Untitled Note

African

Fractals

MODERN COMPUTING

AND INDIGENOUS DESIGN

RON EGLASH

Contents-

Acknowledgments

ix-

PART I

Introduction

CHAPTER I

Introduction to fractal geometry

CHAPTER 2

Fractals in African settlement architecture

CHAPTER 3 Fractals in cross-cultural comparison

CHAPTER 4

Intention and invention in design

PART Il

African fractal mathematics

CHAPTER 5 Geometric algorithms

CHAPTER 6

Scaling

CHAPTER 7

Numeric systems

CHAPTER 8 Recursion

109

CHAPTER 9 Infinity

CHAPTER 10 Complexity

147

151

vii

viii

Contents

PART 1I1

Implications

CHAPTER I1 Theoretical frameworks in cultural studies of knowledge

CHAPTER 12 The politics of African fractals

192

CHAPTER 13 Fractals in the European history of mathematics

203

CHAPTER 14 Futures for African fractals

216

179

APPENDIX

Measuring the fractal dimension of African settlement architecture

Notes 235

References

Index

253

243

231

- Acknowledgments

Thanks to go first to my wife, Nancy Campbell, who has tolerated my obsessions

with grace, and to Evelyn, Albert, and Joanne Eglash, who inspired many of them.

I am grateful for the assistance of my professors ar UCSC: Ralph Abraham, Steve

Caton, James Clifford, Donna Haraway, Gottfried Mayer-Kress, Ken Norris, Heinz-

Onio Peiigen, Carolyn iviartin Shaw, and Patricia Zavella. Equally important were

my fellow UCSC graduate srudents, in particular David Bitin, Julian Bleecker,

Peter Broadwell, Kirby Bunas, Claudia Castenada, Giovanna Di Chiro, Joe

Dumit, Vincente Diaz, Paul Edwards, Linda Garcia, Jennifer Gonzales, Chris Gray,

John Hartigan, Sharon Helsel, Laura Kang, Lorraine Kenny, Matthew Kobbe, Angie

Rosga, Warren Sack, Meg Satterthwaite, Sandy Stone, Marita Sturken, Bernt Wahl,

and Sarah Williams. Thanks also to Billie Harris, Miranda Hays, Rebecca Lyle,

Ken Martin, Sheila Peuse, Adolph Smith, Joshua Weinstein, and Paul Yi.

Research funding from the Institute for Intercultural Studies and the Ful-

bright program made possible my fieldwork in west and central Africa. As chap-

ter 10 makes clear, I owe much to my Senegalese colleagues, Christine Sina Diatta

and Nfally Badiane. Also of great help in Senegal were Abdouli Ba, Real Basso,

Charles Becker, Kolado Cisse, Ibnou Diagne, Pathé Diagne, Souleymane Bachir

Diagne, Mousse Diop, Waly Coly Faye, Max, Marie-Louise Moreau, Margot

Ndiaye, Victor Sagna, Ousman Sen, Fatou Sow, Yoro Sylla, Sakir Thaim, and Riene

Acknowledginents

Tôje. From the West African Research Center I received the expert advice of

American professors Eileen Julien and Janis Mays. Thanks also to Shamita

Johnson, Paul and Betsey Harney, Jane Hale, Lisa McNee, and Liz Mermin.

1 am also grateful to Issiaka Isaac Drabo and the brilliant Canadian pho-

tography team, Michel et Didi, in Burkina Faso. Thanks also to Amadou

Coulibaly, Kalifa Koné and Abdoulaye Sylla in Mali. In Cameroon I received the

generosity of Ireke Bessike, Ngwa Emmanuel, Noife Meboubo, the late Engel-

bert Mveng, and Edward Njock. My work in Benin would not have been possible

without the assistance of Tony Hutchinson; thanks also to Kake Alfred, Natheli

Roberts, and Martine de Sousa for their expertise in vodun. In Ghana Michael

Orlansky graciously introduced me to the many cultural resources available. Many

of the local folks I spoke to in west and central Africa, while extending great gen-

erosity and enthusiasm, asked that their names remain unrecorded, and I thank

them as well.

On my return to the United States I received a fellowship from the Cen-

ter for the Humanities at Oregon State University, which also offered the oppor-

tunity to work with anthropologists Joan Gross, David Gross, and Cort Smith,

as well as Kamau Sadiki from the Portland Black Educational Center. Thanks also

to Michael Roberson for his geometry advice, and David and Barbara Thomas

(now math teachers at Hendersonville High, North Carolina) for investigating

owari patterns. A two month fellowship at the University of Oregon got me

through the summer, and into my current position at The Ohio State Univer-

sity. Here I have been thankful for help from Patti Brosnan, Wayne Carlson,

Jacqueline Chanda, Cynthia Dillard, David Horn, Lindsay Jones, Okechukwu

Odita, Egondu Rosemary Onyejekwe, Robert Ransom, Dan Reff, Rose Kapian,

Carolyn Simpson, Daa'iyah 1 Saleem, Jennifer Terry, Cynthia Tyson, and

Manjula Waldron.

There are also many collegues, geographically scattered, whose feedback has

been invaluable. In particular I would like to thank Madeleine Akrich, Jack

Alexander, Mary Jo Arnokli, George Arthur, Marcia Ascher, Jim Barta, Silvio

Bedini, TQ Berg, Jean-Paul Bourdier, Geof Bowker, Michael T. Brown, Pat

Caplin, Brian Casey, Jennifer Croissant, Don Crowe, Jim Crutchfield, Ubiratan

D'Ambrosio, Ronald Bell, Osei Darkwa, Marianne de Laet, Gary Lee Downey,

Munroe Eagles, Arturo Escobar, Florence Fasanelli, James Fernandez, Marilyn

Frankenstein, Rayvon Fouché, Paulus Gerdes, Chonat Getz, Gloria Gilmer,

David Hakken, Turtle Heart, Deborah Heath, David Hess, Stefan Helmreich, Dar-

ian Hendricks, David Hughes, Sandy Jones, Esmaeli Kateh, Roger P. Kovach, Gelsa

Knijnik, Bruno Latour, Murray Leaf, Bea Lumpkin, Robin Mackay, Carol Malloy,

Benoit Mandelbrot, Mike Marinacci, Joanna Masingila, Lynn McGee, James

Acknowledgments

Morrow, David Mosimege, Brian M Murphy, Diana Baird N'Diaye, Nancy Nooter,

Karen Norwood, Spurgeon Ekundayo Parker, Clifford Pickover, Patricia Poole,

Arthur Powell, Dean Preble, Dan Regan, Jim Rauff, Sal Restivo, Pierre Rondeau,

John Rosewall, Rudy Rucker, Nora Sabelli, Jaron Sampson, Doug Schuler, Patrick

(Rick) Scott, Rob Shaw, Enid Schildkrout, David Williamson Shaffer, Larry

Shirley, Dennis Smith, George Spies, Susan Leigh Star, Paul Stoller, Peter Tay-

lor, Agnes Tuska, Gary Van Wyk, Donnell Walton, D M Warren, Dorothy Wash-

burn, Helen Watson-Verran, Mark W. Wessels, Patricia S. Wilson, and Claudia

Zaslavsky. Last and not least, thanks to my editors at Rutgers, Doreen Valentine

and Martha Heller.

—Introduction

PART

CHAPTER

-Introduction

"tO

fractal-

geometry

- Fractal geometry has emerged as one of the most exciting frontiers in the

fusion between mathematics and information technology. Fractals can be seen

in many of the swirling patterns produced by computer graphics, and they have

become an important new tool for modeling in biology, geology, and other nat-

ural sciences. While fractal geometry can indeed take us into the far reaches.

of high-rech science, irs patterns are surprisingly common in traditional African

designs, and some of its basic concepts are fundamental to African knowledge

systems. This book will provide an easy introduction to fractal geometry for

people without any mathematics background, and it will show how these same

categories of geometric pattern, calculation, and theory are expressed in

African cultures.

Mathematics and culture

For many years anthropologists have observed that the patterns produced in dif-

ferent cultures can be characterized by specific design themes. In Europe and Amer-

ica, for example, we often see cities laid out in a grid pattern of straight streets

and right-angle corners. Another grid, the Cartesian coordinate system, has

long been a foundation for the mathematics used in these societies. In many works

Introduction

of Chinese art we find hexagons used with extraordinary geometric precision—

a choice that might seem arbitrary were it not for the importance of the num-

ber six in the hexagrams of their fortunetelling system (the I Ching), in the anatomy

charts for acupuncture (liù-qi or "six spirits"), and even in Chinese architecture.'

Shape and number are not only the universal rules of measurement and logic;

they are also cultural tools that can be used for expressing particular social ideas

and linking different areas of life. They are, as Claude Lévi-Strauss would put it,

"good to think with."

Design themes are like threads running through the social fabric; they are

less a commanding force than something we command, weaving these strands

into many different patterns of meaning. The ancient Chinese empires,

for

example, used a base-zo counting system, and they even began the first univer-

sal metric system.? So the frequent use of the number 6o in Chinese knowledge

systems can be linked to the combination of this official base 1o notation with

their sacred number six. In some American cities we find that the streets are num-

bered like Cartesian coordinates, but in others they are named after historical

figures, and still others combine the two. These city differences typically corre-

spond to different social meanings—an emphasis on history versus efficiency, for

example.

Suppose that visitors from another world were to view the grid of an

American city. For a city with numbered streets, the visitors (assuming they could

read our numbers) could safely conclude that Americans made use of a coordi-

• nate structure. But do these Americans actually understand coordinate mathe-

matics? Can they use a coordinate grid to graph equations? Just how sophisticated

is their mathematical understanding? In the following chapter, we will find our-

selves in a similar position, for African settlement architecture is filled with remark-

able examples of fractal structure. Did precolonial Africans actually understand

and apply fractal geometry?

As I will explain in this chapter, fractals are characterized by the repeti-

tion of similar patterns at ever-diminishing scales. Tradicional African settle-

ments typically show this "self-similar" characteristic: circles of circles of

circular dwellings, rectangular walls enclosing ever-smaller rectangles, and

streets in which broad avenues branch down to tiny footpaths with striking geo-

metric repetition. The fractal structure will be easily identified when we com-

pare aerial views of these African villages and cities with corresponding fractal

graphics simulations.

What are we to make of this comparison? Let's put ourselves back in the

shoes of the visitors from another planet. Having beamed down to an American

settlement named "Corvallis, Oregon," they discover that the streets are not num-

Fractal geometry

bered, but rather titled with what appear to be arbitrary names: Washington, Jef-

ferson, Adams, and so on. At first they might conclude that there is nothing mathe-

matical about it. By understanding a bit more about the cultural meaning,

however, a mathematical pattern does emerge: these are names in historical suc-

cession. It might be only ordering in terms of position in a series (an "ordinal"

number), but there is some kind of coordinate system at work after all. African

designs have to be approached in the same way. We cannot just assume that African

fractals show an understanding of fractal geometry, nor can we dismiss that pos-

sibility. We need to listen to what the designers and users of these structures have

to say about it. What appears to be an unconscious or accidental pattern might

actually have an intentional mathematical component.

Overall, the presence of mathematics in culture can be thought of in

terms of a spectrum from unintentional to self-conscious. At one extreme is the

emergence of completely unconscious structures. Termite mounds, for example,

are excellent fractals (they have chambers within chambers within chambers)

but no one would claim that termites understand mathematics. In the same way,

patterns appear in the group dynamics of large human populations, but these are

generally not patterns of which any individual is aware. Uncorscious structures

do not count as mathematical knowledge, even though we can use mathematics

to describe them.

Moving along this spectrum toward the more intentional, we next find

examples of decorative designs which, although consciously created, have no

explicit knowledge attached to them. It is possible, for example, that an artist

who does not know what the word "hexagon" means could still draw one with

great precision. This would be a conscious design, but the knowledge is strictly

implicit? In the next step along our spertrum, people make rhese components

explicit—they have names for the patterns they observe in shapes and numbers.

Taking the intention spectrum one more step, we have rules for how these pat-

rerns can be combined. Here we can find "applied mathematics." Of course

there is a world of difference berween the applied math of a modern engineer and

the applied math of a shopkeeper--whether or not something is intentional tells

us nothing about its complexity.

Finally we move to "pure mathematics," as found in the abstract theories

of modern academic mathematicians. Pure math can also be very simple-for

example, the distinction between ordinal numbers (first, second, third) and car-

dinal numbers (one, two, three) is an example of pure math. But it would not

be enough for people in a society simply to use examples of both types; they

would have to have words for these two categories and explicitly reflect on a

comparison of their properties before we would say that they have a theory of

Introduction

the distinction between ordinal and cardinal numbers. While applied mathe-

matics makes use of rules, pure math tells us why they work— and how to find

new ones.

— This book begins by moving along the spectrum just described. We will start by

showing that African fractals are not simply due to unconscious activity. We will

then look at examples where they are conscious but implicit designs, followed

by examples in which Africans have devised explicit rules for generating these

patterns, and finally to examples of abstract theory in these indigenous knowl-

edge systems. The reason for taking such a cautious route can be expressed in terms

of what philosopher Karl Popper called "falsifiability." Popper pointed out that

everyone has the urge to confirm their favorite theories; and so we have to take

precautions not to limit our attention to success--a theory is only good if you

try to test it for failure. If we only use examples where African knowledge sys-

tems successfully matched fractal geometry, we would not know its limitations.

There are indeed gaps where the family of theories and practices centered around

fractal geometry in high-tech mathematics has no counterpart in traditional Africa.

Although such gaps are significant, they do not invalidate the comparison, but

rather provide the necessary qualifications to accurately characterize the indige-

nous fractal geometry of Africa.

Overview of the text

Following the introduction to fractal geometry in the next section, in chapter

2 we will explore fractals in African settlements. It will become clear that the

explanation of unconscious group activity does not fit this case. When we talk

to the indigenous architects, they are quite explicit about those same fractal

features we observe, and use several of the basic concepts of fractal geometry in

discussing their material designs and associated knowledge systems. Termites

may make fractal architectures, but they do not paint abstract models of the

structure on its walls or create symbols for its geometric properties. While these

introductory examples won't settle all the questions, we will at least have estab-

lished that these architectural designs should be explained by something more

than unintentional social dynamics:

In chapter 3 we will examine another explanation: perhaps fractal settlement

patterns are not unique to Africa, and we have simply observed a common charac-

teristic of all non-Western architectures. Here the concept of design themes

become important. Anthropologists have found that the design themes found

in each culture are fairly distinct--that is, despite the artistic diversity within

Fractal geometry

each society, most of the culture's patterns can be characterized by specific geo-

metric practices. We will see that although fractal designs do occur outside of

Africa (Celtic knots, Ukrainian eggs, and Maori rafters have some excellent

examples), they are not everywhere. Their strong prevalence in Africa (and in

African-influenced southern India) is quite specific.

Chapter 4 returns to this exploration with fractals in African esthetic

design. These examples are important for two reasons. First, they remind us that

we cannot assume explicit, formal knowledge simply on the basis of a pattern.

In contrast to the fractal patterns of African sertleinent architecture, these aes-

thetic fractals, according to the artisans, were made "just because it looks pretty

that way." They are neither formal systems (no rules to the game) nor do the arti-

sans' report explicit thinking ("I don't know how or why, it just came to me").

Second, they provide one possible route by which a particular set of mathematical

concepts came to be spread over an enormous continent. Trade networks could

have diffused the fractal aesthetic across Africa, reinforcing a design theme that

may have been scattered about in other areas of life. Of course, such origin stories

are never certain, and all too easy to invent.

Part is of this book, starting with chapter 5, presents the explicit design meth-

ods and symbolic systems that demonstrate fractal geometry as an African know!-

edge system. As in the introduction to fractals in the first chapter, I will assume

the reader has no mathematics background and provide an introduction to any

new concepts along with the African versions. We will see that not only in archi-

tecture, but in traditional hairstyling, textiles, and sculpture, in painting, carv-

ing, and meralwork, in religion, games, and practical craft, in quantitative

techniques and symbolic systems, Africans have used the patterns and abstract

concepis of fracta? geometry.

Chapter 10, the last in part it, is the result of my collaboration with an

African physicist, Professor Christian Sina Diatta. A sponsor for the Fulbright

fellowship that enabled my fieldwork in west and central Africa, Dr. Diatta took

the idea of indigenous fractals and ran with it, moving us in directions that 1

would never have considered on my own, and still have yet to explore fully.

In the third and final part of this book we will examine the consequences

of African fractal geometry: given that it does exist, what are its social implica-

tions? Chapter 11 will briefly review previous studies of African knowledge sys-

tems. We will see that although several researchers have proposed ideas related

to the fractal concept —Henry Louis Gates's "repetition with revision," Léopold

Senghor's "dynamic symmetry," and William Fagg's "exponential morphology" are

all good examples-there have been specific obstacles that prevented anthropologists

and others from taking up these concepts in terms of African mathematics.

Introduction

Chapter 12 covers the political consequences of African fractals. On the

one hand, we will find there is no evidence that geometric form has any inher-

ent social meaning. In settlement design, for example, people report both oppres-

sive and liberatory social experiences with fractal architectures. Fractat versus

nonfractal ("Euclidean") geometry does not imply good versus bad. On the

other hand, people do invest abstract forms with particular local meanings. To

take a controversial example, recent U.S. supreme court decisions declared that

voting districts cannot have "bizarre" or "highly irregular" shapes, and several of

these fractal contours have been replaced by the straight lines of Euclidean

form. If fractal settlement patterns are traditional for people of African descent,

and Euclidean settlement patterns for Europeans, is it ethnocentric to insist on

only Euclidean voting district lines?

Chapter x3 will examine the cultural history of fractal geometry and its

mathematical precursors in Europe. We will see that the gaps are not one-sided:

just as Africans were missing certain mathematical ideas in their version of

fractal geometry, Europeans were equally affected by their own cultural views

and have been slow to adopt some of the mathematical concepts that were long

championed by Africans. Indeed, there is striking evidence that some of the

sources of mathematical inspiration for European fractals were of African

origin. The final chapter will move forward in time, highlighting the con-

temporary versions of fractal design that have been proposed by African

architects in Senegal, Mali, and Zambia, and other illustrations of possible frac-

tal futures.

But to understand all this, we must first visit the fractal past.

A historical introduction to fractal geometry

The work of Georg Cantor (1845-1918), which produced the first fractal; the

Cantor set (fig. x.1), proved to be the beginning of a new outlook on infinity. Infin-

ity had long been considered suspect by mathematicians. How can we claim to

be using only exact, explicit rules if we have a symbol that vaguely means "the

number you would get if you counted forever"? So many mathematicians, start-

ing with Aristotle, had just banned it outright. Cantor showed that it was pos-

sible to keep track of the number of elements in an infinite set, and did so in a

deceptively simple fashion. Starting with a single straight line, Cantor erased the

middle third, leaving two lines. He then carried out the same operation on

those two lines, erasing their middles and leaving four lines. In other words, he

used a sort of feedback loop, with the end result of one stage brought back as the

starting point for the next. This technique is called "recursion." Cantor showed

Fractal geometry

that if this recursive construction was continued forever, it would create an

infinite number of lines, and yet would have zero length.

Not only did Cantor reintroduce infinity-as a proper object of mathe-

matical study, but his recursive construction could be used as a model for other

"pathological curves," such as that created by Helge von Koch in 1904 (figs. 1.2,

1.3). The mathematical properties of these figures were equally perplexing.

Small portions looked just like the whole, and these reflections were repeated down

to infinitesimal scales. How could we measure the length of the Koch curve? If

Take a line

Erase the middle

output at each

stage of process

Bring each of the resulting

lines back in and do it agair

FIGURE 1.1

output at

each stage

starting shape

first line replaced

all lines replaced

reduced version

Bring each of the resulting

lines back in and do it again

FIGURE 1.2

The Koch curve

Helge von Koch used the same kind of recuisive loop as Cantor, but he added lines instead of

erasing them. He began with a triangular shape made of four lines, the "seed." He then replaced

each of the lines with a reduced version of the original seed shape.

FIGURE 1.3

Koch curve variations

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 sing variations on the seed shape (a). Nor is there

anything special about triangles—-any shape can'undergo this recursivé replacement process.

Machematician Giuseppe Peano, for example, experimented with rectangular seed shapes such as

those in (b).

Introduction

we hold up a six-inch ruler to the curve (fig. 1.4) we get six inches, but of

course that misses all the nooks and crannies. If we use a smaller ruler, we get

greater length, and with a smaller one even greater length, and so or to infin-

ity. Obviously this is not a very useful way to characterize one of these curves.

A new way of thinking about measurement was needed. The answer was to plot

these different measures of ruler size versus length, and see how fast we gain length

as we shrink the ruler (fig. 1.5). This rate (the slope) tells us just how crinkled

or tortuous the curve is. For extremely crinkled curves, the plot will show that

we rapidly gain length as we shrink the ruler, so it will have a steep slope. For

relatively smooth curves, you don't gain much length as you shrink the ruler size,

so the plot has a shallow slope.

To mathematicians this slope was more than just a practical way to char-

acterize crinkles. Recall that when we first tried to measure the length of the Koch

curve, we found that its length was potentially infinite. Yet this infinite length

fits into a bounded space. Mathematician Felix Hausdorff (1868-1942) found that

this paradox could be resolved if we thought of the pathological curves as some-

how taking up more than one dimension, as all normal lines do, but less than two

dimensions, as flat shapes like squares and circles do. In Hausdorff's view, the Koch

curve has a fractional dimension, approximately x.3, which is the slope of our

ruler-versus-length plot. Being pure mathematicians, they were fascinated with

this idea of a fractional dimension and never thought about putting it to prac-

tical use.

