 1 4 1 - 1 4 3 , 1 6 6 . 170, 175 C airo , 3 7 - 3 8 , 2 0 1 -2 0 2 D a n g b e . See D a n C a m e r o o n , 2 1 —2 5 , 2 9 - 3 1 , 3 4 - 3 6 , 1 1 3 , D a u b e n , J. W ., 2 0 8 1 1 9 - 1 2 0 , 1 3 8 - 1 3 9 , 145, 14 9 - 1 5 0 , 182, D a v i s , A n g e l a , 240112 1 9 0 , 2 1 6 , 2 3 9 0 7 , 240114 d e S o u s a , M a r t i n e , 141 C a n t o r , G e o r g , 8 - 1 0 , 1 9 7 , 2 0 6 —2 0 8 d e S o u z a , F r a n c i s c o , 141 C a n to r, M oritz, 208 d e a th , 3 4 , 164, 170, 2 0 4 ,2 1 4 C a n t o r set, 1 2 - 1 3 , 15, 17, 9 3 , 9 9 , 1 4 7 - 1 4 8 , d ecen tralizatio n , 31, 39, 189, 197, 222, 229, 206-208 236111 C a p la n , P at, 195, 2 4 1 0 6 D e la n y , S a m u e l R ., 194 C a rb y , H a z e l, 194 D e m o c r a t ic R e p u b l ic o f C o n g o , 6 1 , 1 2 7 , 166 C a r v e r, G e o r g e W a s h i n g t o n , 194 D e rrid a , Ja cq u es, 1 9 2 -1 9 3 c a r v i n g , 7, 4 3 - 4 4 . 4 5 , 6 2 - 6 3 , 6 8 , 1 0 8 , 1 13, D escartes, R e n e , 1 9 5 -1 9 6 1 1 7 , 1 2 0 ,1 3 8 , 1 4 3 ,1 6 6 , 187, 189 d e s c e n t , 8 , 1 2 4 - 1 3 1 , 1 4 9 , 2 0 6 , 237118 C a s a m a n c e , 162, 164 d esig n th em e s, 3, 4 , 6, 27, 3 9 - 4 0 c a sc a d e , 1 0 9 - 1 1 0 , 1 n - i 14, 145 D e stn , G e h r e K ristos, 2 1 6 C ay ley tree, 222 d e te r m in is tic c h a o s. Sec c h a o s Cayuga, 186 dev elo p m en t, 2 2 5 -2 3 0 ce llu la r a u to m a ta , 1 0 2 -1 0 8 , 143, 1 5 4 -1 5 5 , diaspo ra, 55, 180, 199 i 58, 162, 164, 168, 170 D in rta , C h r i s t i a n S i n a , 7, 1 6 1 - 1 6 2 , 164 C e l t i c d e s i g n , 7, 4 8 D iaz, R o g c lin , 4 3 - 4 4 C e s a i r e , A i m e , 1 8 8 , 240115 differen tial e q u a tio n s , 2 3 6 0 2 Index diffusion limited aggregation, 49 digital, 101, 104, 151-152, 156-158, 166, 190, 192-194, 200, 211-213, 229, 23811m, 2, 2411112 dimension, 12, 15, 18-19, 32, 43, 81, 83-84, 93, 104, 113, 115, 154, 170-172, 176, 209, 23808, 239007, 1 ^ disease, 17, 227 disequilibrium, 170 divination, 31, 93-101, 108, 122, 124, 133, 143, 151, 183, 190, 209, 237n4 Dogon, 131-134, 138, 140, 146, 170,175 doubling. See counting: base two Du Bois, W.E.B., 200 dynamical systems theory, 23906 East Africa, 86, 99, 216 economics, 189, 196, 211, 217, 223, 227, 229, 240115 education. See art education: mathematics education Egypt. 37-38, 87-89, 9 9 . ‘ 3 4 - 1 3 5 . ‘ 3 7 . 140-141,188-189,‘91.204-208,23604 Ellison, Ralph, 194 engineering, 5, 73-74, 85, 143, 230 Eno, Brian, 101 environment, 20, 39, 50-51, 133, 219, 227-229,24on6 Epimenides of Crete, 111, 137 epistemology, 180, 189, 193, 225, 230 Eshu,174,175 cssentialism, 180-182 esthetics, 7, 38, 50, 52, 53-57, 62—63, 81, 113, 209 ethics, 192, 194-195, 2 10, 24004 ethnography, 28, 31, 45, 127, 131, 181 —184, 200, 203, 223, 235115. See a lso anthropol­ ogy ethnophilosopliy, 149, 189-190 Ethiopia, 101,1 35-136 Euclidean construction method, 65, 68-69, 113, 118 Eulerian path, 48. 68, 70, 186 evolution, 161, 187, 189-190, 240116 Fagg, Will vam, 7, 84, 139, 190 falsili.ibiiity, 6, 179, 240111 Fang, 127, 129, 149, 210, 23707 fetus. See birth Fibonacci series, 87-89, 11o - 1 11, 156, 205-206 finite state automaton, 156-158, 237m fluid flow, 47-48, 77-78, 97, 104, 209, 213 F o n ,190 Foucault, Michel, 189, 194-195, 209, 24109 Fourier transform, 231, 233-234 fractal dimension. See dimension fractal geometry: definition of, 8-19; European history of, 8-17, 203-215. See also com­ puter: simulation; dimension; infinity; recursion; scaling; self-similarity fractions, 204, 205, 23905 free will, 97, 199,24109 Fulani, 29, 113, 119 Fuller, Thomas, 122, 23705 functionalism. See anthropology: functionalist funepal rituals, 164 Gabon, 127 Gambia, 121, 182, 23705 game of life. See cellular automata game theory, 101 Garcia, Linda, 93 Garvey, Marcus, 200 Gates, Flenry Louis, 90, 190, 219, 24204 Gauss, Carl Friedrich, 206 Geertz, Clifford, 181-182 gender, 190, 212-213, 227 genetics, 124, 161, 180, 188, 228, 24006 geometry. See affine transformation; computer: simulation; coordinate systems; dimension; Euclidean construction method; Eulerian path; fractal geometry; graphing; helix; hexagon; iterated function systems; nondifferenciable curve; pentagon; Poincan§ slice; quincunx; scaling; self-similarity; Sierpinski gasket; sinusoidal waves; spiral; tiling; trigonometry geomancy, 98-101 Gerdes, Paulus, 68, 122, 186, 222 Getz, Chonut, 222 Ghana, 74, 77-80, 101, 104-108, 113, 115, 124, 182, 226-227, 237n5 Gilmer, Gloria, 224 Gleick, James, 182 Gcidel, Kurt, 199, 214, 238mo graphing, 4, 12, 14, 47, 73-74, 79, 81,83-85 graphics. See computer: simulation Greek culture, 76, 89, 99, 141, 147—148, 203-206,210, 225 Griaule, Marcel, 131, 133 griot, 164 Guinea-Bissau, 44, 121 hairstyles, 7, 63, 81-84, 112-114 Hausdorff, Felix, 12 Hausdorff-Besicovirch measure. See dimension Heaver, Hannan, 38, 200, 202 Heighway, John, 113 helix, 112, 114 Hermes Trismegistus, 99, 134, 141,238ml HerskovitS, Melville, 107 hexagon, 4, 5, 121—122, 214, 222, 235m "■hierarchy, 39, 120, 122, 156-158, 189, 197, 2 1o, 230, 236m Hindu culture, 99, 185, 187, 225 Hofstadter, Douglas, 1io, 