The conceptual leap to practical application was created by Benoit Mandel-

brot (b. 1924), who happened upon a study of long-term river fluctuations by British

civil servant. H. E. Hurst. Hurst.had found that the yearly floods of rivers did not

have any one average, but rather varied over many different scales--there were

flood years, flood decades, even flood centuries. He concluded that the only way

to characterize this temporal wiggliness was to plot the amount of fluctuation at

each scale and use the slope of this line. Mandelbrot realized that this was

equivalent to the kind of scaling measure that had been,used for Cantor's patho-

logical curves. As he began to apply computer graphics (figs. 1.6, 1.7), he found

that these shapes were not pathological at all, but rather very common through-

out the natural world. Mountain ranges had peaks within peaks, trees had

branches made of branches, clouds were puffs within puffs--even his own body.

was full of recursive scaling structures.

The fractal simulations for natural objects in figure 1.7 were created just

like the Cantor set, Koch curve, and other examples we have already seen, with

a seed shape that undergoes recursive replacement. The only difference is that

some of these simulations require that certain lines in the seed shape do not get

When the

ruler size is:

The length

measured is:

6 inches

2 inches

8 inches

½ inch

12 inches

FIGURE 1:4

Measuring the length of fractal curves

The new curves of Cantor, Koch, and others represented a problem in measurement cheory.

The length of the curve depends on the size of the ruler. As we shrink the ruler down, the length

approaches infinity.

measured length of curve

slope = 1.1

smaller ruler size —

measured length of curve •

slope = 1.3

smaller ruler size -

measured lengch of curve —

slope = 1.5

smaller ruler size -

FIGURE 1.5

A better way to measure fractal curves

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 steep slope for its plot.

Note for math sticklers: These figures are plotted on a logarithmic graph.

Fractal geometry

replaced. This is illustrated for the lung model at the bottom of figure 1.7. The

lines that get replaced in each iteration are called "active lines." Those that do

not get replaced are called "passive lines." We will be using the distinction between

active and passive lines in simulations for African designs as well.

Mandelbrot coined the term "fractal" for this new geometry, and it is now

used in every scientifc discipline from astrophysics to zoology. It is one of the

most powerful tools for the creation of new technologies as well as a revolutionary

approach to the analysis of the natural world. In medicine, for example, fractal

South Africa

Fractal dimension = 1.00

Smooth Koch curve

Fractal dimension = 1.1

Orcar Britain

Fractal dimension = 1.25

Rough Koch curve

Fractal dimension = 1.3

Norway

Fractal dimension = 1.52

fracturenso cuts

FIGURE 1.6

Measuring nature with fractal geometr

Although the curves of Cantor and others were introduced as abstractions without physica

meaning, Benoit Mandelbrot realized that 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 the fractal dimension is our plot slope from figure 2.5; 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.

--.

After the first iteration

ve see that only the active

lines were replaced:

the passive line remains

the sane. Now there

re three passive lnes

(center) and four activi

lines (the ends).

he similarity it heis ling stuce

f the human lung

FIGURE 1.7

Simulating nature with fractal geometry

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). The 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." The ones which do go through

replacement are called "active lines." Each step is referred to as an "iteration."

Fractal geometry

dimension can be used as a diagnostic cool. A healthy Jung has a high fractal dimen-

sion, but when black lung disease begins it loses some of the fine branching-a

condition that can be detecred by measuring the fractal dimension of the X ray.

For this reason, Benoit Mandelbrot was récently named an honorary member of

the French Coal Miners Union.

Of course, no revolution is without its counterrevolutionaries. It was not

long before some scientists started objecting that Mandelbrot was ignoring the

presence of the natural objects that could be described by Euclidean geometry,

such as Crystals or eggs. It's true that not all of nature is fractal—and this will be

an important point for us to keep in mind. Some writers have mistakenly

attempted to portray Africans as "more natural"—a dangerous and misleading

claim, even when made by well-meaning romantics. Since fractals are associated

with nature, a book about "African fractals" could be misinterpreted as support

for such romantic organicists. Pointing out that some Euclidean shapes exist in

the realm of nature makes it easier to understand that African fractals are from

the artificial realm of culture. Before moving on to these African designs, let's

review the basic characteristics of fractal geometry.

Five essential components of fractal geometry

RECURSION

We have seen that fractals are generated by a circular process, a loop in which

the output at one stage becomes the input for the next. Results are repeatedly

returned, so that the same operation can be carried out again. This is often referred

to as "recursion," a very powerful concept. Later we will distinguish between three

differeni types of recursion, but for now just think of it in terms of this iterative

feedback loop. We've already seen how iteration works to create the Cantor set

and the Koch curve. Although we can create a mathematical abstraction in which

the recursion continues forever, there are also cases where the recursion will "bot-

tom out." In our generation of the Koch curve, for example, we quit once the lines

get too small to print. In fact, any physically existing object will only be fractal-

within a particular range of scales.

SCALING

If you look at the coastline of a continent-take the Pacific side of North Amer-

ica for instance-you will see a jagged shape, and if you look at a small piece of

that coastline-say, California—we continue to see similar jaggedness. In fact,

a similar jagged curve can be seen standing on a cliff overlooking a rocky Cali-

fornia shore, or even standing on that shore looking at one rock. Of course, that's

Introduction

only roughly similar, and it's only good for a certain range of scales, but it is aston-

ishing to realize how well this works for many natural features. It is this "scal-

ing" property of nature that allows fractal geometry to be so effective for

modeling. To have a "scaling shape" means that there are similar patterns at dif-

ferent scales within the range under consideration. Enlarging a tiny section will

produce a pattern that looks similar to the whole picture, and shrinking down

the whole will give us something that looks like a tiny part.

SELF-SIMILARITY

Just how similar do these patterns have to be to qualify as a fractal? Mathe-

maticians distinguish between statistical self-similarity, as in the case of the coast-

line, and exact self-similarity, as in the case of the Koch curve. In exact

self-similarity we need to be able to show a precise replica of the whole in at

least some of its parts. In the Koch curve a precise replica of the whole could

be found within any section of the fractal ("strictly self-similar"), but this isn't

true for all fractals. The branching fractals used to model the lungs and acacia

tree (fig. 1.7), for example, have parts (e.g., the stem) that do not contain a tiny

image of the whole. Unlike the Koch curve, they were not generated by replac-

ing every line in the seed shape with a miniature version of the seed; instead,

we used some passive lines that were just carried though the iterations without

change, in addition to active lines that created a growing tip by the usual

recursive replacement.

INFINITY

Since fractals can be limited to a finite range of scales, it may seem like infinity

is just a historical artifact, at best a Holy Grail whose quest allowed mathematicians

serendipitously to stumble across fractals. It is this kind of omission that has made

many pure mathematicians rather nonplussed about the whole fractal affair,

and in some cases downright hostile (cf. Krantz 1989). There is no way to con-

nect fractals to the idea of dimension without using infinity, and for many math-

ematicians that is their crucial role.

FRACTIONAL DIMENSION

How can it be that the Koch curve, or any member of its fractal family, has infi-

nite length in a finite boundary? We are used to thinking of dimension as only

whole numbers-the one-dimensional line, the two-dimensional plane-but

the theory of measurement that governs fractals allows dimensions to be fractions.

Consider, for example, the increasing dimension of the Koch curves in figure 1.6.

Above the top, we could go as close as we like to a one-dimensional line. Below

Fractal geometry

the bottom, we could make the curve so jagged that it starts to fill in two-

dimensional areas of the plane. In between, we need an in-berween dimension.

Looking for fractals in African culture

As we examine African designs and knowledge systems, these five essential

components will be a useful way to keep track of what does or does not match

fractal geometry. Since scaling and self-similarity are descriptive characteristics,

our first step will be to look for these properties in African designs. Once we estab-

lish that theme, we can ask whether or not these concepts have been intentionally

applied, and start to look for the other three essential components. We will now

turn to African architecture, where we find some of the clearest illustrations of

indigenous self-similar designs.

CHAPTER

Fractals-

-in

-African

-settlement-

architecture-

- Architecture often provides excellent examples of cultural design themes,

because anything that is going to be so much a part of our lives--a structure

that makes up our built environment, one in which we will live, work or play—

is likely to have its design informed by our social concepts. Take religious archi-

tecture for example. Several churches have been built using a triangular floor

plan to symbolize the Christian trinity; others have used a cross shape. The

Roman Pantheon was divided into three vertical levels: the bottom with

seven niches representing the heavenly bodies, the middle with the 12 zodiac

signs, and on top a hemisphere symbolizing the order of the cosmos as a

whole.' But we don't need to look to grandiose monuments; even the most inun-

dane shack will involve geometric decisions—should it be square or oblong?

pitched roof or flat? face north or west?—and so culture will play a role here

as well.

At first glance African architecture might seem so varied that one would

conclude its structures have nothing in common. Although there is great diver-

sity among the many cultures of Africa, examples of fractal architecture can be

found in every corner of the African continent. Not all architecture in Africa

•is fractal--fractal geometry is not the only mathematics used in Africa--but its

repeated presence among such a wide variety of shapes is quite striking.

Fractals in African settlement architecture

In each case presented here we will compare the aerial photo or architec-

tural diagram of a settlement to a computer generated fractal model. The frac-

tal simulation will make the self-similar aspects of the physical structure more

evident, and in some cases it will even help us understand the local cultural mean-

ing of the architecture. Since the African designers used techniques like itera-

tion in buitding these structures, our virtual construction through fractal graphics

will give us a chance to see how the patterns emerge through this process.

2 I

Rectangular fractals in settlement architecture

If you fly over the northern part of Cameroon, heading roward Lake Chad along

the Logone River, you will see something like figure 2. ya. This aerial photo shows

the city of Logone-Birni in Cameroon. The Kotoko people, who founded this city

centuries ago, use the local clay to create huge rectangular building complexes.

The largest of these buildings, in the upper center of the photo, is the palace of

the chief, or "Miarre" (fig. 2.1b). Each complex is created by a process often called

"architecture by accretion," in this case adding rectangular enclosures to preexisting

rectangles. Since new enclosures often incorporate the walls of two or more, of

the old ones, enclosures tend to get larger and larger as you go outward from the

center. The end result is the complex of rectangles within rectangles within rec-

tangles that we see in the photo.

Since this architecture can be described in terms of self-similar scaling--it

makes use of the same pattern at several different scales— it is easy to simulate using

a computer-generated fractal, as we see in figures 2. 1c-e. The seed shape of the model

is a recrangle, but each side is made up of both active lines (gray) and passive lines

(black), After the first iteration we see how a small version of the original rectangle

is reproduced by each of the active lines. One more iteration gives a range of scales

that is about the same as that of the palace; this is enlarged in figure 2.1e.

During my visit to Logone-Birni in the summer of 1993, the Miarre kindly

allowed me to climb onto the palace roof and take the photo shown in figure 2.if.

I asked several of the Kotoko men about the variation in scale of their architecture.

They explained it in terms of a combination of patrilocal household expansion,

and the historic need for defense. "A man would like his sons to live next to

him," they said, "and so we build by adding walls to the father's house." In the

past, invasions by northern marauders were common, and so a larger defensive

wall was also needed. Sometimes the assembly of families would outgrow this

defensive enclosure, and so they would turn that wall into housing, and build an

even larger enclosure around it. These scaling additions created the tradition of

self-similar shapes we still see today, although the population is far below the

a. An aerial view of the city of Logone-Birni in Cameroon.

The largest building complex, in the center, is the palace

of the chief.

Photo courtesy Musée de l'Homme, Paris.

b. A closer view of the palace.

The smallest rectangles, in the

center, are the royal chambers.

c. Seed shape for the fractal

simulation of the palace.

The active lines, in gray,

will be replaced by a scaled-

down replica of the entire

seed.

d. First three iterations of the fractal simulation.

e. Enlargment of the third

iteration.

FIGURE 2.1

Logone-Birni

(figure continues)

f. Photo by the author taken from the roof of the palace.

Yene ada

g. The guti, the

royal insignia,

painted on the

palace walls.

By permission

of Lebeuf 1969.

Le chemin de la lumière

h. The spiral path taken by visitors to the throne.

By permission of Lebeuf 1969.

FIGURE 2. 1 (continued)

Inside Logone-Birni

Introduction

original 180,000 estimated for Logone-Birni's peak in the nineteenth century. At

that time there was a gigantic wall, about 1o feet thick, that enclosed the

perimeter of the entire settlement.

The women I spoke with were much less interested in either patrilineage

or military history; their responses concerning architectural scaling were primarily

about the contrast between the raw exterior walls and the stunning waterproof

finish they created for courtyards and interior rooms. This began by smoothing

wet walls flat with special stones, applying a resin created from a plant extract,

and then alding. beautifully austere decorative lines.

The most important of these decorative drawings is the guti, a royal insignia

(fig. 2.1g). The central motif of the guti shows a rectangle inside a rectangle inside

a rectangle; it is a kind of abstract model that the Kotoko themselves have cre-

ated. The reason for choosing scaling rectangles as a symbol of royalty becomes

clear when we look at the passage that one must take to visit the Miarre (fig. 2.1h).

The passage as a whole is a rectangular spiral. Each time you enter a smaller scale,

you are required to behave more politely. By the time you arrive at the throne

you are shoeless and speak with a very cultured formality.? Thus the fractal

scaling of the architecture is not simply the result of unconscious social dynam-

ics; it is a subject of abstract representation, and even a practical technique applied

to social ranking.

To the west near the Nigerian border the landscape of Cameroon becomes

much greener; this is the fertile high grasslands region of the Bamileke. They too

have a fractal settlement architecture based on rectangles (fg. 2.2a), but it has

no cultural relation to that of the Kotoko. Rather than the thick clay of Logone-

Birni, these houses and the attached enclosures are built from bamboo, which.

is very strong and widely available. And there was no mention of kinship,

defense, or politics when I asked about the architecture; here I was told it is pat-

terns of agricultural production that underlie the scaling. The grassland soil and

climate are excellent for farming, and the gardens near the Bamileke houses typ-

ically grow a dozen different plants all in a single space, with each taking its char-

acteristic vertical place. But this is labor intensive, and so more dispersed

plantings--rows of corn and ground-nut—are used in the wider spaces farther

from the house. Since the same bamboo mesh construction is used for houses,

house enclosures, and enclosures of enclosures, the result is a self-similar archi-

tecture. Unlike the defensive labyrinth of Kotoko architecture, where there

were only a few well-protected entryways, the farming activities require a lot of

movement between enclosures, so at all scales we see good-sized openings. The

fractal simulation in figures 2.2b,c shows how this scaling structure can be mod-

eled using an open square as the seed shape.

fields

servant's room

wite's r

gran

nan's room

bamboo tence

wife's room

first wite's room

Jaranary

granary

10 ... 201

CHE

FIGURE 2.2

Bamileke settlement

(a) Plan of Bamileke settiement from about 196o. (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.

(ia, Beguin 1952; reprinted with permission from OrstoM).

Introduction

Circular fractals in settlement architecture

Much of southern Africa is made up of arid plains where herds of cattle-and other

livestock are raised. Ring-shaped livestock pens, one for each extended family, 3

can be seen in the aerial photo in figure 2.za, a Ba-ila settlement in southern Zam-

bia. A diagram of another Ba-ila settlement (fig. 2.3d) makes these livestock enclo-

sures ("kraals") more clear. Toward the back of each pen we find the family living

quarters, and toward the front is the gated entrance for letting livestock in and

out. For this reason the front entrance is associated with low status (unclean, ani-

mals), and the back end with high status (clean, people). 4 This gradient of sta-

tus is reflected by the size gradient in the architecture: the front is only fencing,

as we go toward the back smaller buildings (for storage) appear, and toward the

very back end are the larger houses. The two geometric elements of this struc-

ture-a ring shape overall, and a status gradient increasing with size from front

to back—-echoes throughout every scale of the Ba-ila settlement.

The settlement as a whole has the same shape: it is a ring of rings. The set-

tlement, like the livestock pen, has a front/back social distinction: the entrance

is low status, and the back end is high status: At the settlement entrance there

are no family enclosures at all for the first 20 yards or so, but the farther back we

go, the larger the family enclosures become.

At the back end of the interior of the settlement, we see a smaller detached

ring of houses, like a settlement within the settlement. This is the chief's

extended family. At the back of the interior of the chief's extended family ring,

the chief has his own house. And if we were to view a single house from above,

we would see that it is a ring with a special place at the back of the interior: the

household altar.

Since we have a similar structure at all scales, this architecture is easy to

model with fractals. Figure 2 3b shows the first three iterations. We begin with

a seed shape that could be the overhead view of a single house. This is created

by active lines that make up the ring-shaped walls, as well as an active line ar

the position of the altar at the back of the interior. The only passive lines are

those adjacent to the entrance. In the next iteration, we have a shape that could

be the overhead view of a family enclosure. At the entrance to the family enclo-

sure we have only fencing, but as we go toward the back we have buildings of

increasing size. Since the seed shape used only passive lines near the entrance

and increasingly larger lines toward the back, this iteration of our simulation has

the same size gradient that the real family enclosure shows. Finally, the third iter-

ation provides a structure that could be the overhead view of the whole settle-

ment. At the entrance to the settlement we have only fencing, but as we go toward

FIGURE 2.3

Ba-ila

(a) Aerial photo of Ba-ila sertlement before 1944. (b) Fractal generation of Ba-ila simulation.

Note that the seed shape has only active lines (gray) except for those near the opening (black).

(a, American Geographic Institute.)

Introduction

the back we have enclosures of increasing size. Again, by having the seed shape

use only passive lines near the entrance and increasingly larger lines toward the

back, this iteration of our simulation has the same size gradient that the real settle-

ment shows.

I never visited the Ba-ila myself; most of my information comes from the

classic ethnography by Edwin Smith and Andrew Dale, published in 1920.

While their colonial and missionary motivations do not inspire much trust,

they often showed a strong commitment toward understanding the Ba-ila point

of view for social structure. Theit analysis of Ba-ila settlement architecture

points out fractal attributes. They too noted the scaling of house size, from

those less than 12 feet wide near the entrance, to houses more than 40 feet wide

at the back, and explained it as a social status gradient; "there being a world of

difference between the small hovel of a careless nobody and the spacious dwelling

of a chief" (Smith and Dale 1968, 114).

It is in Smith's discussion of religious beliefs, however, that the most strik-

ing feature of the Ba-ila's fractal architecture is illuminated. Unlike most mis-

sionaries of his time, Smith was a strong proponent of respect for local religions.

He was rio relativist-understanding and respect were strategies for conver-

sion- but his delight in the indigenous spiritual strength comes across clearly in

his writings and provided him with insight into the subtle relation of the social,

sacred, and physical structure of the Ba-ila architectural plan.

In this village there are about 250 huts, built mostly on the edge of a circle four

hundred yards in diameter. Inside this circle there is a subsidiary one occupied

by the chief, his family, and cattle. It is a village in itself, and the form of it in

the plan is the form of the greater number of Ba-ila villages, which do not attain

to che dimensions of Shaloba's capital. The open space in the center of the vil-

lage is also broken by a second subsidiary village, in which reside important mem-

bers of the chief's family, and also by three or four miniature huts surrounded

by a fence: these are the manda a mizhimo ("the manes' huts") where offerings

are made to the ancestral spirits. Thus carly do we see traces of the all-pervading

religious consciousness of the Ba-ila.

(Smith and Dale 1968, 113)

In the first iteration of the computer generated model there is a detached

active line inside the ring, at the end opposite the entrance. This was motivated

by the ring comprising the chief's family, but it also describes the location of the

sacred altar within each house. As a logician would put it, the chief's family ring

is to the whole settlement as the altar is to the house. It is not a status gradient,

as we saw with the front-back axis, but rather a recurring functional role between

different scales: "The word applied to the chiel's relation to his people is kulela:

in the extracts given above we translate it 'to rule,' but it has this only as a sec-

Fractals in African settlement architecture

ondary meaning. Kulela is primarily to nurse, to cherish; it is the word applied

to a woman caring for her child. The chief is the father of the community; they

are his children, and what he does is lela them" (Smith and Dale 1968, 307).

This relationship is echoed throughout family and spiritual ties at all

scales, and is structurally mapped through the self-similar architecture. The

nesting of circular shapes—ancestral miniatures to chief's house ring to chief's

extended family ring to the great outer ring-—was not a status gradient, as we saw

for the enclosure variation from front to back, but successive iterations of lela.

A very different circular fractal architecture can be seen in the famous stone

, buildings in the Mandara Mountains of Cameroon. The various ethnic groups

of this area have their own separate names, but collectively are often referred to

as Kirdi, the Fulani word for "pagan," because of their strong resistance against

conversion to Islam. Their buildings are created from the stone rubble that

commonly covers the Mandara mountain terrain. Much of the stone has natural

fracture lines that tend to split into thick flat sheets, so these ready-made

bricks-along with defensive needs—-helped to inspire the construction of their

huge castlelike complexes. But rather than being the Euclidean shapes of Euro-

pean castles, this African architecture is fractal.

' Figure 2.4a shows the building complex of the chief of Mokoulek, one of

the Mofou settlements. An architectural diagram of Mokoulek, drawn by French

researchers from the oRstom science institute, shows its overall structure (fig. 2.4b).