213, 2381110 homosexuality, 213-214 homunculus, 127, 242115 Hughes, David, 218-222 humanism, 194-195, 209 Hurst, H. £., 12, 208-209 255 In d e x 256 Hurston, 2 orn Neale, 188 hybrids, 187, 200, 230, 24in 11 Ibo, 197 lfa, 9 3 ~ 9 5 India, 7, 47-48 infinity, 8-9, 12, 13, 18, 34, 4J-42, 70, 76-77, 91,111,135,138-139,146-150, 153, 157—759, j 76, 790, 204-207, 2jo, 222, 239007,1,24inn9,3 information technology. S e e computer initiation, 68, 87, 100, 121—123, 133, 23706 intentionalt'ty, 5-6,19,49-57,81, 113, 123,. 162, 165, 174, 184-187, 219-220, 225 inunlion, 53, 56-57, 68, 71, 113, 154, 24102 iron work, 61, 89-90, 141, 143 irrational numbers, 97, 204, 24innt, 2 lshango bone, 89, 91 Islam, 29, 3j , 38, 93, 162, 202, 205 iteration, 15, 17, 18, 21, 22, 25, 26, 28, 29, 30, 31*3 4 . 37. 38. 4 5 . 48, 61, 67, 68, 69, 76, 79, 8i, 86-88, 91, 95, 103-104, 1jo- 130, 132-137, 146, 155, 170,172, 176, 210, 2J2, 222, 2370;, 238030, 24207 iterated function systems, 76, 222 ivory sculpture, 62, 63, 65-68 japan,47-48 jew e lry, 5 3 - 5 4 Jews, 99, 101, 200, 202, 207-208, 24m!o Jola, 162-165 juma, Calestous, 228-229 Kabbalah, 99 Kamil, abu, 205 Karnak, 88 Kauffman, Stuart, 170 kente cloth, 74-76, 226-227 Kenyecta, Jomo, j 88 Kepler, Johannes, 206 Kikuyu.209,23706 King, Martin Luther jr., 199 kinship, 24, 113, 124, 127, 130-131, 145, 164, 186,209, 23503 Kirdi, 29 knot theory, 48 Koch, Helge von, 9-15, 17-18 Kolmogorov, A. N., 152-1 53. t 55 knra, 2! 7-218 Kotoko, 21, 24, 32 Kronecker, Leopold, 208 Kuba. See Bakuba Knti, Feta, 200 Kwele, 722-123, 127-328 Labbezanga, 3 1-32, 231-232, 234 labor, 24, 39, 113, 187, 189, 196, 227 Leaky, Louis, 23706 Legba, 143-144, '66, 175, 216,23704, 238013 Leibnitz, Gottfried, 100-101 lightning, 9 ’ ~ 9 3 limit cycle, 106, 143, 228 lineage, 24, 124, 127 linearity, 40-42, 71, 74, 76-77, 86, 121, 129-130, 196, 197, 2i i , 222, 237 linguistics, 193 *’• logic, 4, 28, 70, 98, 111-112, 135, 204, 1 31, 213 Logone-Birni, 21-24 lotus, 335, 137 Lourde, Audre, 24003 Lovelace, Ada, 211-232 Luba. See Bnhiba Lucas, Edouard, 206 Lull, Raymond, 99—101 tangs, *5-17, 34 Malagasy, 98 Malawi, 196 Mali, 8. 31-32, 71-72, 131, 182 mancala, 101 Mandelbrot, Benoit, 12, 15, 17, 47, 51, 93, 176, 197,208-209, 2 ,4 Mandiack, 44, 52 Mangbetu, 61-68, 70 marriage, 119, 124 masks, 80-81, 84, 121-123 mathematics education, 222, 223-225, 236002, 3 Mauritania, 113, 115,218-219 May, Robert, 159, 168 Mayer-Kress, Gottfried, 23906 Mbuti, 54, 23909 measurement, 4, 9, 12-18,31, 72—74, 79, 89, 122,151,153-155,159-160, 172, 174-175, 239m medicine, 17, 127, 196, 24205 memory, 34, 97, 156-159, 161, 166, 174, 228, 229, 238113 metalwork, 7, 112, 216. S e e a ls o bronze sculp tore; iron work Mezzrow, Mezz, 2 4 7 0 1 0 migration, 121, 227 mimesis, 50-53. 