It is primarily composed of three stone enclosures (the large circles), each of which

surrounds tightly spiraled granaries (small circles). The seed shape for the sim-

ulation requires a circle, made of passive lines, and two different sets of active

lines (fg, 2.4c). Inside the circle is a scaling sequence of small active lines; these

will become the granaries. Outside the circle there is a large active line; this will

replicate the enclosure filled with granaries. By the fourth iteration we have cre-

ated three enclosures filled with spiral clusters of granaries, plus one unfilled. The

real diagram of Mokoulek shows several unfilled circles-evidence that not

everything in the architectural structure can be accounted for by fractals. Nev-

ertheless, an important feature is suggested by the simulation.

In the first iteration we see that the large external active line is to the left

of the circle. But since it is at an angle, the next iteration finds this active line

above and to the right. If we follow the iterations, we can see that the dynamic

construction of the complex has a spiral pattern; the replications whorl about a

central location. This spiral dynamic can be missed with just a static view—| cer-

tainly didn't see it before I tried the simulation-but our participation in the vir-

tual construction makes the spiral quite evident. The similarity between the small

spirals of granaries inside the enclosures and this large-scale spiral shape of the

FIGURE 2.4

Mokoulek

(a) Mokoulek, Cameroon. The small buildings inside the stone wall are granaries. The rectangular

building (top right) holds the sacred altar. (b) Architectural diagram of Mokoulek. (c) First three

iterations of the Mokoulek simulation. The seed shape is composed of a circle drawn with passive

lines (black) and with gray active lines both inside and outside the circle. (d) Fourth iteration of

the Mokoulek simulation.

(a and b, by permission from Seignobos 1982.)

Fractals in African settlement architeczure

complex as a whole indicates that the fractal appearance of the architecture is

not merely due to a random accumulation of various-sized circular forms. The

idea of circles of increasing size, spiraling from a central point, has been applied

at two different scales, and this structural coherence is confirmed by the archi-

rects' own concepts.

In our simulation the active line became located toward the center of the

spiral. The Mofou also think of their architecture as spiraling from this central

location, which holds their sacred altar. The altar is a kind of conceptual "active

line" in their schema; it is responsible for the iterations of life. Seignobos (1982)

notes that this area of the complex is the site of both religious and political author-

ity; it is the location for rituals that generate cycles of agricultural fertility and

ancestral succession. This geometric mapping between the scaling circles of the

architecture and the spiritual cycles of life is represented in their divination

("fortunetelling") ritual, in which the priest creates concentric circles of stones

and places himself at the center. As in the guti painting in Logone-Birni, in which

the Kotoko had modeled their scaling rectangles, the Mofou have also created

their own scaling simulation.

By the time I arrived at Mokoulek in 1994 the chief had died, and the own-

ership of this complex had been passed on to his widows. The new chief told me

that the design of this architecture, including that of his new complex, began with

a precise knowledge of the agricultural yield. This volume measure was then con-

verted to a number of granaries, and these were arranged in spirals. The design

is thus not simply a matter of adding on granaries as they are needed; in fact, it

has a much more quantitative basis than my computer model, which 1 simply did

by eyeball.

Not all circular architectures. in Africa have the kind of centralized

location that we saw in Mokoulek. The Songhai village of Labbezanga in Mali .

(fig. 2.5a), for example, shows circular swirls of circular houses without any

single focus. But comparing this to the fractal image of figure 2.5b, we see that

a lack of central focus does not mean a lack of self-similarity. It is important to

remember that while "symmetry" in Euclidean geometry means similarity within

one scale (e.g., similarity between opposite sides in bilateral symmetry), fractal

geometry is based on symmetry between different scales. Even these decentral-

ized swirls of circular buildings show a scaling symmetry..

Paul Stoller, an accomplished ethnographer of the Songhai, tells me that

the rectangular buildings that can be seen in figure 2.ga are due to Islamic influ-

ence, and that the original structure would have been completely circular.

Thanks to Peter Broadwell, a computer programmer from Silicon Graphics Inc.,

we were able to run a quantitative test of the photo that confirmed what our eyes

Introduction

FIGURE 2.5

Labbezanga

(a) Aerial view of the village of Labbezanga in Mali. (b) Fractal graphic.

(a, photo by Georg Gerster; b, by permission of Benoit Mandelbror.)

were telling us: the Songhai architecture can be characterized by a facial dimen-

sion similar to that of the computer-generated fractal of figure 2.5b. 6

This kind of dense circular arrangement of circles, while occurring in all

sorts of variations, is common throughout inland west Africa. Bourdier and

Trinh (1985), for example, describe a similar circular architecture in Burkina Faso.

The scaling of individual buildings is beautifully diagrammed in their cover

illustration (fig. 2.6a), a portion of one of the large building complexes created

by the Nankani society. As for the Songhai, foreign cultural influences have now

introduced rectangular buildings as well. In the Nankani complex the outermost

enclosure (the perimeter of the complex) is socially coded as male. As we move

in, the successive enclosures become more female associated, down to the cir-

cular woman's dégo (fig. 2.6b), the circular fireplace, and finally the scaling

stacks of pots (fig. 2.6c).

Using a technique quite close to that of the Kotoko women, the women

of Nankani also waterproof and decorate these walls. The recurrent image of a

Fractals in African seulement architecture

triangle in these decorations (see walls of dégo) represents the zalanga, a nested

stack of calabashes (circular bowls carved from gourds) that each woman keeps

in her kitchen (fig. 2.6d). Since these calabashes are stacked from large to small,

they (and the rope that holds them) form a triangle-thus the triangular

decorations also represent scaling circles, just in a more abstract way. The small-

est container in a woman's zalanga is the kumpio, which is a shrine for her soul.

When she dies, the zalanga, along with her pors, is smashed, and her soul is released

to eternity. The eternity concept, associated with well-being, is symbolically

FIGURE 2.6

Nankani home

(a) Drawing of a Nankani home. (b) The woman's main room (dégo)

inside the Nankani home. (c) A scaling stack of pots in the fireplace.

(d) The zulanga.

(a, Bourdier and Trinh 1985; courtesy of the authors; b-d, photos from

Bourdier and Trinh 1985, by permission of the authors.)

Introduction

represented by the coils of a serpent of infinite length, sculpted into the walls

of these homes.

From the 20-meter diameter of the building complex to the o.2-meter

kumpio-and not simply at one or two levels in between, but with dozens of self-

similar scalings-the Nankani fractal spans three orders of magnitude, which is

comparable to the resolution of most computer screens. Moreover, these scaling

circles are far from unconscious accident: as in several other architectures we have

examined, they have made conscious use of the scaling in their social symbol-

ism. In this case, the most prominent symbolism is that of birthing. When a child

is born, for example, it must remain in the innermost enclosure of the women's

, dégo until it can crawl out by itself. Each successive entrance is— spatially as well

as socially—a rite of passage, starting with the biological entrance of the child

from the womb. It leaves each of these nested chambers as the next iteration ini

life's stages is born. The zalanga models the entire structure in miniature, and its

destruction in the event of death maps the journey in reverse: from the circles

of the largest calabash to the tiny kumpio holding the soul-from mature adult

to the eternal realm of ancestors who dwell' in "the earth's womb." There is a

conscious scheme to the scaling circles of the Nankani: it is a recursion which

bottoms-out at infinity.

Branching fractals

While African circular buildings are typically arranged in circular clusters, the

paths that lead through these settlements are typically not circular. Like the

bronchial passages that oxygenate the round alveoli of the lungs, the routes that

nourish circular settlements often take a branching form (e.g., figure 2.7). But

despite my unavoidably organicist metaphor, these cannot be simply reduced to

unconscious traces of minimum effort. For one thing, conscious design criteria

are evident in communities in which there is an architectural transition from cir-

cular to rectangular buildings, since they can choose to either maintain or erase

the branching forms.

Discussion concerning such decisions are apparent in the settlement of Banyo,

Cameroon, where the transition has a long history (Hurault 1975). I found that

few circular buildings were left, but those that were still intact served as an

embodiment of cultural memory. This role was honored in the case of the chief's

complex and exploited in the case of a blacksmith's shop, which was the site of

occasional tourist visits. After passing approval by the governinent officials

and the sultan, I was greeted by the official city surveyor, who--considering

the fact that his raison d'être was Euclideanizing the streets—showed surprising

Fractals in African settlement architecture

FIGURE 2.7

Branching paths in a Senegalese settlement

(a) Aerial photo of a traditional settlement in northeast Senegal. The space between enclosure

walls, serving as roads and footpaths, creates a branching pattern. (b) A branching fractal can be

created by the background of a scaling set of circular shapes.

(a, courtesy Institut Geographique du Senegal.)

appreciation for my project and helped me locate the most fractal area of the

city (fig. 2.8a). At the upper left of the photo we see a portion of the Euclidean

grid that covers the rest of the city, but most of this area is still fractal. The loca-

tion of this carefully maintained branching—-fanning out from a large plaza

that is bordered by the palace of the sultan and the grand mosque-is no

coincidence. By marking my position on the aerial photo as I traveled through

(fg. 2.8b), I was later able to create a map by digitally altering the photo image

(fig. 2.8c). This provides a stark outline-looking much like the veins in a

leaf--of the fractal structure of this transportation network. I may have plunged

through a wall or two in creating this map, but it certainly underestimates the

fine branching of the footpaths, as I did not attempt to include their extensions

into private housing enclosures.

How does the creation of these scaling branches interact with the kinds of

iterative construction and social meaning we have seen associated with other

examples of fractal architecture? A good illustration can be found in the

Position I—outside palace

Position 2-road below mosque

Position 3--narrow walkway

FIGURE 2.8

Branching paths in Banyo

(a) Aerial photo of the city of Banyo,

Cameroon. (6) 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, courtesy National Institute of Cartography, .

Cameroon.)

8 17937.

EET

FIGURE 2.9

Streets of Cairo

(a) Map of streets of Cairo, 1808. (b) Fractal simulation for Cairo streets. (c) Enlarged view

of fourth iteration.

Introduction

branching streets of North African cities. Figure 2.ga shows a map of Cairo, Egypt,

in 1898. The map was created by an insurance company, and I have colored the

streets black to make the scaling branches more apparent. Figure 2.gb shows its

computer simulation. Delaval (1974) has described the morphogenesis of Saha-

ran cities in terms of successive additions similar to the line replacement in the

fractal algorithms we have used here. The first "seed shape" consists of a mosque

connected by a wide avenue to the marketplace, and successive iterations of con-

struction add successive contractions of this form.

Since these fractal Saharan settlement architectures predate Islam (see

Devisse 1983), it would be misleading to see them as an entirely Muslim inven-

tion; but given the previous observations about the introduction of Islamic

architecture as an interruption of circular fractals in sub-Saharan Africa, it is impor-

tant to note that Islamic cultural influences have made strong contributions to

African fractals as well. Heaver (1987) describes the "arabesque" artistic form in

North African architecture and design in terms that recall several fractal con-

cepts (e.g., "cyclical rhythms" producing an "indefinitely expandable" struc-

ture). He discussed these patterns as visual analogues to certain Islamic social

concepts, and we will examine his ideas in greater detail in chapter 12 of this book.

Conclusion

Throughout this chapter, we have seen that a wide variety of African settlement

architectures can be characterized as fractals. Their physical construction makes

use of scaling and iteration, and their self-similarity is clearly evident from com-

parison to fractal graphic simulations. Chapter 3, will show that fractal architecture

is not simply a typical characteristic of non-Western settlements. This alone does

not allow us to conclude an indigenous African knowledge of fractal geometry;

in fact, I will argue in chapter 4 that certain fractal patterns in African decora-

tive arts are merely the result of an intuitive esthetic: But as we have already seen,

the fractals in African architecture are much more than that. Their design is linked

to conscious knowledge systems that suggest some of the basic concepts of frac-

tal geometry; and in later chapters we will find more explicit expressions of this

indigenous mathematics in astonishing variety and form.

CHAPTER

-Fractals-

in~

cross-cultural-

comparison-

- The fractal settlement patterns of Africa stand in sharp contrast to the Carte-

sian grids of Euro-American settlements. Why the difference? One explanation

could be the difference in social structure. Euro-American cultures are organized

by what anthropologists would call a "state society." This includes not just the

modern nation-state, but refers more generally to any society with a large

political hierarchy, labor specialization, and cohesive, formal controls--what is

sometimes called "top-down" organization. Precolonial African cultures included

many state societies, as well as an enormous number of smaller, decentralized

social groups, with little political hierarchy-that is, societies that are organized

"bottom-up" rather than "top-down."' But if fractal architecture is simply the

automatic result of a nonstate social organization, then we should see fractal settle-

ment patterns in the indigenous societies of many parts of the world. In this chap-

ter we will examine the sertlement patterns found in the indigenous socieries

of the Americas and the South Pacific, but our search will turn up very few frac-

tals. Rather than dividing the world between a Euclidean West and fractal

non-West, we will find that each society makes use of its particular design

themes in organizing its built environment. African architecture tends to be frac-

tal because that is a prominent design theme in African culture. In fact, this cul-

tural specificity of design themes is true not only for architecture, but for many

Introduction

other types of material design and cultural practices as well. We will begin our

survey with a brief look at the design themes in Native American societies, which

included both hierarchical state empires as well as smaller, decentralized tribal

cultures.

Native American design

The Ancestral Pueblo society dwelled in what is now the southwestern United

States around 1 100 c.e. Aerial photos of these sites (fig. 3.1) are some of the most

famous examples of Native American settlements. But as we can see from this

vantage point, the architecture is primarily characterized by an enormous circular

form created from smaller rectangular components-- certainly not the same shape

at two different scales. This juxtaposition of the circle and the quadrilateral (rec-

tangle or cross-shaped) form is not a coincidence; the two forms are the most impor-

tant design themes in the material culture of many Native American societies,

including both North and South continents.

As far as architecture is concerned, there are no examples of the nonlinear

scaling we saw in Africa. The only Native American architectures that come

close are a few instances of linear concentric figures (fig. 3.2a). Shapes approx-

imating concentric circles can be seen in the Poverty Point complex in north-

FIGURE 3.1

Euclidean geometry in Native American architecture

(a) Aerial photo of Bandelier, one of the Ancestral Puehlo settlements (starting around 1 100 c.E.)

in norhtwestern New Mexico. (b) Aerial photo of Pueblo Bonito, another Ancestral Pueblo .

settlement (starting around 950 c.E.). Note that they are mostly rectangular at the smallest scale

and circular at the largest scale.

(a, phoro by Tom Buker; b, photo by Georg Gerster.)

Fractals in cross-cultural comparison

ern Louisiana, for example, and there were concentric circles of tepees in the

Cheyenne camps. The step-pyramids of Mesoamerica look like concentric

squares when viewed from above. But linear concentric figures are not fractals.

First, these are linear layers: the distance between lines is always the same, and

thus the number of concentric circles within the largest circle is finite. The non-

linear scaling of fractals requires an ever-changing distance between lines,

FIGURE 3.2

Linear concentric forms in Native American architecture

(a) Native American architecture is cypically based on quadrilateral grids or a combination of

circular and grid forms. The only examples of scaling shapes are these linear concentric forms. In

the Poverty Point complex, for example, concentric circles were used, and concentric squares can be

seen if we look at the Mexican step pyramids from above. These forms are better characterized as

Euclidean than fractal for two reasons: (b) First, they are linear. Here is an example of a nonlinear

concentric circle. While the linear version must have a finite number of circles, this figure could

have an infnite number and still fit in the same boundary. (c) Second, they only scale with respect

10 one point (the center). Here is an example of circles with more global scaling symmetry.

Introduction

which means there can be an infinite number in a finite space (fig. 3.2b). Sec-

ond, even nonlinear concentric circles are only self-similar with respect to a

single locus (the center point), rather than having the global self-similarity of

fractals (fig. 3.2c).

The importance of the circle is detailed in a famous passage by Black Elk

(1961), in which he explains that "everything an Indian does is in a circle, and

that is because the Power of the World always works in circles, and everything

tries to be round." But he goes on to note that his people thought of their world

as "the circle of the four quarters." A similar combination of the circle and quadri-

lateral form can be seen many Native American myths and artifacts; it is not their

only design theme, but it can be found in a surprising number of different soci-

eties. Burland (1965), for example, shows a ceremonial rattle consisting of a wooden

hoop with a cross inside from southern Alaska, a Navajo sand painting showing

four equidistant stalks of corn growing from a circular lake, and a Pawnee buffalo-

hide drum with four arrows emanating from its circular center. Nabokov and

Easton (1989) describe the cultural symbolism of the tepee in terms of its com-

bination of circular hide exterior and the four main struts of the interior wood

supports. Waters (1963) provides an extensive illustration of the cultural sig-

nificance of combining the circular and cross form in his commentary on the Hopi

creation myth.

The fourfold symmetry of the quadrilateral form has lead to some sophis-

ticated conceptual structures in Native American knowledge systems. In Navajo

sand painting, for example, the cruciform shape represents the "four directions"

concept, similar to the Cartesian coordinate system. While orderly and consis-

tent, it is by no means simple (see Witherspoon and Peterson 1995). The four

Navajo directions are also associated with corresponding sun positions (dawn,

day, evening, night), yearly seasons (spring, summer, fall, winter), principal

colors (white, blue, yellow, black), and other quadrilateral divisions (botanical

categories, partitions of the life cycle, etc.). These are further broken into inter-

secting bipolarities (e.g., the east/west sun path is broken by the north/south direc-

tions). Combined with circular curves (usually representing organic shapes and

processes), the resulting schema are rich cultural resources for indigenous mathe-

matics (see Moore 1994). But, except for minor repetitions (like the small circular

kivas in the Chaco canyon site of fig. 3.1) there is nothing particularly fractal

about these quadrilateral designs.

Many Mesoamerican cities, such as the Mayans' Teotihuacán, the Aztec's

Tenochtitlán, and the Toltec's Tula, embedded a wealth of astronomical knowl-

edge in their rectangular layouts, aligning their streets and buildings with heav-

enly objects and events (Aveni 1980). J. Thompson (1970) and Klein (1982)

Fractals in cross-cultural comparison

describe the quadrilateral figure as an underlying theme in Mesoamerican geo-

metric thinking, from small-scale material construction techniques such as

weaving, to the heavenly cosmology of the four serpents. Rogelio Díaz, of the

Mathematics Museum at, the University of Querétaro, points out that the skin

patterns of the diamondback rattlesnake were used by the Mayans to symbolize

this concept (fig. 3-3a).

Comparing the Mayan snake pattern with an African weaving based on the

cobra skin pattern (fg. 3.3b), we can see how geometric modeling of similar nat-

ural phenomena in these two cultures results in very different representations.

The Native American example emphasizes the Euclidean symmetry within one

size frame ("size frame" because the term "scale" is confusing in the context of

snake skin). This Mayan pattern is composed of four shapes of the same size, a

fourfold symmetry. But the African example emphasizes fractal symmetry, which

is not about similarity between right/left or up/down, but rather similarity

between different size frames. The African snake pattern shows diamonds within

diamonds within diamonds. Neither design is necessarily more accurate: cobra

skin does indeed exhibit a fractal pattern-the snake's "hood," its twin white

patches, and the individual scales themselves are all diamond shaped-and yet

snake skin patterns (thanks to the arrangement of the scales) are also charac-

teristically in diagonal rows, so they are accurately modeled as Euclidean struc-

tures as well. Each culture chooses to emphasize the characteristics that best fit

its design theme.

There are a few cases in which Native Americans have used scaling geo-

metries in artistic designs. Several of these were not, however, part of the tra-

ditional repertoire.? Navajo blankets, for example, were originally quite linear;

.. it was-only cn examining Persian ruga that Navajo weavers began to use more

scaling styles of design (and even then the designs were much more Euclidean

than the Persian originals; see Kent 1985). The Pueblo "storyteller" figures have

some scaling properties, but they are of recent (196os) origin. Pottery and cala-

bash (carved gourd) artisans in Africa often create scaling by allowing the

design adaptively to change proportion according to the three-dimensional form

on which it is inscribed (see "adaptive scaling" in chapter 6), but this was quite

rare in Native American pottery until the 196os.

Finally, there are three Native American designs that are both indigenous

and fractal. The best case is the abstract figurative art of the Haida, Kwakiutl,

Tlingut, and others in the Pacific Northwest (Holm 1965). The figures, primarily

carvings, have the kind of global, nonlinear self-similarity necessary to qualify

as fractals and clearly exhibit recursive scaling of up to three or four iterations.

They also make use of adaptive scaling, as illustrated by the shrinking series of

FIGURE 3.3

Snakeskin models in Native American and African cultures

(a) Rogelio Díaz of the Mathematics Museum at the University of Querétaro shows how the skin

patterns of the diamondback rattlesnake were used by the Mayans to symbolize a cosmology based

on quadrilateral structure. (b) The Mandiack weavers of Guinea-Bissau have also created an

abstract design based on a snakeskin patrern, but chose to emphasize the fractal characteristics.

Fractals in cross-cultural comparison

figures on the diminishing handles of soup ladles. Researchers since Adams

(1936) have pointed to the similarity with early Chinese art, which also has some

beautiful examples of scaling form, and its style of curvature and bilateral sym-

metry could indeed be culturally tied to these New World designs through an

ancient common origin. However, I doubt that is the case for the scaling char-.

acteristics. The Pacific Northwest art appears to have developed its scaling

structure as the result of competition between artisans for increasingly elaborate

carvings (Faris 1983). Although some researchers have attributed the competi-

tion pressure to European trading influences, the development of the scaling designs

was clearly an internal invention.