56 Mitsogbo, 127-1 29, 149, 2 10 mod two, 95, 98-99 Mofon, 29-31 morphogenesis. See biology Morse, Mnrstnn, 97—98, 237114 Mozambique, 222 Mudimbe. V. Y., 149, 180, 189-190, 194, 24005 multiculturalistn, 206, 225 music, 64-65, 143, 149, 154, 174. 193. 194, 200, 204, 209, 23804, 241010, 242012 Mveng, Engelbert, 149-150. 190 Nanknni, 32—34, 148-149. 210 narrative, 93, 95, 96, 133, 137, 146, 148, 149, 179, 186, 202, 206, 23704, 238119 Native American culture, 40-46, 48, 116, 184, 186,229, 23704 In d e x n a t u r e , 1 7 , 1 8 , 4 7 , 48, 5 0 - 5 3 , 5 6 - 5 7 , 6 2 , 1 4 1 , 149, 180, 1 8 1 , 1 9 0 , 1 93 , 228 , 2 3 6 0 2 , 2.3902 N a z a r e a - S a n d o v a l , V i r g i n i a , 22 9 “ n e g r i t u d e , " 188, 1 9 0 , 1 9 1 , 2 4 o n 5 n e u r a l n e ts , 1 5 2 , 1 5 4 , 1 65 n e u r o b i o lo g y , 1 5 2 , 1 5 6 , 1 8 7 , 1 9 9 , 2 3 8 , 2400 6, N e w A g e m y s ti c is m , 187 N i g e r i a , 24 , 94 , 1 3 7 , 1 7 3 - 1 7 5 , 1 8 7 , 200, 2 2 7 , 230 257 p o w e r law , 7 1 - 7 4 , 8 9 - 9 3 , 1 5 9 - 1 6 1 p r i m i t i v i s m , 53 , 89, 180 , 1 8 8 - 3 8 9 , ! 94> 1 9 6 -19 7 , 224-225 p r o b a b i l i t y , 94. See also c h a o s ; r a n d o m n e s s ; s ta tis ti cs; s t o c h a s t i c v a r i a t i o n p r o g r a m m i n g . See c o m p u t e r : p ro g ra m s pseudorandom n um ber gen eratio n , 9 7 - 9 8 p u s h - d o w n a u t o m a t o n , 1 5 7 —1 5 9 Pythagoras, 203-204 N i l e river, 99 , 2 0 8 - 2 0 9 nomads, 115 n o n d t f f e t e n t i a b l e c u r v e , 239117 Q u e e n L a ti fa , 240115 q u in c u n x ,5 5 ,18 2 n o n l i n e a r i t y , 4 0 - 4 3 , 70, 7 1 , 7 6 - 7 7 , 8 0 - 8 2 , 84, 8 6 - 8 6 , 9 7 , 108, 1 1 3 , 1 1 8 , 1 2 2 , 1 4 3 , 1 6 2 , r a ci s m , 180 , 1 8 7 , 188 1 8 2 , 1 9 0 , 200, 2 1 6 , 2 22 , 2 3 6 0 2 , 2 3 7 0 4 , ra n do m n e s s, 3 1 , 93—99 , 1 5 2 - 1 5 5 , 1 5 8 - 1 6 : , 1 74 , 238 08 n u m b er s , 4, 5, 6, 8, 18, 3 1 , 4 1 , 4 2 , 7 6 , 8 6 - 1 0 8 , 1 22 , 1 5 3 , 1 5 7 , 1 5 9 , 1 8 6 , 190, 2 0 3 - 2 0 6 , 2 1 2 , 2 2 9 , 235112 n u m e r o l o g y , 4, 20, 9 5 , 1 2 1 —1 2 2 , 1 3 4 - 1 3 5 , 204, 235n2 N um m o , 131, 133, 175 1 8 6 , 1 96 , 1 9 7 , 1 9 8 , 2 3 7 0 4 , 2 38 11 13 , 2 3 9 0 7 ra tio s, 204 r e bi rt h . See bi r th re c u rs io n , 8 - 1 2 , 1 6 - 1 7 , 34 , 4 3, 4 5 , 4 7 - 4 8 , 5 5 , 7 7 , 8 6 , 8 