The other two traditional Native American designs do not qualify as frac-

tals quite as well. One involves the saw-tooth pattern found in several basket

and weaving designs. When two saw-tooth rows intersect at an angle, they cre-'

ate a triangle made from triangular edges. But these typically have only two iter-

ations of scale, and there is no indication in the ethnographic licerature that

they are semantically tied to ideas of recursion or scaling (see Thomas and Slock-

ish 1982, 18). The other is an arrangement of spiral arms often found on

coiled baskets. Again, this is self-similar only with respect to the center point,

but there are some nonlinear scaling versions (chat is, designs that rapidly get

smaller as you move from basket edge to center). However, these designs

generally appear to be a fusion between the circular form of the basket and the

cruciform shape of the arms: again more a combination of two Euclidean

shapes than a fractal.

One of the most common examples of this fusion between the circle and

the cross is the "bifold rotation" pattern in which the arms curve in opposite

directions, as shown in figure 3-4a. Figure 3 4b shows the figure of a bat from-

Mimbres pottery with a more complex version of the bifold rotation. Euclidean

syminetry has been emphasized in this figure; for example, the ears and mouth

of the bat have been made to look similar to increase the bilateral symmetry, and

the belly is drawn as a rectangle. Figure 3 4c shows the figure of a bat from an

African design; it is a zigzag shape that expands in width from top to bottom, rep-

resenting the wing of the bat. Here we see neglect of the bilateral symmetry of

the bat, and an emphasis on the scaling folds of a single wing. Again, the Native

American representation makes use of its quadrilateral/circular design theme, just

as the African representation of the bat emphasizes scaling design.

There is plenty of complexity and sophistication in the indigenous geom-

etry and numeric systems of the Americas (see Ascher 1991, 87-94; Closs 1986;

Eglash 1908b), but with the impressive exception of the Pacific Northwest carv-

ings, fractals are almost entirely absent in Native American designs.

) +-

Arkansas pottery

Pima basket

Southwestern portery motif

(a) The circular and quadrilateral forms were often combined in Native American designs as a

fourfold or bifold rotation.

(b) This image of a bat, from a Mimbres pottery in Southwestern

Native American tradition, shows an emphasis on circular and

quadrilateral form. The ear and the mouth, for example, are made

to look similar to emphasize bilateral symmetry, and the belly is

drawn as a rectangle. It also shows the wing bones as a bifold

rotation pattern.

little attention to the bilateral symmetry of the bat's body but gives an

emphasis on the scaling symmetry of the wing folds, shown as an

expanding zigzag pattern.

FIGURE 3-4

The bifold rotation in Native American design

(a: Left, from Miles 1963. Center, from Southwest Indian Craft Arts by Clara Lee Tanner. Copyright

1968 by the Arizona Board of Regents. Reprinted by permission of the University of Arizona Press.

Right, courtesy Don Crone. b, from Zaslow 1977, courtesy of the author. c, courtesy of Danicl Biebuyck.)

Fractals in cross-cultural comparison

Designs of Asia and the South Pacific

Several of the South Pacific cultures share a tradition of decorative curved and

spiral forms, which in certain Maori versions—-particularly their rafter and tat-

• too patterns-—would certainly count as'fractal (see Hamilton 1977). These are

strongly suggestive of the curvature of waves and swirling water. Classic Japan-

ese paintings of water waves were also presented as fractal patterns in Mandel-

brot's (1982) seminal text (plate Cr6). These may have some historic relation

to scaling patterns in Chinese art (see Washburn and Crowe 1988, fig. 6.9), which

are based on swirling forms of water and clouds, abstracted as spiral scaling

structures. While both the Japanese and Chinese patterns are explicitly associ-

ared with an effort to imitate nature, these Maori designs are reported to be more

about culture in particular, they emphasize mirror-image symmetries, which are

associated with their cultural themes of complimentarity in social relations

(Hanson 1083).

In almost all other indigenous examples, however, the Pacific Islander pat-

terns are quite Euclidean. Settlement layout, for instance, is typically in one

or two rows of rectangular buildings near the coasts, with circular arrangements

of rectangles also occurring inland (see Fraser 1968). The building construc-

tion is generally based on a combination of rectangular grids with triangular

or curved arch roofs. Occasionally these triangular faces are decorated with tri-

angles, but otherwise nonscaling designs dominate both structural and deco-

rative patterns.3

Again, it is important to note that this lack of fractals does not imply a lack

of sophistication in their mathematical thinking. For example, Ascher (1991)

has analyzed some of the algorithmic properties of Warlpiri (Pacific Islander) sand

drawings. Similar structures are also found 'in Africa; where they are called

lusoria. But while the lusona tend to use similar patterns at different scales (as

we will see in chapter 5), the Warlpiri drawings tend to use different patterns at

different scales. Ascher concludes that the Waripiri method of combining dif-

ferent graph movements is analogous to algebraic combinations, but the African

lusona are best described as fractals.

Complicating my characterization of the South Pacific as dominated by

Euclidean patterns is the extensive influence of India. It is perhaps no coinci-

dence that the triangle of triangles mentioned above is most common in Indone-

sia. In architecture, a famous exception to the generally Euclidean form is that

of Borobudur, a remple of Indian religious origin in Java. Although northern India

tends toward Euclidean architecture, explicit recursive design is seen in several

remples in southern India-the Kandarya Mahadeo in Khajuraho is one of the

Introduction

clearest examples--and is related to recursive concepts in religious cosmology.

These same areas in southern India also have a version of the lusona drawings,

and many other examples of fractal design. Interestingly, these examples from south-

ern India are the products of Dravidian culture, which is suspected to have sig-

nificant historical roots in Africa.

European designs

Most traditional European fractal designs, like those of Japan and China, are due

to imitation of nature—-a topic we will take up in the following chapter. There

are at least two stellar exceptions, however, that are worth noting. One is the

scaling iterations of triangles in the floor tiles of the Church of Santa Maria in

Costedin Rome (see plate 5-7 in Washburn and Crowe 1988). I have not been

able to determine anything about their cultural origins, but they are clearly

artistic invention rather than imitation of some natural form. The other can be

found in certain varieties of Celtic interlace designs. Nordenfalk (1977) suggests

that these are historically related to the spiral designs of pre-Christian Celtic reli-

gion, where they trace the flow of a vital life force. They are geometrically

classified as an Eulerian path, which is closely associated with mathematical knot

theory (cf. Jones 1990, 99).

Conclusion

Fractal structure is by no means universal in the material patterns of indigenous

societies. In Native American designs, only the Pacific Northwest patterns show

a strong fractal characteristic; Euclidean shapes otherwise dominate the art and

architecture. Except for the Maori spiral designs, fractal geometry does not

appear to be an important aspect of indigenous South Pacific patterns either. That

is not to say that fractal designs appear nowhere but Africa-southern India is

full of fractals, and Chinese fluid swirl designs and Celtic knot patterns are

almost certainly of independent origin. The important point here is that the frac-

tal designs of Africa should not be mistaken for a universal or pancultural phe-

nomenon; they are culturally specific. The next chapter will examine the

question of their mathematical specificity.

....z.

CHAPTER

-Intention-

and

invention-

design

- Before we can discuss the fractal shapes in African settlement architectures as

geometric knowledge, we need to think carefully about the relation berween mate-

rial designs and mathematical understanding. Designs are best seen as positioned

on a range or spectrum of intention. At the bottom of the range are uninten-

tional pasterns, created accidentally as the by-product of some other activity.

In the middle of the range are designs that are intentional but purely intuitive,

with no rules or guidelines to explain its creation. At the upper end of the range,

we have the intentional application of explicit rules that we are accustomed to

associating with mathematics. The following sections will examine the fractal

designs that occur in various positions along this intentionality spectrum.

Fractals from unconscious activity

An excellent example of unintentional fractals can be found in the work of Michael

Batty and Paul Longley (1989), who examined the shape of large-scale urban sprawl

surrounding European and American cities (fig. 4.1). While the blocks of these

cities are typically laid out in rectangular grids, at very large scales--around 100 ' •

square miles-we can see that the process of population growth has created an

irregular pattern. This type of fractal, a "diffusion limited aggregation," also

5°

Introduction

FIGURE 4.1

Urban sprawl in London

Large-scale urban sprawl

generally has a fractal

structure. The urban sprawl

fractals only exist at very

large scales--about 100 sq.

miles—-and result from che

unconscious accumulation

of arlian popularion dynamics.

At levels of conscious intent

(e.g., the grid of city blocks),

European cities are typically

Euclidean. Area is 10 X 10

kilometers.

(Reprinted with permission from

Batty et al. 1989.)

occurs in chemical systems when particles in a solution are attracted to an elec-

trode. Fractal urban sprawl is clearly the result of unconscious social dynamics,

not conscious design. At the smaller scales in which there is conscious planning,

European and American settlement architectures are typically Euclidean.

Fractals from nature: mimesis versus modeling

It might be tempting to think that the contrast between the Euclidean-designs

of Europe and the fractal designs of Africa can be explained by the important role

of the natural environment in African societies. But this assumption turns out

to be wrong; if anything, there is a tendency for indigenous societies to favor Euclid-

ean shapes. Physicist Kh. S.,Mamedov observed such a contrast in his reflections

on his youth in a nomadic culture:

My parents and countrymen... up to the second world war had been

nomads. ... Outside our nomad tents we were living in a wonderful kingdom

of various curved lines and forms. So why were the aesthetic signs not formed

from them, having instead preserved geometric patterns... ? [I)n the cities

where the straight-line geometry was predominant the aesthetic signs were formed

... with nature playing the dominating role.... [The nomad did not need the

"portrait" of an onk to be carried with him elsewhere because he could view all

sorts of oaks every day and every hour ... while for the townsfolk their inclina-

tion to nature was more a result of nostalgia.

(Mamedov 1986, 512-513)

Intention and invention in design

Contrary to romantic portraits of the "noble savage" living as one with

nature, most indigenous societies seem quite interested in differentiating them-

selves from their surroundings. It is the inhabitants of large state societies, such

as those of modern Europe, who yearn fo mimic the natural. When European

designs are fractal, it is usually due to an effort. to mimic nature. African fractals

based on mimicry of natural form are relatively rare; their inspiration tends to

come from the realm of culture.

How should we place such nature-based designs in our intentionality spec-

trum? That depends on the difference between mimesis and modeling.

< Mimesis)

is an attempt to mirror the image of a particular object, a goal explicitly stated

by Plato and Aristotle as the essence of art, one that was subsequently followed

in Europe for many centuries (see Auerbach 1953). A photograph is a good example

of mimesis. A photo might capture the fractal image of a tree, but it would be

foolish to conclude that the photographer knows fractal geometry. If artisans are

simply trying to copy a particular natural object, then the scaling is an unintended

by-product.

The most important attributes that separate mimesis from modeling are

abstraction and generalization. (Abstraction is an attempt to leave out many of

the concrete details of the subject by creating a simpler figure whose structure

is still roughly analogous to the original-often called a "stylized" representation:

in the arts? Generalization means selecting an analogous structure that is com-

mon to all examples of the subject; what is often referred to as an "underlying"

form or law.' For example, Mandelbrot (1981) points to the European Beaux Arts

style as an attempt not merely to imitate nature, but to "guess its laws." He notes

that the interior of the Paris opera house makes use of scaling arches-within-arches;,

a patremn that generalizes some of the scaling characteristics of nature, but is not

a copy of any one particular natural object.

Since the ultimate generalization is a mathematical model, why didn't

design practices such as the Beaux Arts style result in an early development of

fractal geometry? For Europeans, such lush ornamentation was presented-and

generally accepted-as embodying the opposite of mathematics; it was an effort

to create designs that could only be understood in irrational, emotional, or intu-

itive terms. Even some movements against this attempt, such as the use of dis-

tinctly Euclidean forms in the high modern arts style, simply reinforced the

association because it only offered a reversal, suggesting that "mathematical"

shapes (cubes, spheres, etc.) could be esthetically appreciated. With rare

exceptions (e.g., Thompson 1917), mimesis of nature in pre-WW 11 European

art traditions merely furthered the assumption that Euclidean geometry was the

only true geometry.?

Introduction

The difference between mimesis and modeling provides two different posi-

tions along the intentionality spectrum. The least intentional would be merely

holding a mirror to nature--for example, if someone was just shooting a cam-

era or painting a realistic picture outdoors and happened to include a fractal object

(cloud, tree, etc.). This mimesis does not count as mathematical thinking. More

intentional is a stylized representation of nature. If the artist has reduced the nat-

ural image to a structurally analogous collection of more simple elements, she has

created an abstract model. Whether or not such abstractions move toward more

mathematical models is a matter of local preference.

The two examples of African representations of natute we observed in

the previous chapter certainly show that the artisans have gone beyond

mere mimesis. The Mandiack cobra pattern we saw in figure 3.2. shows a strictly

systematic scaling pattern. This textile design conveys the scaling property

of the natural cobra skin pattern-diamonds at many scales— in a stylized or

abstract way. We can take this idea a step further by examining another

Bwami bat sculpture (fig. 4.2). This spiral pattern is also a stylized repre-

sentation of the natural scaling of the bat's wing, but it.is a different geometric

design than the expanding zigzag pattern we saw in figure 3.4c. It is more styl-

FIGURE 4.2

Stylized sculpture of a bat

Another Lega 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.

(By permission of the Museum of African Art, N.Y.)

Intention and invention in design

ized in the sense of being further abstracted from the original natural bat's

wing. This provides further evidence that the sculptors were focused on the

scaling properties--the generalized underlying feature-and not particular con-

crete details.

The greatest danger of this book is that readers might misinterpret its?

meaning in terms of primitivism. The fact that African fractals are rarely the result

of imitating natural forms helps remind us that they are not due to "primitives

living close to nature." But even for those rare cases in which African fractals

are representations of nature, it is clearly a self-conscious abstraction, not a mimetic

reflection. The geometric thinking that goes into these examples may be simple,

but it is quite intentional.

The fractal esthetic

Just as we saw how designs based on nature range from unconscious to inten-

tional, artificial designs also vary along a range of intention, with some simply

the result of an intuitive inspiration, and others a more self-conscious applica-

tion of rules or principles. The examples of African fractals in figure 4-3 did not

appear to be related to anything other than the artisan's esthetic intuition or

sense of beauty. As far as i could determine from descriptions in the literature

and my own fieldwork, there were no explicit rules about how to construct these

designs, and no meaning was attached to the particular geometric structure of

the figures other than looking good. In particular, I spent several weeks in

Dakar wandering the streets asking about certain fractal fabric patterns and jew-

elry designs, and the insistence that these patterns were "just for looks" was so

adamant that if someone finally had offered an explanation, I would have

viewed it with suspicion.

Since some professional mathematicians report that their ideas were pure

intuition—a sudden flash of insight, or "Aha!" as Martin Gardner puts it-we

shoukdn't discount the geometric thinking of an artisan who reports "I can't tell

you how I created that, it just came to me." Esthetic patterns clearly qualify as

intentional designs. On the other hand, there isn't much we can shy about the

mathematical ideas behind these patterns; they will have to remain a mystery unless

something more is revealed about their meaning or the artisan's construction tech-

niques, It is worth noting, however, that they do contribute to the fractal design

theme in Africa. Esthetic patterns help inspire practical designs, and vice versa.

Since ancient trade networks were well established, the diffusion of esthetic pat-

terns is probably one part of the explanation for how fractals came to be so wide-

spread across the African continent.

FIGURE 4-3

Esthetic fractals

(a) Meurant (quoted in Reif 1996) reports

that the Mbuti women who created this

fractal design, a bark-cloth painting, told

him the design was not "telling stories,"

nor was it "representing any particular

object." (b) Scaling patterns can be

found in many African decorative designs

that are reported to be "just for beauty."

Upper left, Shoowa Raffta cloth; lower left,

Senegalese tie dye; right, Senegalese

pendant.

(a, courtesy Georges Meruant.b: Upper left,

British Museum; lower left, from Musée Royal

de l'Afrique Central, Belgium; right, photo

courtesy IFAN, Dakar.)

Intention and invention in design

FIGURE 4.4

The quincunx fractal

A customer in Touba, Senegal, selects a fractal quincunx pattern for his leather neck bag. The

quincunx is historically important because of its use by early African American "man of science"

Benjamin Banneker.

Of course, there are plenty of African designs that are strictly Euclidean,

but even these can occur in "fractalized" versions. One particularly interesting

example is the quincunx (fig. 4-4). The basic quincunx is a pattern of five squares,

with one at the center and one at each corner. The design is common in Sene-

gal, where it is said to represent the "light of Allah." The quincunx is histori-

cally important because the image was recorded as being of religious significance

to the early African American "man of science" Benjamin Banneker. Since evi-

dence shows that Banneker's grandfather (Bannaka) came from Senegal, the

quincunx is a fascinating possibility for geometry in the African diaspora (see

Eglash 1997c for details). Because of the fractal esthetic, this religious symbol'

is often arranged in a recursive pattern-five squares of five squares as shown

in figure 4.4 in the design for a leather neck bag.

Finally, there are also examples of the fractal esthetic in common house-

hold furnishings. Euro-American furniture is differentiated by form and func-

tion--stools are structured differently from chairs, which are structured

differently from couches. But in African homes one often sees different sizes

of the same shape (fig. 4.5). A similar difference has been noted in cross-cultural

comparisons of housing. Whereas Euro-Americans would never think to have

a governer's mansion shaped like a peasant's shack (or vice versa), precolonial

African architecture typically used the same form at different sizes (as we saw

for the status distinctions in the Ba-ila settlement in chapter 2). It is unfortunate

that this African structural characteristic is typically described in terms of a

lack--as the absence of shape distinctions rather than as the presence of a scal-

ing design theme.

Introduction

FIGURE 4.5

The fractal esthetic in household objects

African stools, chairs, and benches are often created in a scaling series.

(Photo courtesy of Africa Place, Inc.)

Conclusion

We now have some guidelines to help determine which fractal designs should count

as mathematics, which should not, and which are in between. Figure 4:6 sum-

marizes this spectrum. Fractals produced by unconscious activity, or as the unin-

tentional by-product from some other purpose, cannot be attributed to indigenous

concepts. But some artistic activities, such as the creation of stylized represen-

Unintentional

Unconscious activity

•urban spraw!

Accidental fractals

•"mirror" portrait of nature

(mimesis; e.g., photography)

Intentional

but implicit

Conscious use of natural scaling

•stylistic abstraction of natural scaling

Esthetic design

• intuitive fractal design theme

FIGURE 4.6

From unconscious accident to explicit design

Intentional

and explicit

Construction techniques

Knowledge systems

Intention and invention in design

tations of nature or purely esthetic designs, do show intentional activity focused

on fractals. Such examples may be restricted in terms of geometric thinking-

the-artisans may only report that the design suddenly came to them in a flash of

intuition -but these are clearly distinguished from those which are unconscious

or accidental. The following chapters will consider examples that are not only

intentional, but also include enough explicit information about design techniques

and knowledge systems to be easily identifable as mathematical practice and ideas.

•African

fractal

mathematics

PART

CHAPTER

Geometric-

-algorithms

— The word "algorithm

derives from the name of an Arab mathematician,

Al-Khwarizmi (с. 780-850 C.E.), whose book Hisab al-jabr w'al-muqabala (Cal-

culation by Restoration and Reduction) also gave us the word "algebra."

Although Al-Khwarizmi focused on numeric procedures for solving equations,

the modern term "algorithm' applies to any formally specified procedure. A geo-

metric algorithin gives explicit instructions for generating a particular set of spa-

tial patterns. We have already seen how iterations of such pattern-generating

procedures can produce fractals on a computer screen; in this chapter we will

examine two indigenous algorithis that also use iteration to produce scaling

designs: the 45-degree- angle constructions of the Mangbetu, and the lusona draw-

ings of the Chokwe.

Geometry in Mangbetu design

The Mangbetu occupy the Uele River area in the northeastern part of the

Democratic Republic of Congo (formally Zaire). Archaeological evidence shows

iron smelting in the area since 2300 B.C.E., but the Mangbetu, coming from drier

lands around present-day Uganda, did not arrive until about xooo c.E. Through

both conflict and cooperation, they exchanged cultural traditions with other

6 x

African fractal mathematics

societies of the area: Bantu-speaking peoples such as the Buda, Bua and Lese, and

Ubangian-speaking peoples such as the Azande, Bangba, and Barambo. Around

1800 a number of small chiefdoms were consolidated into the first Mangberu king-

dom. Although it lasted only two generations, a tradition of courtly prestige con-

tinued even in small villages and spread to many of the Mangbetu's trading partners.

This combination of cultural diversity, exchange, and prestige resulted in a

thriving artistic tradition.

A detailed account of Mangbetu history and traditions can be found in

African Reflections: Art from Northeastern Zaire. Schildkrout and Keim (1990) begin

their analysis by showing that the most famous aspect of Mangbetu art, the

"naturalistic look," was actually quite rare in the traditional Mangbetu society

of the nineteenth century. During a research expedition to the Congo in x9x4

(the origin of the photos used here), mammalogist

Herbert Lang became fascinated with lifelike carvings

of human figures, and as word spread that he was pay-

ing high prices for them, more of these carvings were

produced. Other collectors came to buy these pieces, and

eventually the economic rewards for producing natu-

ralistic Mangbetu art became so strong that it replaced

other styles.