9 , 9 3 , 9 5 , 9 8 , 9 9 , 108 , 1 0 9 - 1 4 7 , 1 4 9 , 1 5 1 , 1 5 5 - 1 5 9 , 1 6 1 , 1 7 6 , 1 8 7 , 190, 1 92 , 1 9 4 - 1 9 5 , 1 9 9 - 2 0 0 , 2 0 2, 20 5, 2 0 9 - 2 1 4 , 2 1 7 , N u p e, 137 2 3 7 0 1 , 238113, 2 3 9 m , 2 4 1 0 7 , 2 4 2 0 0 6 , 7. N y a n g u l a , A l e x , 2 2 0 - 2 2 2 , 242112 Se e also ca s ca d e ; it e r a t io n ; s e l f - r e f e r e n c e r e f l e x i v e a n t h r o p o l o g y . See a n t h r o p o l o g y : O du m , H ow ard, 214 O g o n i , 228 O g o t e m m e l i , 131 r e f l e x iv e r e l i g i o n , 7, 20, 28, 3 1 , 4 7 , 48, 5 3 , 7 8 , 9 0 , 92 , 9 3 , 9 9 , 1 24 , 1 2 7 , 1 2 9 , 1 3 1 - 1 3 2 , 1 3 5 , i/F n o is e , 1 5 9 , 1 6 1 , 1 66 1 4 1 - 1 4 3 , 1 64 , 1 6 6 , 1 7 0 , 180 , 1 8 9 , 1 94 , O n y e j e k w e , E g o n d u , 23 0 20 2, 204, 20 5, 2 0 7 , 20 8, 2 1 1 , 2 4 2 n 3 optim ization, 7 3 -7 4 o r i e n t a l i s m , 188 r e p r o d u c t i o n , 1 0 7 - 1 0 8 , 1 24 , 1 2 5 , 1 3 4 , 1 3 8 , 1 4 0 , 2 0 9 - 2 1 0 , 2 1 2 - 2 1 4 . See also b i r t h O R S T O M , 25, 29 owari, 1 0 1 - 1 0 8 rite o f p ass age , 34 P a le s t in e , 89 r o m a n t i c o r g a n i c i s m , 1 94 p a r a d o x , 1 2 , 1 1 1 —1 1 2 , 1 64 , 2 0 3 - 2 0 5 R o s i c r u c i a n i s m , 9 5 , 208 p a r t i c i p a n t s i m u l a t i o n , 29, 182 — 1 8 4 , 2 3 5 n s p e n t a g o n , 204, 2 4 1 m R o u s s e a u , Jea n J a c q u e s , 1 9 2 - 1 9 3 p e r i o d i c i t y , 103 , 106, 1 4 1 - 1 4 3 , 1 5 3 , 1 5 6 , Russell, Bertrand, 211 r it ua l, 3 1 , 68, 9 9 , 1 2 1 , 1 2 3 , 1 2 6 , 1 2 7 , 1 6 2 , 1 6 4 , 1 6 5 , 1 8 0 , 186 R u c k e r , R u d y, 104, 162 158-160 , 1 7 2 - 1 7 3 ,2 2 8 Pe ter, Rdz sa , 2 1 2 - 2 1 3 S a h a r a , 38 , 71 p h a s e s p a c e , 239116 Sah el, 7 1 - 7 4 p h i l o s o p h y , 1 49 , 1 7 9 , 1 8 9 - 1 9 0 , 2 0 3, 235112 S a m p s o n , Ja ro n , 224 p h y s ic s , 7 , 1 5, 50 , 1 1 3 , 1 5 1 - 1 5 5 , 1 5 8 - 1 7 6 , 1 9 4 Su r o -W iw a , K e n , 228 pi, 206 s c a l i n g , 1 2 , 1 7 - 1 9 , 2 1 , 26 , 28—29 , 3 1 - 35> 3^. P l a to , 20 3 —205, 2 1 0 , 2 4 m m , 2, 2 4 2 0 3 p l o t t i n g . See g r a p h i n g 4 1 , 4 3 , 4 3 - 4 8 , 5 2 , 5 4 , 5 6 , 6 1 - 6 3 , 6 5 , 68, 7 0 ,7 1-8 5 ,8 6 , 8 9 , 1 0 4 ,1 i o , 112 -114 , P o i n c a r e sl ic e , 238118 1 1 6 - 1 1 8 , 12 0 - 1 2 4 , 1 2 6 - 1 2 8 , 1 3 0 - 1 3 5 , p o i n t a t tr a c t o r, 106 13 7 , 1 4 1 , 1 4 8 - 1 4 9 , 156, 166, 174 , 175, p ol ar c o o r d i n a t e s , 2 3 1 - 2 3 3 1 90 , 1 96 , 200, 2 0 2, 208, 2 1 6 , 2 2 5 , 22 6 , p o l it ic s , 3 1 , 3 4 , 1 0 1 - 1 0 2 , 120, 1 2 4 , 1 4 5 , 1 7 4 , 2 2 7 , 228 , 2 3 5 0 2 , 2 3 9 m 1 79 , 180, 1 8 9 - 1 9 0 , 1 9 2 - 2 0 2 , 2 2 7 - 2 3 0 , 24 0 0 5 , 24111118, 9 Po pp e r, K a r l, 6, 1 7 9 , 2 3 9 m p o p u l a t i o n , 5, 2 5 , 4 9 - 5 0 , 9 7 , 1 5 9 , 1 6 8 , 1 96 , 1 9 7 , 20 5, 22 9 , 236116 S e h i n n a k e r , E. F., 228 S c h y l e r , G e o r g e , 1 94 s c u l p t u r e , 7, 5 2 , 6 3 , 6 6 , 68, 79 , 80, 8 1 , 84, i t 2 , 113 ,12 7 ,13 3 ,1 3 4 ,1 3 8 -1 3 9 ,2 1 6 se cr et s , 9 3 , 9 7 , 1 2 1 - 1 2 2 , 200, 204 P o r tl a n d B a s e li n e Essays, 1 8 8 - 1 8 9 s e lf - g e n e r a t io n , 9 5 , 9 7 , 100, 1 3 5 , 140, 20 6 , 209 p o s i t i v i s m , 1 79 s e lf - o r g a n i z a t io n , 1 0 1 , 104, 1 0 7 - 1 0 8 , 1 6 1 , p o s t m o d e r n i s m , 1 9 3 - 1 9 4 , 1 99 , 2 1 6 , 2 3 6 n s , 241 n n y , 1 1 , 242116 1 6 4 - 1 6 6 , 168 , 1 70 , 1 7 6 , 1 9 5 - 1 9 7 , 2 22 o , 22 6 , 2 2 8 - 2 3 0 , 2 4 2 0 6 j 8, Index 258 self-organized criticality, 1 6 1 , 1 7 0 , 2 2 6 Tnhwa, 127, 130,237118 s e l f - r e f e r e n c e , 1 1 0 - 1 1 2 , 1 3 5 , 1 3 7 - 1 4 0 , 1 46 ta ll ie s , 1 2 1 - 1 2 2 s e lf - s im il a r it y , 4 , 1 8 - 1 9 , 2 1 > 2 4 > 2 9 >3 1 - 34 > 3 ^. 4 2 , 4 3 , 9 3 , 1 0 0 , 1 2 4 , 1 2 5 , 1 40 , 1 7 6 , 1 9 5 , T a n g , C h a o , 161 209, 2 1 8 S e n e g a l , 8, 5 5 , 8 1 , 9 3, 1 4 0 , 1 6 1 - 1 6 2 , 1 7 4 , 18 2, t a r u m h e t a , 8 6 - 8 7 , 106, 108 .,, 1 83 , 1 9 0 , 1 1 7 - 1 1 8 , 2 3 7 0 5 T a n z a n i a , 8 9 , 1 95 ta tto o patterns, 47 t e x t i l e s , 7, 1 7 2 —1 7 3 S e n g h o r , L e o p o l d , 7, 190 T h o m p s o n , D ' A r c y , 1 90 s e x u a l i t y , 2 0 9 - 2 14 T im e , A x e l, 23704 S h a m m a s , A n t o n , 200, 202 t i l i n g , 1 72 S h a n g o , 90 —9 3 , 1 7 5 T o g o , 124 Shaw, C a ro ly n M artin, 2 0 9 -2 10 , 23706 to u r is m , 3 4 , 2 1 7 - 2 1 8 S i e r p i n s k i g a s k e t (o r t r i a n g l e ) , 1 1 3 , 1 1 5 , triangular num bers, 8 6 - 8 7 , t r ib e , 4 0, 1 8 9 , 203 218- 219 tr ic k st er , 9 9 . i t 6, 1 3 7 , 1 7 4 , 1 7 5 , 182 , 2 1 6 S i m s , j o h n , 2 22 s in u s o i d a l w a v e s , 1 4 1 - 3 4 2 t r i g o n o m e t r y , 68 s la v e s , 108 , 1 2 2 , 200, 235 114 , 2 3 7 0 5 S o l o m o n o f f , R a y , 15 3 T r i n h , M i n - h a , 3 2~3 3 T s w a n a , 200 S o n g l i a i , 3 1 - 3 2 , 1 95 T u r i n g , A l a n , 2 1 3 - 2 14 S o t h o , 200 Turing m achine, s o ul , 3 3 - 3 4 , 1 2 4 , 1 26 tw ins, 8 9 -9 0 , 1 8 1 - 1 8 2 S ou th A frica, t 5, , 0 '^ 5 7 — 1 5 9 , 238112 t 184 , 200 S o u t h P a c i f i c , 3 9 , 4 7 - 4 8 , 186 S o w , F a t o u , 18 3 spectru m , 5 - 6 , 49, 5 1 - 5 2 . 