Schildkrout and Keim show that originally the

most important esthetic was not naturalism, but abstract

geometric design. The indigenous fascination with arti-

fice and abstraction was ignored by colonizers, and

their preconceptions of Africans as nature-loving

"children of the forest" became a self-fulfilling expec-

tation. But the artifacts and photographic records from

the x914 expedition provide us with excellent examples

of traditional Mangbetu patterns, as well as an oppor-

tunity to infer some of their techniques.

Figure 5.1 shows the decorative end of an ivory

hatpin. Like the architecture and esthetic patterns we

have seen, this is clearly a scaling design, but the pre-

cision of the pattern suggests that there may be a more

FIGURE 5.1

Mangebetu ivory sculpture

(Transparency no. 3935, photograph by Lynton Gardiner, courtesy

American Museum of Natural History.)

Geometric algorithms

formal geometric process at work. Similar design can be seen at work in the Mang-

betu's geometric style of personal adornment. Figure 5.za shows a Mangbetu hair-

style, popular during the time that this carving was,created (about 1914), which

featured a disk angled to the vertical at 4'5 degrees. Men often wore a hat with

the top flattened, forming the same angle, as seen in figure 5 2b. Just as a plane

cuts diagonally through the top of the heads in the ivory sculpture of figure g.1,

real Mangbetu headdresses also terminated in a 45-degree angle.

This was only one part of an elaborate geometric esthetic based on mul-

riples of the 45-degree angle. Figure 5.2b shows an ivory hatpin, ending in a disk

perpendicular to it, inserted perpendicular to the hat. To its right, a small ivory

arrow pinned to the hat points horizontally, thus forming an angle of 135 degrees

with the hatpin. Each part of the ensemble was aligned by a multiple of the

45-degree angle. This adornment style included artificial elongation of the head,

which is clearly visible in the photograph in figure 5.2b. Elongation was accom-

plished by wrapping a cloth band around the head of infants; the woman in

figure 5.za is weaving one of these bands. Head elongation resulted in an angle

of 135 degrees between the back of the head and the neck.

FIGURE 5.2

Geometric design in Mangbetu personal adornment

(a) Mangbetu woman weaving headband. (b) Mangbetu chief.

(a, negative no. 111919, photograph by H. Lang, courtesy American Museum of Natural History;

b, negative no. 224105, photograph by H. Lang, courtesy American Museum of Natural History.)

African fractal mathematics

While the Mangbetu geometric conception of the body may have inspired

the 45 degree sangle design theme, those designs were certainly not limire ros pre

mimicry of anatomy. We can clearly see this in their musical instruments.

drum in figure 5-3a, for example, has its upper surface cut at a 45-degree angle

to the vertical. The stringed instrument shown in figure 5. 3b has a resonator that

meets the vertical tuning stem at a 135-degree angle. Even in the case of anthro-

pomorphic designs, the artisans elaborated on the human form in ways that show

FIGURE 5•3

Geometric design in Mangbetu musical instruments

(a) Drum. (b) Harp.

(a, negative no. 1 1 1896, photograph by H. Lang, courtesy American Museun of Natural History;

b, couriesy Rietberg Museum Zurich, photograph by Weustein and Kauf.)

Geometric algorithms

creative—and not merely imitative-applications of geometrical thinking.

For example, there is an anthropomorphic decorative motif at the end of the

tuning stem shown in figure 5. 3b, but these human heads are not simply mim-

icking human form. In figure 5.2b we saw that the Mangbetu had a 135-degree

angle between the back of the head and the neck. The carved heads in figure

5.3b have a go-degree angle between the back of the head and the neck. Such

distortions indicate active geometric thinking rather than passive reflection of

natural anatomical angles (which, recalling the artificial head elongation, were

not so natural to begin with).

There are also purely abstract designs that make use of multiples of 45 degrees,

as we see in figure 5.4. Modern Mangbetu report that the creation of a design

reflected the artisan's desire to "make it beautiful and show the intelligence of

¿the creator" (Schildkrout and Keim 1990,

100). This suggests another reason for arti-

sans to achere to angles that are multiples of

45 degrees: if there were no rules to follow,

then it would have been difficult to compare

designs and demonstrate one's ingenuity. By

restricting the permissible angles to a small

set, they were better able to display their

geometric accomplishments:

Combining this 45-degree-angle con-

struction technique with the scaling prop-

erties of the ivory carving in figure 5.1 can

reveal its underlying structure. The carving

has three interesting geomenic features:

first, each head is larger than the one above

it and faces in the opposite direction. Sec-

ond, each head is framed by two lines, one

formed by the jaw and one formed by the

hair; these lines intersect at approximately

Do degrees. Third, there is an asymmetry:

the left side shows a distinct angle about

120 degrees from the vertical.

FIGURE 5-4

Mangebetu ivory sculpture

(Transparency no. 3929, photograph by Lynton Gardiner,

courtesy American Museum of Natural History.)

FIGURE 5-5

Geometric analysis of an ivory sculpture

tan 0, =

0, = arctan

3 = 180

FIGURE 5.6

Geometric relations in the Mangbetu iterative squares structure

Since 0, and 0, are the alternate interior angles of a transversal intersecting two parallel lines,

6, = 02.

African fractal mathematics

All of these features can be accounted for by the structure shown in fig-

ure 5.5. This sequence of shrinking squares can be constructed by an iterative

process, bisecting one square to create the length of the side for the next

square, as indicated in the diagram. We will never know for certain if this iter-

ative-squares construction was the concept underlying the sculpture's design, but.

it does match the features identified above. In the ivory sculpture, the left side

is about 20 degrees from the vertical. In the iterative-squares structure, the left

side is about 18 degrees from the vertical, as shown in figure 5.6. Here we see

that the construction algorithin can be continued indefinitely, and the result-

ing structure can be applied to a wide variety of math teaching applications, from

(simple procedural construction to trigonometry (Eglash 1998a).

Lusona

The Chokwe people of Angola had a tradition of creating patterns. by drawing

lines called "lusona" in the sand. Gerdes (1991) notes that the lusona sand

drawings show the constraints necessary to define what mathematicians call an

"Eulerian path": the stylus never leaves the surface and no line is retraced. The

lusona also tend to use the same pattern at different scales, that is, successive iter-

ations of a single geometric algorithm. Figure 5.7 shows the first three iterations

of one of the dozens of lusona that were recorded by missionaries during the nine-

teenth century, when the lusona tradition was still intact.

As in the case of the Mangbetu 45-degree constructions, the restriction to

an Eulerian path provides the Chokwe with a means to compare designs within

a single framework, and to show how increasing complexity can be achieved within

these constraints of space and logic. But unlike the competitive basis for com--

parison that the Mangbetu describe, the Chokwe made use of these figures to cre-

ate group identity. The reports indicate that the lusona were used in an age-grade

initiation systern; rituals that allowed each member to achieve the status of

reaching the next, more senior level of identity. By using more complex lusona,

the iterations of social knowledge passed on in the initiation become visualized

by the geometric iterations. In chapter 8 we will see other examples of iterative

scaling patterns in initiation rituals. This tradition of group identity through knowl-

edge of the lusona was also deployed by the Chokwe as a way to deflate the ego

of overconfident European visitors, who found themselves unable to replicate the

lusona of many children.

Conclusion

These two examples, the Mangbetu ivory carving and the lusona drawings, help

us see that. African fractals are not just the result of spontaneous intuition; in some

Geometric algorithms

cases they are created under rule-bound techniques equivalent to Western

mathematics. And their cultural significance makes it clear that all mathe-

matical activity-no matter in which sociery it is found—is produced through

an interaction between the freedom of local human invention and the univer-

sal constraints we discover in space and fogic.

"Path of a hunted bird."

FIGURE 5•7

Lusona

(a) These figures, "lusona," were tradicionally 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 frst and chird iterations of another lusona algorithm carved into a

wooden box lid.

(a, based on drawings in Gerdes 1995.)

African fractal mathematics

Recall that in both examples the role of "constraint" was crucial to the devel-

opment of their scaling geometry. For the Mangbetu's design it was the constraints

of straight-edge construction with angles at multiples of 45 degrees. For the

Chokwe's lusona it was the constraints of an Eulerian path. Büt in each case the

choice of particular objective constraints-deciding which of the infinite laws

of space and logic we are concerned with— was established by and for the social

relations of the community. In the case of the Mangbetu it was artistic compe-

tition, and in the case of the Chokwe it was age-grade identity. In other words;

the invention and discovery components of mathematics are inextricably linked

through social expression.

Philosophic perspectives on the relation of culture and mathematics will

be further discussed in part i1, but to do so we need a fuller portrait of African

fractal geometry. The next chapter will examine African conceptions of the most

fundamental characteristic of fractals: nonlinear scaling.

CHAPTER

Scaling-

- We have already seeri many examples of scaling in African designs. In the settle-

ment architecture of chapter 2, for example, the computer simulations clearly show

that we can think about these patterns in terms of fractal geometry. How do the

African artisans think about scaling? Is it just intuition, or do they use explicit

mathematical practices in thinking about similarity at different sizes? By exam-

ining varieties of designs with different scaling properties, and comparing these

with the artisans' discussions of the patterns, we can gain some insight into scal-

ing as a mathematical concept in African cultures.

Power law scaling in windscreens from the Sahel

The Sahel is a broad band of arid land between the Sahara Desert and the rest

uf sub-Saharan Africa. Since there are few trees and a great deal of millet cul-

tivation, it is not surprising chat artisans use millet stalks to weave fences, walls,

and other constructions. But the consistent use of a nonlinear scaling pattern in

rhese straw screens (fig. 6.za) is a bit odd. Rather than uniform lengths, the rows

of millet straw get shorter and shorter as they go up. In the United States we are

used to the image of "the white picket fence" as a symbol of unchanging, linear

repetition, yet here the fences are distinctly nonlinear. While I was in Mali on

b Windscreen under construction

in Mali.

The straw windscreen in Niger.

Step 1: Lay a new

bundle across eight

of the first-layer

bundles.

first-layer

bundles.

FIGURE 6.1

An African windscreen

(a) The diagonal lengchs of chese rows from boccom 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. The length of each diagonal row--how high up you go before doing the interlace step-—is

determined by counting a certain number of 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 double 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, photo by permission of Gardi 1973.)

(figure continues)

Scaling

the outskirts of the capital city of Bamako, i had the opportunity to interview

some of the artisans who create these screens and was provided with a striking

example of indigenous application of the scaling concept.

The artisans began by explaining that in "fertile areas" such as the forests

of the south, the screens are not made with scaling rows but rather with rows of

long, uniform length. This is because the long rows use less straw and take less

time to make. But here in the Sahel, they said, we have strong winds and dust.

The shortest rows are the ones that keep out dust the best, because they are the

tightest weave. But they also take more materials and effort. "We know that the

wind blows stronger as you go up from the ground, so we make the windscreen

to match—that way we only use the straw needed at each level."

The reasoning the artisans reported is equivalent to what an engineer

would call a "cost-beneft" analysis; developing the maximum in function (keep-

ing out dust) for a minimum of cost (effort and materials). My primary interest

here is in showing that the scaling concept in Africa can be much more sophis-

ticated than just an observation, "the same thing in different sizes." The creation

Assuming decrease in wind

penetration is reciprocal of length:

a =1

(wind engineers:

a = 1/3)

Gradiant wind

Vg = V constant

• .

-0.4

₴ -0.6

• •

- V

Boundary-layer wind

V = V (z)

-1.0

Power law: Vt) = V, (L_)ª

Log (H)

1t0

les)

FIGURE 6.1 (continued)

(d) The relation between wind speed and vertical height as shown in the Wind Engincering

Handbook. (e) The African windscreen makers say that they have scaled the rows of straw to

match the change of wind speed with height. If we assume, just for simplicity, that the decrease in

wind penetration is the reciprocal of the length, then we can get the African estimate for a by

measuring the slope of row length versus height on a log-log graph. This gives a = 1, whereas the

engineers use a = ⅓— not bad for a ballpark estimate.

Note that the graph is in a very straight line, except where the discrete nature of the screen

(the screen makers must count in whole-number units due to the straw bundles) forces an approxi-

mation by repeating the same length twice.

African fractal mathematics

of the windscreen as an optimal design required matching the scaling variation

of wind speed versus height to a scaling variation in lengths of straw. By trans-

ferring this concept between two completely, different domains, the artisans

have demonstrated that they understand scaling in the abstract; indeed, the design

essentially plots the relation of wind speed to height on a straw graph.

Although I was concerned only with the overall relation of scaling and

reasoning, I measured the rows just to see how close they came to what a West-

ern engineer would develop for an optimal match with wind speed. If the straw

screen had linear scaling, then each row would decrease in length by the same

amount (e.g., 12 inches, 1o inches, 8 inches, etc.). But the rows decrease less and

less with height; it turns out that the screen design shows a close fit to what is

called a "power law"---that is, it scales according to an exponent (fig. 6.rc).

Figure 6.1b, reprinted from the Wind Engineering Handbook, shows the equation

of wind speed with height most commonly used by engineers-—also a power law.

So the Sahel windscreen is not only a practical application of the abstract scal-

ing concept, it is also a fairly accurate one. Of course, one might object that the

indigenous engineers did not actually set up the algebra and perform the opti-

mizing calculation. But I asked three American mathematicians how they would

set up these equations to determine the optimal design, and all three said the same

thing: "I wouldn't solve it analyticaily, l'd just graph the equations on the com-

puter and see where the functions peaked." Whether we make our graphs on a

computer screen or a straw screen doesn't matter, as long as we get the right answer.

Stretching space in kente cloth

If someone in America were asked to think of an African textile, kente cloth would

be the most likely image. Its combination of strong colors. bold designs, and asso-

ciations with ancient kingdoms of West. Africa has made it a favorite for imports.

But most of the imported kente cloth is created by automated machine, and while

I would fiercely defend it as "authentic," the need for pattern repetition in

automation has eliminated a wonderful scaling transformation that can be seen

in the older patterns created on hand looms (fig. 6.za). The scaling-change is not

just small and large versions of the same thing; rather, it is as if the design was

drawn on a rubber sheet, which was half stretched and half contracted. In

Ghana I traveled to the village of Bonwire, where hand-loom weaving is still prac-

ticed, and asked the artisans there why this scaling transformation was created.

The weavers replied that they think of the compressed version as the orig-

inal pattern, and said they call it "spreading" when they create the stretched ver-

sion. The reason they gave for the spreading pattern can best be understood with

(O,

(6. 1)

(1,4)

(0, 0)

(1, 0)

FIGURE 6.2

Kente cloth

(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 concinuous, 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 pastern, to match the scaling of the saccadic eye

movements. (c) The compression of space is used in mathematics to model scaling patterns, like

char of the saccadic eye movements. Mathematicians call chis a "contractive affine transformation."

African fractal mathematics

the following experiment. Hold your finger in front of your face, and without mov-

ing your head, track the finger with your eyes as you move it slowly across the

visual field. Now try the same thing again, smoothly tracking the visual field, but

without the finger to guide your eyes. You'll find that it can't be done! Your eye

moves involuntarily in little jumps, called "saccadic" movements. When a per-

son comes into your visual field, those same saccadic movements densely cover

the face, and then make a few glances at the body (fig. 6.2b). The weavers in Bon-

wire reported the same idea: "When you see a person you give heavy looks to the

face, and light looks to the body:" They explained that the purpose of the scal.

ing change is to match this visual scaling: the compressed part of the pattern is

the cloth worn over the shoulder, and the stretched part is worn down the 1

length of the body.

The mathematical term for this operation is "contractive affine transfor-

mation" (fig. 6.2c), which can be used for creating fractals through a method called

"iterated function systems" (see Wahl 1995, 156-157). In kente cloth there is

no iteration-the operation is done only once-but it does show active think-

ing about a scaling transformation. As in the case of the windscreen, the weavers

are taking a rather abstract observation about a time-varying quantity and map-

ping this model into a material design.

Logarithmic spirals

In chapter 3 (fig. 3.2) we examined the contrast between nonlinear concentric

circles and linear concentric circles. In the same way, nonlinear spirals are easy

to understand if we contrasi them widi linear spirals (fig. 6.za). The linear spi-

ral, also called an Archemedean spiral in honor of the Greek mathematician who

favored it, is in the shape of a coiled rope or watch spring. Each revolution brings

you out by the same distance (just as each layer in the linear concentric circle

was the same thickness). For that reason, a linear spiral of a finite diameter can

have only a finite number of turns. A nonlinear spiral of finite diameter can have

an infinite number of turns, because even though there is less and less space remain-

ing as one goes toward the center, the distance between each revolution can get

smaller and smaller.

A good example of this nonlinear scaling can be seen in the logarithmic

spiral (fig. 6.3b). Logarithmic spirals are typical structures in two different cat-

egories of natural phenomena. On the one hand, they are found in astonishing

varieties of organic growth. Theodore Cook's The Curve of Life (1914), for

example, shows dozens of logarithinic spirals from every branch of the evolutionary

tree: snail and nautilus shells; the horns of rams and antelope; algae, pinecones,

Scaling

FIGURE 6.3

Spirals

(a) In the linear spiral of Archimedes, there is a

constant distance between each revolution

Snly a finite number of turns can fit in thi.

finite space. (b) In the logarithmic spiral, there

is an increasing distance between each

revolution. An infinite number of turns can fi

in this finite space.

г = 0

*= 1.19

and sunflowers; and even anatomical parts of the human ear and heart. Many

researchers have speculated on why this is so; their answer is cypically that liv.

ing systems need to keep the same proportions as they grow, and so a scaling curve

allows the same form to be maintained. I prefer to think of it as recursion: if we

look at the chambered nautilus, for example, we can think of each new cham-

"ber as the next iteration through the same scaling algorithm.

On the other hand, logarithmic spirals are also found in fluid turbulence.

We become aware of this when we watch a hurricane from space, or simply admire

the swirls of water along a riverbank. Explanations for these fluid curves are much

less speculative, since we can write equations for turbulence and show them pro-

ducing logarithmic spirals in computer simulations (as we will see in chapter 7).

But the Euro-American tradition is not the only one interested in simulacra. The

artists of what is now Ghana-particularly those of the Akan society-long ago

abstracted the logarithmic spiral for precisely these two categories. Their sym-s

bols for the life force (hg. 6.4a) are clearly related to the "curves of life," and icons

for Tanu, the river god (fig. 6.4b), show the logarithmic swirls of turbulence.

African fractal mathematics

FIGURE 6.4

Logarithmic spirals

(a) Several Chanaian iconic figures, such as this goldweight, link a spiritual force with the

structure of living systems through logarithmic spirals. This example is particularly striking since

it shows how spirals can be combined with bilateral symmetry to create other self-similar shapes

(the large diamond shape created by the meeting of the large spiral arms is repeated on either side

by the small diamond at the meeting of the small spiral arms). (b) This figure, again based on

logarichmic spirals, appears on the temples of Tanu, the river god, and links this spiritual force to

the geometric structure of fluid turbulence.

(a, photo courtesy Doran Ross.)

Again, we need to avoid the assumption that the Ghanaian log spirals are

simply minetic "reflections" of nature, and examine how they. are used and

designed. The Akan and other societies of Ghana created a collection of specific

icons that several researchers have compared to a written language. But rather than

composed of the vast number of symbols we call "words," the Ghanaian symbolic

vocabulary is much smaller, and each symbol refers not to a single word but an

entire social, religious, or philosophical concept. Moreover, in many cases the

structure of the symbol is not arbitrary (as Gregory Bateson said, "There is noth-

ing 'sevenish' about the numeral z"), but rather is shaped so that each icon's geo-

metric structure recalls the concept it represents. In other words, they are not

only abstractions in the sense of being stylized, but also generalizations in the sense

of the designers' intent to find an underlying structure that all examples have in

common. For this reason we can accurately describe the Ghanaian log spiral icons

as geometric models for the phenomena of organic growth and fluid turbulence.

Some aspects of these designs illustrate a conscious reflection on their geo-

metric properties. Figure 6.4a, for example, not only displays the log spiral's Euclid-

Scaling

ean symmerry-for we can see how clockwise and counterclockwise spirals com-

pare-but also experiments with other kinds of scaling symmetry: note that the

large diamond shape created by the meeting of the large spiral arms is repeated

on both sides by the small diamond at the meeting of the small spiral arms. Can

this scaling be continued in further iterations? I will leave that question as an

exercise for the readers.

•There are hints that the precolonial Ghanaian designers were headed

toward a quantitative approach in their log spiral designs. Figure 6.5a shows

the sculpture of a water buffalo in which they have inscribed uniform discrete

steps. I don't think this was motivated by numeric measures, but rather the

reverse. By cutting these steps we can clearly gauge the nonlinear nature of the

spiral-the way steps of a constant increment show an increasing amount of

curve generated-and this practice could have led to quantitative measures.

Another move in that direction would generalize such discretized logarithmic

FIGURE 6.5

Logarithmic scaling

in Ghanaian designs

(a) Logarichmic scaling can be demonstrated

in a three-dimensional curve by showing

how discrete steps of the same vertical

increment lead to rapidly increasing area.

(b) Overhead view of pyramid-shaped

goldweight. (c) Logarithmic plot of

goldweight triangle lengths.

(a, photo from the Metropolitan Museum of Art.

b, photo courtesy Geurge Arthur, Marshall

University.)

length from base to apex of triangle

( log scale)

step of pyramid

FIGURE 6.6

Adaptive scaling with triangles

(a) Antelope headdress created by the Krumba of Burkina Faso. (b) Mask sold in Accra, Ghana,

based on design used in the Sakara-Bounou religious dances. (c) Representation of the water spirit

created by the Baga of Guinea. (d) Sculpture from the Congo. (e) A Kikuyu wooden shield.

The wood has a nonlinear curve toward the center, and the triangles are scaled to match.

(a, contesy Musée de l'Horme, c, Metropolitan Museum of Are; photo by Elior Elisofon. d, Detroit

Museum of Art. e, British Museum; from Zaslausky 1973.)

Scaling

scaling to forms other than spirals, and that did indeed occur, as we can see in

figure 6.5b, one of the Akan gold weights. A plot of the length of these triangles

(fig. 6.5c) indicates that reasonable accuracy was achieved in this indigenous

logarithmic scaling practice.

Adaptive scaling

So far this chapter has focused on questions of intentionality, precision, and mathe-

matical reasoning in African scaling designs. Adaptive scaling has little mathe-

matical sophistication, but it too is an important part of the African fractal design

theme. By adapting the scale of a pattern to fit various forms, a number of,

esthetic and practical effects can be achieved. These examples fall into two cat-

egories. In conformal mapping, the pattern simply fits along the contours of a con-

crete, preexisting structure. In global mapping, the pattern is distorted by

compression or expansion-as we saw happen along one dimension in kente

cloth-according to a more universal, abstract transformation.

Figure 6.6 shows several examples of conformal mapping on triangles. My

search of the facial markings of antelope of the western Sudan did not turn up

anything like the scaling pattern of figure 6.6a; these triangles are decorative

auditions, sized to fit into the shape of the sculpture. Other examples (fig. 6.6b-e)

show a series of triangles conforming to the scaling contours of a mask, a sin.

nous curve, a carved human figure, and a shield. Figure 6.7a shows conforma!

mapping in the hairstyle Americans call "corn-rowing"; its simulation is shown

in figure 6.7b. The Yoruba name for this style is ipako elede, which means the

nape of the neck of a boar-—because the boar's bristles show a similar nonlinear

scaling: Figure 6.7c shows a hairstyle that combines conformal mapping with

iteration. Adaptive scaling of circles can be seen in the Senegalese textile in

figure 6.7d.

A practical application of conformal mapping appears in figure 6.7e, an

aerial photo of the Nkong-mondo quarter in the city of Edéa in southern

Cameroon, where we see a scaling series of houses. As explained to me by one

member of the neighborhood, Mr. Sosso, the houses were constructed along a nar-

rowing ridge, and the scaling was simply conforming to the natural landscape.

However, the oldest inhabitant of this Bassa neighborhood, Mr. Bellmbock,

told me that the pattern was created because people wanted neighbors of a sim-

ilar economic class next door, so that the range in house size reflected an eco-

nomic gradient, from poorest to wealthiest. Mr. Bellmbock lived in the smallest

house, and Mr. Sosso in the largest, so I would not discount the possibility that

there was an economic scaling as well.

Y l

seed shape using one active line

(gray) and two passive lines.

Fractal model for middle Ipako Electe braid.

FIGURE 6.7

Adaptive scaling based on various shapes

(a, b) A Yoruba hairstyle, Ipako Elede, adapts the scaling of the braids to the nonlinear contours of

the head. (c) This hairstyle begins by braiding a small horseshoe shape in the top center, and then

tracing the contour in increasing perimeters-a combination of adaptive scaling and iteration.

(d) Fitting circles between intersecting curves creates a scaling series in this textile design from

Guinea. (e) An aerial photo of the Nkong-mondo quarter in the city of Edéa in southern

Cameroon, where we see a scaling series of houses.

(a, from Sagay 1983. c, from Sagay 1983. d, photo courtesy IFAN, Dakar.)

Scaling

+ infinity

- infinity

FIGURE 6.8

Mapping from the plane

to a spherical surface

(a) Mapping bars of infinite

length from the plane to a sphere.

(b) A Yoruba hairstyle, Koroba

("hncker.").

(b, from Sagay 1983.)

It is possible to misread these examples of conformal mapping as being the

product of artisans who are strongly guided by concrete forms rather than

abstract thought. But adaptive scaling can also be seen in more abstract

examples: global transformations in which space itself is distorted. This is a com-

mon operation in Western geometry, the most frequent example being a map-

ping between the plane and a sphere (fig. 6.8a). Figure 6.8b shows a hairstyle

that appears to have a planar design mapped onto a spherical surface. Figure 6.9

provides an even more abstract illustration, the inverse of the previous mapping-

now going from spherical to rectangular-and utilizing three dimensions instead

African fractal mathematics

of two. In this Chokwe sculpture, the entire human figure is distorted as if its

spherical volume had been mapped to a cubic volume; the resulting nonlinear

scaling is dramatically illustrated by the discrete steps in the headdress. Art his-

torian William Fagg (1955) made a similar suggestion about other African

designs, which he compared to the drawings of natural growth by biologist

D'Arcy Thompson: "I believe that the morphology of African sculpture may be

usefully studied ... by reference to mathematics... For example in certain masks

FIGURE 6.9

Mapping from a spherical volume

to a rectangular volume

(a) Bastin (1992, 68) shows that this Chokwe crown,

the Cipenya-Mutwe, is made up of linear bands in real

life. The nonlinear scaling we see in this sculpture can

be explained as the inverse of the transformation we saw

in figure 6.8a. Rather than shrinking as we move from

che center to the margins, the inverse mapping causes

expansion from center to margins. This is not only the

inverse of the previous mapping, but also operates on

three-dimensional volume rather than surface. Similar

transformations are used in neuroscience to model the

ways that tactile receptors are mapped from body to

brain, since there is a much greater density of sensory

neurons at the extremes. (b) The reason for this transformation is to invoke the impression of

power and stability (Chanda 1993). The meaning has nothing in particular to do with geometric

mapping, other than achieving the desired effect, but it is interesting to note that the transfor-

mation is uniformly applied to all external areas, even to the extent of deforming the forehead.

(a, courtesy Jacques Kerchache and Museum of Mankind, London. b, courtesy Musern of the Philadelphia

Civic Center.)

Scaling

for the Gelde society the natural... physiognomy is "blown up,' so to speak, in

a way which could be plotted on a set of flaring exponential coordinates."

(1917, 43).

Conclusion

The examples of scaling designs in this chapter vary greatly in purpose, pattern,

and method. While it is not difficult to invent explanations based on unconscious

social forces for example, the flexibility in conforming designs to material sur-

faces as expressions of social flexibility—1 do not think that any such explana-

tion can account for this diversity. From optimization engineering, to modeling

organic life, to mapping between different spatial structures, African artisans have

developed a wide range of tools, rechniques, and design practices based on the

conscious application of scaling geometry. In the next chapter, we will see that

African numeric systems also share many fractal characteristics.

CHAPTER

-Numeric

"systems

- So far we have focused on geometric structures rather than numeric systems. The

only exception was in the windscreen, where the nonlinear scaling was created

by counting a specific sequence of diagonal straw rows. But there are many

other instances in which the African approach to fractal geometry makes use of

numbers.

Nonlinear additive series in Africa

The counting numbers (1,2,3 ...) can be thought of as a kind of iteration, but

only in the most trivial way. ' It is true that we could produce the counting num-

bers from a recursive loop, that is, a function in which the output at one stage

becomes the input for the next: X,+, = X, + 1. But this is a strictly linear series,

increasing by the same amount each time-the numeric equivalent of what we

saw in the linear concentric circle and linear spiral. Addition can, however, pro-

duce nonlinear series,? and there are at least two examples of nonlinear additive

series in African cultures. The triangular numbers (1,3,6,10,15 ...) are used in

a game called "tarumbeta" in east Africa (Zaslavsky 1973, 131). Figure 7.1 shows

how these numbers are derived from the shape of triangles of increasing size, and

how the numeric series can be created hy a recursive loop. As in the case of cer-

Numeric systems

number

of stones:

number

of iterations:

A game called "tarumbera" in East Africa makes use of che triangular numbers, starting with 3

(3, 6, 10, 15 ... ). In chis game, one player calls out a count as he removes stones consecutively,

left to right and bottom to top, while the other player, with his back turned, must signal whenever

the first stone in a row has been removed.

The stones in each triangular array can be built up in an iterative fashion, that is, the next

riangle can be created by adding another layer to any side of the previous triangle. The number to

se added in each additional layer is simply the number of iterations. For each iteration i, and tota

number of stones N, we have:

Ni+s = N; + i (starting with No = 0)

• 1 = 0 + 1 (a trivial array, not used in the game)

•. 3 = 1 + 2

*6=3+3

In other words, the next number will be given

by the last number plus the iteration count:

10 = 6 + 4

• Next

15 - 10 + 5

Neurrent

Increase count

of iterations by :

FIGURE 7.1

The triangular numbers in an East African game

tain formal age-grade initiation practices (see chapters 5 and 8), the simple

versions are used by smaller children, and the higher iterations are picked up with

increasing age. While there is no indication of a formal relationship in this instance,

there is still an underlying parallel berween the iterative concept of aging com-

mon to many African cultures— each individual passing through multiple turns

of the "life cycle"-and the iterative nature of the triangular number series.

Another nonlinear additive series was found in archaeological evidence from

North Africa. Badawy (1965) noted what appears to be use of the Fibonacci series

in the layout of the temples of ancient Egypt. Using a slightly different approach,

The Fibonacci series

(1, 1, 2, 3, 5, 8, 13 ... ) was

found by Badawy (1965)

in his study of the layout

of the temples of Egypt.

His analysis was quite

complex, but it is:not

difficult to create a simple

visualization. Here we see

the series in the successive

chambers of the temple of

Karnak.

The Fibonacci series is

produced by adding the

previous number to the

current number to get the

next number, starting with

1 + 1 = 2. For each iteration

i, the number N in the series

is given by:

Ni+! = N; + Ni-s

that is,

Nnext = Ncurtent + Nprevious

l+1=

• 13

Gray rectangles added

for measurement &

Nprevious

Neurrent

FIGURE 7.2

The Fibonacci series in ancient Egypt

Numeric systems

I found a visually distinct example of this series in the successive chambers of

the remple of Karnak, as shown in figure 7.2a. Figure 7.2b shows how these num-

bers can be generated using a recursive loop. This formal scaling plan may have

beer derived from the nonnumeric versions of scaling architecture we see

throughout Africa. An ancient set of balance weights, apparently used in Egypt,

Syria, and Palestine circa 1200 B.C.e., also appear to employ a Fibonacci sequence :

(Petruso 1985). This is a particularly interesting use, since one of the striking

mathematical properties of the series is that one can create any positive integer

through addition of selected members— a property that makes it ideal for appli-

cation to balance measurements (Hoggatt 1969, 76). There is no evidence that

ancient Greek mathematicians knew of the Fibonacci series. There was use of

the Fibonacci series in Minoan design, but Preziosi (1968) cités evidence indi-

cating that it could haye been brought from Egypt by Minoan architectural

workers employed at Kahun.

Doubling series in Africa

Some accounts report that Africans use a "primitive" number system in which

they count by multiples of two. It is true that many cases of African arithmetic

are based on multiples of two, but as we will see, base-2 systems are not crude

artifacts from a forgorten past. They have surprising mathematical significance,

not only in relation to African fractals, but to the Western history of mathematics

and computing as well.

The presence of doubling as a cultural theme occurs in many different African

societies and in many different social domains, connecting the sacredness of twins,

'spirit coubies, and double vision with material objects, such as the biacksmith's

twin bellows and the double iron hoe given in bridewealth (fig. 7•3). Figure 7.4a

shows the Ishango bone, which is around 8,000 years old and appears to show a

doubling sequence. Doubling is fundamental to many of the counting systems of

Africa in modern times as well. It is common, for example, to have the word for

i an even number 2N mean "N plus N" (e.g., the number 8 in the Shambaa lan-

guage of Tanzania is "ne na ne," literally "four and four"). A similar doubling takes

place for the precisely articulated system of number hand gestures (fig. 7.4b), for

example, "four" represented by two groups of two fingers, and "eight" by two groups

of four. Petitto (1982) found that doubling was used in multiplication and

division techniques in West Africa (fig. 7-4c). Gillings (1972) détails the per-

sistent use of powers of two in ancient Egyptian mathematics as well, and

Zaslavsky (1973) shows archaeological evidence suggesting that ancient Egypt's

use of base-2 calculations derived from the use of base-2 in sub-Saharan Africa.

African fractal mathematics

Doubling practices were also used by African descendants in the Ameri-

cas. Benjamin Banneker, for example, made unusual use of doubling in his cal-

culations, which may have derived from the teachings of his African Tather and

grandfather (Eglash 1997c). Gates (1988) examined the cultural significance of

doubling in West African religions such as vodun and its transfer to "voodoo"

in the Americas. In the religion of Shango, for example, the vodun god of thun-

der and lightning is represented by a double-bladed axe (fig. 7.5a), used hy

Shango devotees in the new world as well (R. Thompson 1983). Figure 7.gh shows

FIGURE 7•3

Doubling in African social practices

(a) This figure is used by women in Ghana to encourage the

birth of twins. (b) A double iron hoe is sometimes used as part

of the bride price ceremony. (c) The double bellows of the

blacksmith. (d) Double vision; a common theme in several

African spiritual practices, often implying that one can see

both the material world and the spirit world.

(b, Marc and Evelyn Bernheim from Rapho Guillonette; courtesy

of Uganda Nacional Museum, c, photo courtesy IFAN, Dakar.

d, fron Berjonneau and Sonnery 1987.)

Numeric systems

(a) The Ishango bone, estimated to be over 8,000 years old, shows

what appears to be use of doubling: 3 + 3 = 6, 4 + 4 = 8,10 = 5 + 5.

(b) Even numbers are

typically represented by

doubling in the precisely

articulated system of

African hand gestures.

(e) Doubling was traditionally used by cailors in West Africa when doing large mental

multiplications; it is essentially based on what we would call factoring.

For example, 3 x 273 ("3 taken 273 times") would be calculated by successively

doubling 3 (6, 12, 24...) while keeping crack of the counterpart in powers of two (2, 4,8...).

When the next power of two would overshoot 273, he then has to memorize the number

reached so far through doublings of 3 (268), while subtracting the power of two that was

reached (273 - 256 = 17). Then he starts again, doubling 3, and keeping track of the powers

of rwo. When the next power of two would overshoot 17, he again memorizes the number

reached through doublings of 3 (48) and subtracts the power of two (17 - 16 = 1). Since one

is luft over, he just needs to add an additional 3. The answer is then given by the sum of the

underlined terms: 768 + 48 + 3 = 819.

Despite the complexity of the method, the cailors were quite fast at performing these silent

mental operations.

FIGURE 7.4

Doubling in African arithmetic

(a mel b, from From Zaslawsky 1973-)

the use of a doubling sequence in the structure of a Shango temple and in reli-

gious ceremonies (ritual choreography aligning two priests, four children,

eight legs). A curator at the Musée Ethnographique in Porto Novo, Benin, who

specialized in Shango explained to me that these doubling structures were used

because the god of lightning required a portrait of the forked structure of a light-

ning bolt. The model is particularly interesting in that the lengths of each iter-

ation are shortened, so that one could have infinite doublings in a finite

(a) Shango, the god of

lightning, is part of the vodun

religion of Benin and was one

of the important components

in the creation of the voodoo

religion in the New World.

Here we see the double-bladed

"thunder axe," with another

double blade within each side.

(b) Shango temple and initiation. Here we see

the doubling sequence carried out further,

using the bilateral symmetry of the human body

itself in the last iteration. This is used to symbolize

the bifurcating pattern of the lightning bolt.

FIGURE 7.5

Doubling in the religion of Shango

fa, courtesy IFAN, Dakar. b: hoch center photos, courtesy IFAN, Dakar; lower right, courtesy Dave

Crowley, www.stornguy.com.)

Numeric systems

space-a true fractal. The self-similar structure of lightning has been a favorite

example for fractal geometry texts (see Mandelbrot 1977). The doubling

sequence used to model the fractal structure of lightning in Shango would not

give an accurate value for the empirical fractal dimension--real lightning rends

to branch much more than doubling allows for—but it's enough to know that the

vodun representation offers a testable quantitative.model.

The most mathematically significant aspect of doubling in African reli-

gion occurs in the divination ("fortunetelling") rechniques of vodun and its reli-

gious relatives (Eglash 1997b). The famous Ifa divination system (fig. 7.6) is based

on tossing pairs of flat shells or seeds split in two. Each lands open-side or closed-

side (like "heads or tails" in a coin toss). They are connected by a doubled chain

to make four pairs. Each group of four pairs gives one of the 16 divination sym-

bols, which tell the future of the diviner's client. The lfa system is what a

mathematician would call "stochastic," that is, it operates by pure chance. But

a closely related divination system, Cedena, has a nonstochastic element—-it is

closer to what mathematicians call "deterministic chaos."

My introduction to cedena, or sand divination, took place in Dakar, Sene-

gal, where the local islamic culture credits the Bamana (also known as Bambara)

with a potent pagan mysticism. Almost all diviners had some kind of physical

deformity-"the price paid for their power."3 One diviner seemed quite willing

to teach me about the system, suggesting that it "would be just like school." The

first few sessions went smoothly, with the diviner showing me a symbolic code

in which each symbol, represented by a set of four vertical dashed lines drawn

in the sand, stood for some archetypical concept (travel, desire, health, etc.) with

which he assembled narratives about the future. But when I finally asked how

he derived the symbols in particulat, the meaning of some of the patterns

drawn prior to the symbol writing—they all laughed at me and shook their

heads. "That's the secret!" My offers of increasingly high payments were met

with disinterest. Finally, I tried to explain the social significance of cross-cultural

mathematics. I happened to have a copy of Linda Garcia's Fractal Explorer with

me and began by showing a graph of the Cantor set, explaining its recursive con-

struction. The head diviner, with an expression of excitement, suddenly stopped

me, snapped the book shut, and said "show him what he wants!"

As it turns out, the recursive construction.of.the.Cantor set was just the

right thing to show, because the Bamana divination is also based on recursion

^ (fig. 77). The divination begins with four horizontal dashed lines, drawn rapidly,

so that there is some random variation in the number of dashes in each. The dashes

are then connected in pairs, such that each of the four lines is left with either

one single dash (in the case of an odd number) or no dashes (all pairs, the case

One open, one closed: 0 + 1 = odd

One closed, the other closed: 1 + 1 = even

One open, one closed: 0 + 1 = odd

One open, the other open: 0 + 0 = even

FIGURE 7.6

Binary codes in divination

(a) This Nigerian priest is telling the future by Ifa divination, in which pairs of flät shells or seeds

split in two are tossed with each landing open-side or closed-side. They are connected by a doubled

chain to make four pairs, giving a total of tó divination symbols. In this version of Ifa (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

either "closed" (meaning we see the rounded outside) or "open" (meaning we see the 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, photo by E. M. McClelland, courtesy Royal Anthropological Institute.)

sled

ere

.OW

Numeric systems.

of an even number). The narrative symbol is then constructed as a column of four

vertical marks, with double vertical lines representing an even number of dashes

and single lines representing an odd number. At this point the system is similar

to the famous Ifa divination: there are two possible marks in four positions, so

16 possible symbols. Unlike Ifa, however, the random symbol production is

repeated four times rather than two. The difference is quite significant. Each of

the Ifa symbol pairs are interpreted as one of 256 possible Odu, or verses. The

Ifa diviner must memorize the Odu; hence, four symbols would be too cumber-

some (65,536 possible verses). But the Bamana divination does not require any

verse memorization; as we will see, its use of recursion allows for verse self-assembly.

As in the additive sequences we examined, the divination code is gener-

ared by an iterative loop in which the ourput of the operation is used as the input

for the next stage. In this case, the operation is addition modulo 2 ("mod 2" for

short), which simply gives the remainder after division by two. This is the same

even/odd distinction used in the parity bit operation that checks for errors on

contemporary computer systems. There is nothing particularly complex about

mod 2; in fact, I was quite disappointed at first because its reapplication

destroyed the potential for a binary placeholder representation in the Bamana

divination. Rather than interpret each position in the column as having some

meaning (as would our binary number torr, which means one x, one 2, zero 4s,

and one 8), the diviners reapplied mod 2 to each row of the first two symbols

and to each row of the last two. The results were then assembled into two new

symbols, and mod 2 was applied again to generate a third symbol. Another four

symbols were created by reading the rows of the original four as columns, and

mod 2 was again recursively applied to generate another three symbols.

The use of an iterative loop, passing outputs of an operation back_as

inputs for the next stage, was a shock to me; I was at least as taken aback by the

sand symbols as the diviners had been by the Cantor set. It would be naive to

claim that this was somehow a leap outside of our cultural barriers and power

differences—in fact, that's just the sort of pretension that the last two decades

of reflexive anthropology has been dedicated against-but it would aiso be

ethnocentric to rule out those aspects that would be attributed to mathematical

collaboration elsewhere in the world: the mutual delight in two recursion

fanatics discovering each other. And the appearance of the symbols laid out in

two groups of seven-the Rosicrucian's mystic number-added some numer-

ological icing on the cake.

The following day I found that the presentation had not been complete:

an additional two symbols were left out. These were also generared by mod 2 recur-

sion using the two bottom symbols to create a fiftèenth, and using that last

African fractal mathematics

symbol with the first symbol to create a sixteenth (bringing the total depth of

recursion to five iterations). The fifteenth symbol is called "this world," and the

sixteenth is "the next world," so there was good reason to separate them from

the others. The final part of the system— creating a narrative fröm the symbols—

was still unclear, but I was assured that it could be learned if I carefully followed

their instructions. I was to give seven coins to seven lepers, place a kola nut on

=-=-

→

(e) After this, the original four

are read sideways to create four

more symbols, and the entire

process is repeated, producing

another group of seven. In the

final step, the first and last from

each group öf seven are paired off

to generate the final two symbols.