56, 1 7 2 - 1 7 3 , 176, 231-234 S p i l l e r s , H o r c e n s e , 194 spiral, 2 3 - 2 4 , 29 , 3 1 , 4 5 , 4 7 - 4 8 , 7 6 - 7 9 , 8 1 , 8 6, 1 0 4 - 1 0 5 , 1 0 7 - 1 0 8 , i i 2 , 1 2 9 - 1 3 0 , 148, 1 62 , 364, 2 j o , 2 1 6 , 22 4 , 226, 23808, 242113 sp i ri t, 4, 28, 3 1 , 8 9 - 9 0 , 1 1 3 , 1 1 9 , 1 2 1 , 1 2 4 , 1 2 6 , 1 2 7 , 1 2 9 , 1 3 1 , 1 4 1 , 148 , 1 74 , 1 7 5 , 1 8 6 , 18 8 , 1 9 3 , 194, 200, 2 0 4, 2 3 7 0 7 U l a m , S t a n i s l a w , 102 V a n W y k , G a r y , 200 v i d e o , 99 , 2 2 6 - 2 2 7 , 2 2 9 , 2 4 2 0 3 v i r t u a l c o n s t 1 net i o n , 2 1 , 29 , 1 83— 1 8 4 , 2 1 3 , 230 , 235115 v o d u n , 9 0 - 9 3 , 9 4 - 9 5 , 1 4 1 - 1 4 3 , 1 44 , 1 6 6 , 170, ■74. ' 7 5 1 t 83, 190, 1 94 , 2 1 6 , 2 3 8 0 1 3 , 24004 v o n N e u m a n n , J o h n , 1 0 1 - 1 0 2 , 108 v o o c l o o . See v o d u n voting, 1 6 4 - 1 6 5 , 229 -230 S p i v a k , G a y a t r i , 184 W a sh b u r n , D orothy, 48, 187 s q u a r e ro o t , 205 W e s t , C o r n e l , 194 s t a t e , 3 9 - 4 0 , 5 1 , 189 , 2 3 6 m w h i t e n o i s e , 1 5 4 — 1 5 5 , 1 5 8 - 1 6 1 , 1 7 3 - 1 7 4 , 228, s ta ti s t ic s , 18, 2 4 1 0 9 sta tus , 2 6 - 2 9 , 5 5 , 68, 2 3 5 6 5 2 39n 7 W i e n e r , N o r h p t t , 21 4 s t o c h a s t i c v a r i a t i o n , 9 3 , 24 111 9 W o l f r a m , S t e p h e n , 1 0 6 , 1 5 s . 1 58 S t o l l e r , Pa u l, 3 1 , 1 95 S t o n e w o r k , 29 , 1 0 3 , 1 1 3 , 1 3 5 - 1 3 7 . 1 8 5 , 1 96 , 2 10 s t o o ls , 5 5 - 5 6 s tr u c t u r a l i s m , i 8 t , 188 S u d a n ,8 1,1 3 5 s y m b o l s , 6, 7, 8, 20, 24, 3 4 , 4 2, 4 3, 5 5 , 7 1 , 7 7 —7 8 , 9 3 - i o t , 1 0 8 - 1 0 9 , 120, 1 2 6 —1 2 8 , W o l o f , 162 w om b, 34, 133, 212 w o m e n , 24 , 3 2 - 3 4 . 9 0 , 1 2 4 , 1 9 5 , 2 0 0 , 204, 2 1 2 - 2 1 3 , 2 2 2 - 2 2 7> 24 ° n 5 , 2 4 2 0 4 Y o r u h a . 8 1 . 8 2 , 1 1 2 , 1 1 3 , 1 1 8 , 1 7 4 . 1 8 3 , 190. 1 96 , 240114, 241117 1 3 1 , 139, 1 4 5 , 1 4 7 , 1 5 1 - 1 5 2 , 1 5 6 - 1 5 8 , Z a i r e . See D e m o c r a t i c R e p u b l i c o f C o n g o 1 6 4 ,17 9 ,18 1-18 2 ,18 6 ,18 8 ,19 2 -19 4 , Z a m b i a , 8, 26 , 2 2 0 - 2 2 2 1 96 , 2 0 8 , 2 1 1 , 2 4 0 0 1 , 2 4 2 n3 Z e n o o f E l ea , 2 0 3 - 2 0 5 sym m etry, 7 , 3 1 , 4 2 - 4 3 , 4 5 - 4 7 , 79, ' ' 3 , ' '8 , 1 8 6 - 1 8 7 , ' 9 0 , 1 9 7 , 2 2 2 , 236113 S y r i a , 89 Z h a h o r i n s k y r e a c t i o n , 1 0 4 , 162 Zim babw e, Z u l u , 222 io o , 196,200