Numeric systems

a pile of sand next to my bed at night, and in the morning bring a white cock,

which would have to be sacrificed to compensate for the harmful energy released

in the telling of the secret. I followed all the instructions, and the next morn-

ing bought a large white cock at the marker. They held the chicken over the div-

ination sand, and I was told to eat the bitter kola nut as they marked divination

symbols on its feet with an ink pen. A little sand was thrown in its mouth, and

then i was told to hold it down as prayers were chanted. There was no action on

the part of the diviner; the chicken simply died in my hands.

While still a bit shaken by the chicken's demise (as well as experiencing

a respectable buzz from the kola nut), I was told the remaining mystery. Each sym-

bol has a "house" in which it belongs-for example, the position of the sixteenth

symbol is "the next world" —but in any given divination most symbols will not

be located in their own house. Thus the sixteenth symbol generated might be

"desire," so we would have desire in the house of the next world, and so on. Obvi-

ously this still leaves room for creative narration on the part of the diviner, but

the beauty of the system is that no verses need to be memorized or books con-

sulted; the system creates its own complex variety.

The most elegant part of the method is that it requires only four random

drawings; after that the entire symbolic array is quickly self-generated. Self-

generated variety is important in modern computing, where it is called "pseudo-

random number generation" (fig. 7.8). These algorithms take little memory,

but can generate very long lists of what appear to be random numbers,

although the list will eventually start over again (this length is called the

"period" of the algorithm). Although the Bamana only require an additional

12 symbols to be generated in this fashion, a maximum-length pseudorandomi

number generator using their initial four symbols will produce 65:535 symbols

before it begins to repeat.

A similar system for self-generated variety was developed as a model for

the "chaos" of nonlinear dynamics by Marston Morse (1892-1977). Before the

19705, mathematicians had assumed that, besides a few esoteric exceptions (the

algorithms for producing irrational numbers such as Va), the output of an equa-

tion would eventually start repeating. That assumption was partly based on

European cultural ideas about free will:-complex behavior could not be the

result of predetermined systems (see Porter 1986). It was not until the redost?os

that mathematicians realized that even simple, common equations describing things

like population growth or fluid flow could result in what they called "determin-

istic chaos"-- an output that never repeats, giving the appearance of random num-

bers from a nonrandom (deterministic) equation. Morse developed the minimal

case for such behavior.

African fractal mathematics

mod 2

1111

0111

0011

0001

1000

0100

0010

1001

0110

1100

0110

101-1

0101

1010

1101

1110

CLK

Din

74LS95

FIGURE 7.8

Pseudorandom number generation

from shift register circuits

(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 the last two bits in the register

and places the result in the first position. The

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" - 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. While school-

teachers are making increasing use of African

culture, in the mathematics classroom, few have

explored the potential applications to

technology education.

The construction of the Morse sequence begins by counting from zero in

binary notation: ooo, oor, 010, 011 .... It then takes the sum of the digits in

each number- 0 + 0 + 0 = 0,0.+,0 + 1 = 1, etc.- and finally mod 2 of each

sum. The result is a sequence with many recursive properties, but of endless

variety. Morse did the same "misreading" of the binary number as did the

/ Bamana-although he did not have an anthropologist scowling at him for

ignoring place value—-and he did it for the same reason: combined with the

mod 2 operation, it maximizes variety.

In my reading of divination literature l eventually came across the dupli-

cate of the Bamana technique 5,000 miles to the east in Malagasy sikidy (Suss-

man and Sussman 1977), which inspired a study of the history of its diffusion.

The strong similarity of both symbolic technique and semantic categories to what

Europeans termed "geomancy" was first noted by Flacourt (r66r), but it was not

until Trautmann (1939) that a serious claim was made for a common source for

this Arabic, European, West African, and East African divination technique. The

commonality was confrmed in a detailed formal analysis by Jaulin (1966). But

where did it originate?

Numeric systems

Skinner (1980) provides a well-documented history of the diffusion evidence,

from the first specific written record—a ninth-century Jewish commentary by Aran

ben Joseph—-to its modern use in Aleister Crowley's Liber 777. The oldest Ara-

bic documents (those of az-Zanti in the thirteenth century) claim the origin of

geomancy (ilm al-raml, "the science of sand") through the Egyptian god Idris (Her-

mes Trismegistus); while we need not take that as anything more than a claim

to antiquity, a Nilotic influence is not unreasonable. Budge (1961) attempts to

connect the use of sand in ancient Egyptian rituals to African geomancy, but it

is hard to see this as unique. Mathematically, however, geomancy is strikingly out /

of place in non-African systems.

Like other linguistic codes, number bases tend to have an extremely long

historical persistence. Even under Platonic rationalism, the ancient Greeks held

10 to be the most sacred of all numbers; the Kabbalah's Ayin Sof emanates by

10 Sefirot, and the Christian West counts on its "Hindu-Arabic" decimal nota-

tion. In Africa, on the other hand, base-2 calculation was ubiquitous, even for

multiplication and division. And it is here that we find the cultural connotations

of doubling that ground the divination practice in its religious significance.

The implications of this trajectory-from sub-Saharan Africa to North Africa

to Europe-- are quite significant for the history of mathematics. Following the

introduction of geomancy to Europe by Hugo of Santalla in twelfth-century Spain,

it was taken up with great interest by the pre-science mystics of those times—

alchemists, hermeticists, and Rosicrucians (fig. 7.9). But these European geo-

mancers-Raymond Lull, Robert Fludd, de Peruchio, Henry de Pisis, and

others-persistently replaced the deterministic aspects of the system with chance.

By mounting the 16 figures on a wheel and spinning it, they maintained their

society's exclusion of any connections between decerminism and unpredictabil-

ity. The Africans, on the other hand, seem to have emphasized such connections.

In chapter so we will explore one source of this difference: the African concept

of a "trickster" god, one who is both deterministic and unpredictable.

On a video recording I made of the Bamana divination, I noticed that the

practitioners had used a shortcut method in some demonstrations (chis may

have been a parting gift, as the video was shot on my last day). As they first taught

me, when they count off the pairs of random dashes, they link them by drawing

short curves. The shortcut method then links those curves with. larger curves, and

those below with even larger curves. This upside-down Cantor set shows that

they are not simply applying mod & again and again in a mindless fashion. The

self-similar physical structure of the shortcut method vividly illustrates a recur-

sive process, and as a nontraditional invention (there is no record of iss use else-

where) it shows active mathematical practice. Other African divination practices

100

African fractal mathematics

Mine to nee

SANTO 10.n01

0000

lic.

пит стабии!

соcоc00s

fer pator optucou vir

ture tell.

nice cirs tutaber falli

Poser ambar fili

That more up

PRILM TIM LUMAR 1LL

o rt muta fur omat

• my muta,

0000

00.00

Coco

0000

tubarbita contaci

nrains. lude matt

va intuna trutter mu

FIGURE 7•9

Geomancy

African divination was caken up under the name "geomancy" by European mystics. This chart was

drawn for King Richard n in 1391.

(From Skinner 1980.)

can be linked to recursion as well; for example Devisch (1991) describes the Yaka

diviners' "self-generative" initiation and uterine symbolism.

Before leaving divination, there is one more important connection to mathe-

matical history. While Raymond Luil, like other European alchemists, created

wheels with sixteen divination figures, his primary interest was in the combi-

natorial possibilities offered by base-a divisions. Lull's work was closely exam-

ined by German mathematician Gottfried Leibniz, whose Dissertatio de arte

combinatoria, published in 1666 when he was twenty, acknowledges Lull's work

as a precursor. Further exploration led Leibniz to introduce a base-z counting

system, creating what we now call the binary code. While there were many other

Numeric systems

influences in the lives of Lull and Leibniz, it is not far-fetched to see a histor-

ical path for base-a calculation that begins with African divination, runs

through the geomancy of European alchemists, and is finally translated into binary

calculation, where it is.now applied inevery digital circuit from alarm clocks

to supercomputers.

In a 1995 interview in Wired magazine, techno-pop musician Brian Eno

claimed that the problem with computers is that "they don't have enough African

in them." Eno was, no doubt, trying to be complimentary, saying that there is

some intuitive quality that is a valuable attribute of African culture. But in doing

so he obscured the cultural origins of digital computing and did an injustice

to the very concept he was trying to convey.

LOI

Discrete self-organization in Owari

Figure 7.10a shows a board game that is played throughout Africa in many dif-

ferent versions variously termed ayo, bao, giuthi, lela, mancala, omweso, owari, rei,

and songo (among many other names). Boards that were cut into stones, some

of extreme antiquity, have been found from Zimbabwe to Ethiopia (see Zaslavsky

1973, fig. 11-6). The game is played by scooping pebble or seed counters from

one cup, and placing one of those counters into each cup, starting with the cup

to the right of the scoop. The goal is to have the last counter land in a cup that

has only one or two counters already in it, which allows the player to capture

these counters. In the Ghanaian game of owari, players are known for utilizing

a series of moves they call a "marching group." They note that if the number

of counters in a series of cups each decreases by one (e.g., 4-3-2-1), the entire

pattern can be replicated with a right shift by scooping from the largest cup, and

that if the pattern is left uninterrupted it can propagate in this way as far as needed

for a winning move (fig. 7.1ob). As simple as it seems, this concept of a self-

replicating pattern is at the heart of some sophisticated mathematical concepts.

John von Neumann, who played a pivotal role in the development of

the modern digital computer, was also a founder of the mathematical theory

of self-organizing systems. Initially, von Neumann's theory was to be based on

self-reproducing physical robots. Why work on a theory of self-reproducing

machines? I believe the answer can be found in von Neumann's social out-

look. Heims's (1984) biography emphasizes how the disorder of von Neumann's

precarious youth as a Hungarian Jew was reflected in his adult efforts to impose

a strict máthematical order on various aspects of the world. In von Neumann's

application of game theory to social science, for example, Heims writes that his

"Hobbesian" assumptions were "conditioned by the harsh political realities of

I02

African fractal mathematics

FIGURE 7.10

Owari

(a) The owari board has 12 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 Hungarian existence." His enthusiasm for the use of nuclear weapons against

the Soviet Union is also attributed to this experience.

During the Hixon Symposium (von Neumann 1951) he was asked if com-

puting machines could be built such that they could repair themselves if "dam-

aged in air raids," and he replied that "there is no doubt that one can design

machines which, under suitable circumstances, will repair themselves." His

work on nuclear radiation tolerance for the Atomic Energy Commission in

1954-1955 included biological effects as well as machine operation. Putting

these facts together, I cannot escape the creepy conclusion that von Neumann's

interest in self-reproducing automata originated in fantasies about having a

more perfect mechanical progeny survive the nuclear purging of organic life

on this planet.

Models for physical robots turned out to be too complex, and at the sug-

gestion of his colleague Stanislaw Ulam, von Neumann settled for a graphic ab-

straction: "cellular automaga," as they came to be called. In this model (fg. 7.11a),

each square in a grid is said to be either alive or dead (that is, in one of two pos-

sible states). The iterative rules for changing the state of any one square 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 cell in the center, surrounded by dead

cells in its eighe nearest neighbors. The state of each celt in the next iteration is

determined by a set of rules. In "classic" life (the rules first proposed by John Horton

Conway), a dead cell becomes a live celf'if it has three live nearest neighbors, and a

cell dies unless it has two or three live neighbors.

-191-191-18

This initial condition produces a fixed pattern after four iterations. The patterns occurring before

it settles down to stability are called the "transienc."

This stable pattern flips back and forth between these

two states. This is called a "period-z" partern.

→

A period 4-pattern. Periods of any lengch can be produced, as we saw in the previous examples

of psendorandom number generation. Dererministic chaos, in which thie pattern never repears

(i.c., a period-inónity pattern, like che Morse sequence), is also possible.

Iteration 49

Iteration 133

Iteration 182

A constant-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 ir has three live nearest neighbors, and

a cell dies only if it has seven or eight live neighbors.

FIGURE 7.11

Cellular automata

104

African fractal mathematics

on the eight nearest neighbors (e.g., if three or more nearest-neighbors are full,

the cell becomes full in the next iteration). At first, researchers carried out on

these cellular automata experiments on checkered tablecloths with poker chips

and dozens of human heipers (Mayer-Kress, pers. comm.), but by 1970 it had been

developed into a simple computer program (Conway's "game of life"), which was

described by Martin Gardner in his famous "Mathematical Games" column in

Scientific American. The "game of life" story was an instant hit, and computer screens

all over the world began to pulsate with a bizarre array of patterns (fg. 7.1 1b).

As these activities drew increasing professional attention, a wide range of mathe-

matically oriented scientists began to realize that the spontaneous emergence

of self-sustaining patterns created in certain cellular automata were excellent

models for-the kinds of self-organizing patterns that had been se elusive in stud-

ies of fluid flow and biological growth

Since scaling structures are one of the hallmarks of both fluid turbulence

and biological growth, the occurrence of fractal patterns in cellular automata

attracted a great deal of interest. But a more simple scaling structure, the log-

arithmic spiral (fig. 7.x2), has garnered much of the attention. Even back in the

19,5os mathematician Alan Turing, whose theory of computation provided von

Neumann with the inspiration for the first digital computer, began his research

on "biological.morphogenesis" with an analysis of logarithmic spirals in growth

patterns. Markus (199x) notes that the application areas for cellular automata

models of spiral waves include nerve axons, the retina, the surface of fertilized

eggs, the cerebral cortex, heart tissue, and aggregating slime molds. In the rext

for cAlAB, the first comprehensive software for experimenting with cellular

automata, mathematician Rudy Rucker (1989, 168) refers to systems that pro-

duce paired log spirals as "Zhabotinsky CAs," after the chemist who first observed

such self-organizing patterns in artificial media: "When you look at Zhabotin-

sky CAs, you are seeing very striking three dimensional structures; things like

paired vortex sheets in the surface of a river below a dam, the scroll pair stretch-

ing all the way down to the river bottom. ... In three dimensions, a Zhabotin-

sky reaction would be like two paired nautilus shells, facing each other with their

lips blending. The successive layers of such a growing pattern would build up very

like a fetus!"

Figure 7.13 shows how the owari marching-group system can be used as a !

one-dimensional cellular automaton to demonstrate many of the dynamic phe-

nomena produced on two-dimensional systems. Earlier we noted that the

Akan and other Ghanaian societies had a remarkable precolonial use of loga-

rithmic spirals in iconic representations for living systems. The Ghanaian four-

fold spiral (fig. 6.4a) and the four-armed computer graphic in figure 7.y2b are

(a) Paired spirals emerge from a three-state cellular ayromation. Black cells are live, white cells are dead,

and gray cells öre in a refractory or "ghost" 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

co create a spical pactern.

(b) This four-armed logarithmic spiral from Markus (1991) was produced by a

six-state cellular automacon in which a sequence of ghost states corresponds

to increasingly dark shades of gray. The system makes use of a very high-

resolution 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 the Ghanaian fourfold spiral in figure 6.4a.

y",

• Bivalve shell.

(From Haeckel 1904.)

Mushroom cut in half.

North African sheep

(From Cook 1914.)

) Recursive line replacement, as we sav for other fractal generations, can also produce such paired spirals.

FIGURE 7.12

Spirals in cellular automata

We can view the owari board as a one-dimensional cellular automaton. One

dimension is not necessarily a disadvantage; in fact, most of the professional

mathematics on cellular automata (see Wolfram 1984, 1986) have been done on

one-dimensional versions, because it is easier to keep track of the results. They can

show all the dynamics of two dimensions.

The patterns noted by traditional owari players offer a great deal of insight into

self-organizing behavior. Their cbservation of a class of self-propagating patterns,

the "marching group," provides an excellent starting point.

- 13 iterations -_

3421→532→43111→4222→3331→442→5311→42211-3322->433-4411->4552→33211→4321

The marching group is an example of a constant pattern. Here we see counters in

the initial sequence 342} converge on their marching formation simply by repeating

the "scoop from the left cup" rule through 13 iterations.

Just as we saw in two-dimensional cellular automata, transients of many different

lengths can be produced. Transients of maximum length are used as an endgame tactic

by indigenous Ghanaian players, who call it "slow motion"-accumulating pieces on

your side to prevent your opponent from capturing them. In nonlinear dynamics, the,

constant pattern is called a "point attractor," and the transients would be said to lie

in the "basin of attraction."

The marching group rule can also produce periodic behavior (a "limit cycle" or

"periodic attractor" in nonlinear dynamics cerms). Here is a period-3 system using

only four counters:

21) >22→31→>235

Which leads to marching groups, and which ones lead to periodic cycles?

Total number

of counters

The numbers which lead to marching groups--

1, 3, 6, 10, 15 ... — should look familiar to readers:

t's the triangular numbers we saw in tarumbeta

"he period of cycles in between each marchin

group is given by one plus the iteracion level of the

previous triangular number reached.

(Note: Some sequences will be truncated for

13, 14, and 15 since there are more counters

than holes.)

Behavior

(afrer transients)

Marching

Period 2

Marching

Period 3

Marching

P'eriod 4

• Period 4

• Perised 4

Marching

Period 5

• Period 5

Period s

Marching

FIGURE 7.13

Owari as one-dimensional cellular automaton

Numeric systems

quite distant in terms of the rechnologies that produced them, but there may

well be some subtle connections between the two. Since cellular automata

model the emergence of such patterns in modern scientific studies of living sys-

tems, and certain Ghanaian log spiralicons were also intended as generalized

models for organic growth, it is not unreasonable to consider the possibility that

the self-organizing dynamics obseryable in owari were also linked to concepts

of biological morphogenesis in traditional Ghanaian knowledge systems.

Rattray's classic volume on the Asante culture of Ghana includes a chap-

ter on owari, but unfortunately it only covers the rules and strategies of the game.

Recently Kof Agudoawu (1991) of Ghana has written a booklet on owari "ded-

icated to Africans who are engaged in the formidable task of reclaiming their her-

itage," and he does note its association with reproduction: wari in the Ghanaian

language Twi means "he/she marries." Herskovits (1930), noting that the "awari"

I07

FIGURE 7.14

Logarithmic curves and owari

The cover of the hinged owari board

we saw in bgure 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 logarichmic curves held in a

fist: God holding the power of life.

recurrion

108

African fractal mathematics

game played by the descendants of African slaves in the New World had retained

some of the precolonial cultural associations from Africa, reports that awari had

a distinct "sacred character" to it, particularly involving the carving of the

board. Owari boards with carvings of logarithmic spirals (fig»7.x4) can be com-

monly found in Ghana today, suggesting that Western scientists may not be the

only ones who developed an association between discrete self-organizing patterns

and biological reproduction. It is a bit vindictive, but I can't help but enjoy the

thought of von Neumann, apostle of a mechanistic New World Order that

would wipe out the irrational cacophony of living systems, spinning in his grave

every time we watch a cellular automaton-whether in pixels or owari cups—

bring forth chaos in the games of life.

Conclusion

Both tarumbeta and owari's marching:group dynamics are governed by the tri-

angular numbers. There is nothing special about the triangular number series--

similar nonlinear growth properties can be found in the numbers that form

successively larger rectangles, pentagons, or other shapes. Nor is there anything

special about the powers of two we found in divination-similar aperiodic prop-

erties can be produced by applications of mod 3, mod 4, etc. What is special is

the underlying concept of recursion— the ways in which a kind of mathematical.

feedback loop can generate new structures in space and new dynamics in time.

In the next chapter, we will see how this underlying process is found in both prac-

tical applications and abstract symbolics of African cultures.

CHAPTER

Recursion

- Recursion is the motor of fractal geometry; it is here that the basic transfor-

mations- whether numeric or spatial- are spun into whole cloch, and the pat-

terns that emerge often tell the story of their whirling birth. We will begin by

defining three types of recursion. ' While it is possible to categorize the examples

in this chapter solely on the basis of these three types, it is more illuminating

to combine the analysis with cultural categories. It is in examining the inter-

action between the two that the use of fractal geometry as a knowledge system,

and not just unconscious social dynamics, becomes evident. The cultural cat-

egories begin with the concrete instances of recursive construction techniques

and gradually move toward the abstractions of recursion as symbolized in

African iconography.

Three types of recursion

The least powerful of the three is cascade recursion, in which there is a pre-(1

determined sequence of similar processes. For example, there is a children's

story in which a man buys a Christmas tree, but discovers it is too tall for his

ceiling and cuts off the top. His dogs find the discarded top, and put it in their

doghouse, but they too discover it is too tall, and cut off the top. Finally the

109

IIO

African fractal mathematics

mice drag this tiny top into their hole, where it fits just fine—the recursion

"bottoms out." Note that these were all independent transformations; it is only

by coincidence, so to speak, that they happened to be the same. Figure 8.1a shows

the numeric version of cascade recursion, in which we divide a number by two

in each part of the sequence. This is not a very powerful type of recursion, for

two reasons. First, it requires that we know how many transformations we want

ahead of time- and that is not always possible. if the mouse was in charge, he

would have said "just keep dividing until it's small enough to fit in my .hole."

Second, we have to know what transformation to make ahead of time, and that

is not always possible, either. Recall, for example, the generation of the

Fibonacci series we saw in chapter 7 (fig. 8.1b). Although the generation is just

using addition, it cannot be created by a recursive cascade, because the

amount to be added in each transformation changes in relation to previous

results. Generating the Fibonacci series requires a feedback loop or, as mache-

maticians call it, iteration.

In iteration, there is only one transformation process, but each time the

process creates an output, it uses this result as the input for the next iteration,

as we've seen in generating fractals. A particularly important variety of itera-

tion is "nesting," which makes use of loops within loops. Hofstadter (1980,

103-129) nicely illustrates nesting with a story in which one of the characters

starts to tell a story, and within that story a character starts to read a passage from

a book. But at that point the recursion "bottoms out": the book passage gets

finished and we start to, ascend back up the stories. Nested loops are very

common in computer programming, and we can illustrate this with a program

for drawing the architecture.of.Mekoulek. (fig. 8.ic).we examined.in.chapter.2

The Ba-ila architecture we saw in chapter 2 can also be simulated this way, using

one loop for the rings-within-rings, and another for the front-back scaling

gradient that makes up each of those rings. In chapter 6 the first corn-row hair-

style (ipako elede) showed braiding as an iterative loop; the second corn-row

example added another iterative loop of successive perimeters of braids? 1 is

common for computer programs to do such nesting several layers deep, and keep-

ing track of all those loops within loops can be quite a chore..

The third type of recursion is "self-reference." We are all familiar with the

way that symbols or icons can refer to something: the stars and stripes flag refers

to America, the skull-and-cross-bones label refers to poison, the group of let-

ters c-a-t refers to an animal. But it's also possible for a symbol to refer to itself.

Kellogg's cornflakes, for example, once came in a box that featured a picture of

a family sitting down to breakfast. In this picture you could see that the family

had a box of Kellogg's cornflakes on their hreakfast table, and you could see that

Recursion

III

this box showed the same picture of the family, with the same box on their table,

and so on to infinity (or at least to as small as the Kellogg company's artisans

could draw).

Self-reference is best known for its rale in logical paradox. If, for example,

you were to accuse someone of lying, it would be an ordinary statement. But sup-

pose you accuse yourself of lying? This is the paradox of Epimenides of Crete, who

declared that "all Cretans are liars." If he's telling the truth, he must be lying,

but if he's lying, then he's telling the truth. The role of self-reference in logical

input

output

* 1

Nprevious

Nnext

Neurrens

WHILE e-count < 4 do:

• draw enclosure

WHILE g-Count < 12 do:

• draw granery

• rocate toward center

• shrink granery size

• increase g-count by 1

END of g-count's loop

• resec g-count to 0

• rotare coward center

• shrink enclosure size

• increase e-count by 1

END of e-count's loop.

FIGURE 8.1

Recursive cascade versus iteration

(a) A recursive cascade, in which the same transformation (division by two) happens to be used

in each part of a sequence. This requires knowing how many times the transformatin should

happen ahead of time. It also requires that the transformation is independent of previous results.

(b) The Fibonacci sequence is produced by adding the previous number to the current number to

ger the next number, starting with 1 + 1 = 2. In the Fibonacci sequence we add a different amount

in each iteration--we could not know how much each transformation should add ahead of time,

so a recursive cascade would not do the job. (c) In some cases it is necessary to put an iterative

kop inside another iterative loop ("nesting"). Here is an example of nesting in a compurer

program for drawing the architecture of Mokoulck we examined in chapter 2. It is written in what

programmers call "pseudocode," a mixture of a programming language and ordinary English. The

hrst koop draws three large enclosures, and the inner loop draws 12 graneries inside each enclosure.

Variable "e-count" tracks the number of enclosures, and g-count tracks the number of graneries."

I12

African fractal mathematics

paradox has been important for mathematical theory, but it has also been put to

practical use in computer programming. Most programming has little routines called

"procedures," and often a procedure will need to call other procedures. In self-

referential programming the procedure calls itself.

Practical fractals: recursion in construction techniques

In his discussion of the metal-working techniques of Africa, Denis Williams gives

a poetic description of recursive cascade in the edan brass sculptures of the

Yoruba: "The image proliferates like lights in a bubble: one edan bears in its lap

another, smaller version of itself, which bears in turn a smaller in its lap, and this

bears another in its lap, etc.— a sort of sculptural relay race" (1974, 245). While

the edan sculptures are unique to the Yoruba, recursive construction techniques

are quite common in Africa. For example, Williams goes on to note that much

African metalwork, unlike European investment casting, uses a "spiral technique"

to build up structures from single strands (whether before casting, as in the lost

wax technique,' or afterwards as wire), resulting in "helical coils formed from

smaller helical coils." A wig made from metal wires (fig. 8.2a) shows a similar

iterative construction using coils made of coils. In chapter 6 we saw some

examples of African hair styles in which either adaptation to contours or

abstract spatial transformation resulted in a scaling pattern. The fractal braids

shown in figure 8.2b have nothing to do with the shape of the head; they are

rather the result of successive iterations that combine strands of hair into

braids, braids into braids of braids, and so on. Figure 8.2c shows another wig,

this one for a sculpture, that features braids of many scales.

This collection of sculpture, metalwork, and hairstyling sounds like a

motley assortment, but once we start looking for recursion we see a close rela-

tion: all examples used a single transformation-stacking, braiding, coiling-

that was applied several times. Looking at the felation between the basie

transformation and its final outcome can help us distinguish among different types

of recursion. The braiding pattern of figure 8.2b, for example, is based on iter-

ation, because the way each stage is braided depends on the braids produced in

previous stages; they are braids of braids. The braids in figure 8.2c, on the

other hand, are of different scales simply because each stage uses different

amounts of single-hair strands—a cascade of predetermined transformations.

Similarly, the coils of coils indicate iteration, because the output of one stage

becomes the input for the next.

Recursive construction techniques are also used for the decorative

designs of African artisans. In our discussion of the fractal esthetic in chap

Recursion

ter 4, we examined decorative patterns which did not provide evidence for a

formal geometric method. That doesn't mean no formal method could possibly

exist; it's just that none could be readily discerned from the design itself, and

the artisans did not report anything beyond intuition or esthetic taste. But there

are some designs that do indicate an explicit recursive technique from the pat-

tern itself. Figure 8.2e shows a Mauritanian textile with two such scaling pat-

terns. Intentional application of iteration as a construction technique is

indicated by the way the X fractal's seed shape is shown on either side, and

by having iteration carried out on two completely different seed shapes in the

same piece: The triangle fractal (close to what mathematicians call the "Sier-

pinski gasket") is also found in Mauritanian stonework (fig. 8.2f). A three-

dimensional version from Ghana (fig. 8.2h) may have been inspired by these

designs.

Both of the above are examples of additive construction, as we saw in the

Koch curve of chapter 1, but subtractive iterations, as we saw for the Cantor

set, are also found in African decorative fractals (fig. 8.2i). Carving designs

include applications of iterative construction, particularly for calabash deco-

rations (fig. 8.2l). A geometric algorithm for producing nonlinear scaling

through folding was invented by the Yoruba artisans who produced the adire

cloth of figure 8.2n. It is not merely a metaphor to refer to a specified series

of folds as algorithmic; in fact, one of the classic fractals, the "dragon curve,"

was discovered in 196o when physicist John Heighway experimented with

iterative paper folding (Gardner 1967). The adire cloth also-shows the appli-

cation of reflection symmetry at every-scale from single-stitch rows, which are

reflected on either side of the fold edges, to the entire fabric, which is created

by the joining of two mirror image cloths.

So far we have only discussed the technical method employed, but of course

cultural meaning is often atcached.to.these techniques as well. Recursive hair-

styles, for example, embed layers of social labor with each iteration, a way to

invest physical adornment with social meaning (such as friendship between styl-

ist and stylee). Figure 8. 3a shows a Fulani wedding blanket, in which spiritual

energy is embedded in the pattern through its iterative construction! Prestige)

can also be associated with increasing iterations, as we find for brass casting

and beadwork in the grassland areas of Cameroon (fig. 8.3b,c). The scaling iter-

ations in one of the brass sculptures (fig. 8.3d) was reported to be symbolic as

well: it showed three generations of royalty. But kinship groups are nor just

static entities; they change across time, and in the following two sections we

will see that African representations of such temporal processes often involve

recursion.

II3

FIGURE 8.2

Recursive construction techniques

(a) Coils of coils are used to create this metal wig from Senegal. (b) A scaling cascade of fra

a mask from the Dan societies of Liberia and Côte d'Ivoire. (c) Iterative braiding in this hain

from Yaounde, Cameroon, la tresse de fil, can be simulated by fractal graphics. (d) Three iter

of the tresse de fil simulation.

(b, from Barbier-Mueller 1988.)

(figure

FIGURE 8.2 (continued)

Iterative construction

in Mauritanian decoration

(e) Recursive construction with triangles and

X-shapes in Tuareg leatherwork. The X-shape

is related to the quincunx discussed in chapter 4.

(f) Designs using several iterations of triangles

can also be found in Mauritanian stonework.

(g) The use of triangies 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 fexible 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, from Jefferson 1973; fand g, photos courtesy

IFAN, Dakar; h, from Phillips 1995. fig 5.103.)

(figure continues)

II6

African fractal mathematics

FIGURE 8.2 (continued)

Scaling pattern from subtractive iteration

(i) A Fante woman posing in front of a painted studio backdrop, Cape Coast, Ghana, 1860.

(f) The Fante pattern can be thought of as two iterations of scaling subtraction (that is, erasing).

Strips are erased from an ail-black background. Where the thick strips intersect, we get large

squares, and where the rhin strips intersect we get small squares.

(i, photo from the National Museum of African Art, Smithsonían Institution:)

• (figure continu

Representing recursion as a process in time: part I, luck and age

A simple example of African representation for recursion as a time-varying process

is shown in figure 8.4, where we see three designs that depict wishes for catches

of everlarger fish. Since the experience of bad luck or good luck in fishing can

occur on a daily basis, it is easy to see how a big fish could become an icon for

good luck. But in these designs the artisans take the concept a step further. Good

fortune is not in terns of a singular chance event, as one sees in the mychs of

the Native American trickster. The wish is for an iterative process- that each

fish is to be successively larger than the last one.

While these good luck icons are often a more informal part of cultural prac-

- tice, other recursive processes are taken much more seriously. Anthropologists

Seed shape, with active

lines in gray.

Fourth iteration.

Fourth iteration enlarged, with adaptive scaling

(mapping from a sphere to a plane) applied to

march the adaptive scaling of the calabash design.

FIGURE 8.2 (continued)

Iteration in carvings

(k) The Bakuba of Zaire created several carvings that feature a self-similar design. This Bakuba

kindlen bottle makes use of hexagons of hexagons as well as adaptive scaling as it narrows into the

nock. (1) Chappel (1977) records a wide variety of calabash designs, many with scaling actributés.

This is probably the best example of iterative construction in these carvings. The design

simulation not only requires recursion but adapcive scaling as well. (m) Seed Shape and fourth

iteration; fourth iteration enlarged, with adaptive scaling applied.

|k, courtsy Musée Royal de l'Afrique Central, Belgium.)

(figure continues)

FIGURE 8.2 (continued)

Adire cloth: scaling from iterative folding

(n) This Yoruba adire cloth is actually two separate pieces attached along the horizontal midline.

The dye pattern is created by sewing along folds before dye is applied and then removing the

threads so that the white lines are left where the dye did not penetrate. (o) The folding method is

based on reflection symmetry across a diagonal. It is easiest to understand by making a paper model.

The adire artisans have not only

developed an algorithm for generating

this nonlinear scaling series, but have

done so in a way that maximizes efficient

production: all folds fall along the same

two edges, so only two edges need be

sewn. Your paper model can imitate this

effect by running a heavy felt marker

along the two edges, so that the ink

bleeds through all the layers (you can

cheat by inking each fold as you unfold

it). Note that the white lines in the adire

are triple--this, too, is created by a

reflection symmetry, sewing next to the

fold to create the two outer lines (one on

each side of the fold), and sewing right

on the edge of the fold to create the

center line.

(n, photo from Picton and Mack 1979.)

First, cut out a paper rectangle with width twice the height,

and fold it in half, making a square.

Second, fold the square along i

diagonal, making a triangle.

Third.

mark points

at ½ and ¼ of

the enter siles

of the triangle.

These points can he

determined by folding, if

one wishes to maintain the

origami equivalent of compass anci

straight-edge construction, but doing

it by eyehall works just fine.

Fourth, fold from the corners on

opposite sides along the line hetween

the ½ and ¼ marks.

Finally, foll in the sma

overlapping corner on

sicle.

ach

FIGURE 8.3

Making meaning

through iterative

construction

(a) This Fulani

wedding blanket from

Mali is based on

diamonds that scale

from either side as

we move toward the

center; a pattern that is

easily simulated using a

fractal (see diagram)?*

The weavers who

created it report that

spiritual energy is

woven into the pattern,

and that each successive iteration shows an increase in

this energy. Releasing rhis spiritual energy is dangerous,

and if the weavers were to stop in the middle they would

risk death. The engaged couple must bring the weaver

food and kola nuts to keep him awake until it is finished.

(b) The prestige bronze of Foumban, Cameroon, often

makes use of self-similar iterations. (c) Prestige is also

symbolized by the labor and artistry required to produce

the many iterations of bead pattens for this elephant

mask. (d) According to Salefou Mbetukom, the leading

castor of Foumban, this sculpture shows the succession of

kings in the royal family.

(c, from agence Hoa-Qui.)

120

African fractal mathematics

have always been interested in the contrast between the elaborate political and

economic hierarchy of European societies and the relatively "classless" (some-

times even rulerless) structure of many precolonial African societies If it is not

political and economic structure that governs their society, then what does?

One part of the answer is age. All human cultures differentiate between chil-

FIGURE 8.4

If wishes were fishes

(a) Scaling scales: this Bamana tattoo,

created with henna, is said to represent the

scales of fish. It is good luck, signifying ever-

larger 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 chan just

luck, e.g., "just as you have swallowed others,

I will swallow you." Other 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, photo courtesy of Traci Roberts and

Ann Campbell.)

Recursion

dren, adults, and elders, but in many African societies the divisions are much

more elaborate and structured. In these age-grade systems, all community mem-

bers born within a given number of years will move together through a series

of ritual initiations. In chapter 5 we saw, one example in which these initiation

stages appeared to be accompanied by an iterative scaling geometry, the lusona.

Figure 8.5a shows another geometric visualization of age-grade initiation: a hexag-

onal mask created by the Bassari of the Senegambian and Guinea-Bissau region.

Although the mask is only a linear-concentric scaling of hexagons, and thus

not a fractal, it does suggest an iterative process, and we might well suspect a link

between stages in age-grade and stages in iteration. The initiation process is a

closely guarded secret, so it is not simply a matter of asking Bassari experts, but

during my visit with the Bassari in 19941 found that the meaning of other

mathematical patterns in Bassari culture can be used to make some educated guesses

about the meaning of the mask. Despite the extensive migrations from the vil-

lages to cities (Nolan 1986), there is still strong participation in the age groups

and transition rituals. The "forest spirit" Annakudi, for example, seems to be

undaunted by the city of Tambacounda, where a local age group hosted him at

a well-attended dance during my stay. Indeed, I found the stereotype of traditional

elders and irreverent youths to be somewhat reversed (which was explained to

121

FIGURE 8.5

Scaling hexagons in a Bassari mask

(a) The Bassari initiation masks frequently feature scaling hexagons in she center. This appears to

be a linear scaling. (b) One of the Bassari elders demonstrates the traditional string talleys, with

knots in groups of six.

la, phoco from agence Hoa-Qui/Michel Renaudeau.)

I22

African fractal mathematics

me as an effect of the strong hierarchy of secret knowledge: the youth are often

more wary about breaking taboos because they are less certain about boundaries

and consequences). This is not to say that there is any overt presence of fear. In

fact, it is the positive aspects of the secrets that are stressed, as became obvious

when elders gleefully refused my questions while emphasizing the wonderful

nature of the information they could not divulge.

The number six is a prominant feature of Bassari mathematics in many

areas of their life. They have a popular game, for example, played with pebbles

on a sand pattern, which makes use of two axes with six holes in each line. In

their traditional calendar there are six months per year, each of 3o (6 x 5) days,

with an initiation about every 12 (6 X 2) years (to a total of nine initiations).

Each of these rites of passage involves a lengthy education in a new level of

traditional knowledge. The most important is the passage to adulthood, which

lasts for six days. In addition to these time measures, the number six also appears

in the Bassari counting system. String tallies, traditionally used for recording

various counts, often used knots grouped by six. The Bassari elder who demon-

strated these tallies to me (fig. 8.gb) told me that he did not know much about

traditional forms of calculation, bur he did know that in precolonial times it was

performed by specialists who were trained in the memorization of sums. This prac-

tice may explain the origins of the famous African American calculating

prodigy, Thomas Fuller. In 1724, at the age of 14, he was captured-quite

possibly from the geographic areas that included the BassariS—-and sold into

slavery in Virginia, where he astonished both popular and professional audiences

with his extraordinary calculating feats (Fauvel and Gerdes 1990).

Finally, there is the Bassari divination system. Although the cast shells are

interpreted by images rather than any numeric reading, they are cast six times.

Each cast provides the answer to a specific question (or verification of a previ-

ous question) relevant to the client's problem; the final sixth cast shows the prob-

lem as a whole. If we compare this divination to the initiation system, the

number six can be seen as a marker for information clusters, a punctuation

point which, like the tally system, allows the distinctions that maintain a com-

prehensive structure. And like the initiation, each cycle of six provides an

expanding view of the whole. Thus it seeins likely that the scaling hexagons of

the initiation mask represent this six-stage iteration of knowledge.

Nonlinear scaling iterations can also be found in African initiation masks.

Figure 8.Ga shows a Bakwele mask in which both size and curvature have a non-

linear increase with each stage. My guess--] have not found any cultural

descriptions that can confirm this— is that it suggests "to open your eyes" as a

metaphor of knowledge, and thus maps the scaling iterations of the mask to iter-

Recursion

ations of knowledge gained in initiation stages. Figure 8.6b shows a Bembe mask

used in the first of a three-stage initiation for a voluntary association, the

bruami (Biebuyck 1973). Before the ceremony, the mask is hidden behind a screen,

and during the ritual the screen is gradually lifted by a high-ranking senior mem-

ber. Both the relation between the number of eyes in the mask and the number

of stages in initiation, as well as this method of visually exposing the pattern

as a sequence, again suggest intentional use of a scaling geometric design to rep-

resent scaling iterations of knowledge.

123

FIGURE 8.6

Ô, photo courtesy Gene Isaacson; b, courtesy Musée de l'Homme.)

124

African fractal mathematics

Representing recursion as a process in time: part II, kinship and descent

If age-grade systems are one part of the standard anthropological explanation

for how "classless" societies are structured, kinship is the other.° Kinship sys-

tems are primarily based on genetic ties ("blood relations") and marriage,

although most societies also have "fictive" kin (e.g, adoption) which are just

as real-kinship is a cultural phenomenon. Descent is also culturally based. Most

Western European and American societies think of descent as biological, but

that is because most of them have bilateral descent, in which both parents

are used to establish kinship. Unilineal descent, where a kin group traces their

lineage through one sex only, is actually more common (in about 6o percent of

the world's cultures). A "Clany is a unilineal kinship group whose members report

that they are descended from a common distant ancestor, often a mythological

figure.¡Claus often have important religious and political functions, although

they are typically spread out across many villages and usually prohibit marriage

between.clan.members.

We have already seen how the Bamana use recursion to generate a binary

code in their divination; here we will look at their representation of descent as

recursion. The antelope figure in Bamana iconography is associated with both

human and agricultural fertility. In the chi wara association, which is open to

both men and women, the antelope appears in a striking headdress (fig. 8.7a),

which represents the recursion of reproduction: mother and child. When seeing

one headdress individually, the scaling seems trivial, but with several examples

together the extraordinary insistence on self-similarity becomes apparent. This

icon acts as the seed transformation in an iterative loop: the child becomes a

mother, who häs a child, who becomes a mother, and so on. Figure 8.7b shows

the descent carried to a third iteration.

In chapter 2 we saw several examples in which descent was tied to scal-

ing architecture. The Batammaliba, who live in the northern parts of Ghana,

Benin, and Togo, have developed an elaborate system for this relationship

(Blier 1987). Figure 8.8 shows a diagram of their two-story house, based on the

circle of circles found ir: much of the West African interior. In front of the house

lies the first of two scaling transformations. It is the "soul mound," a circle of

cylinders representing the spirits of those currently living in the house and

physically structured like a scaled-down version of the house architecture. As

the current family gives way to a new generation, the soul mound undergoes a

second transformation in which it is divided into a single cylinder and is

moved inside. A scaling sequence of these single cylinders-one for each gen-

eration- can be seen wrapped around the central tower inside the house.

FIGURE 8.7

Recursion and reproduction in Bamana sculpture

(a) The chi wara figure, used in ritual dances for agricultural fertility, shows a striking self-similarity:

alhough the fgures vary widely, each one is similar to itself. This can be attributed to the Bamana

View of reproduction as cyclic iterations. (b) Here the cycle is carried out to three iterations.

(a: upper left, from the de Hevenon Collection, Museum of African Art, Smithsonian Institution; upper

Tohr, courtesy Musée de l'Homme; loser, from Carnegie Institute 1970. b, courtesy Musée de l'Homme.)

I 26

African fractal mathematics

FIGURE 8.8

Recursion in Batammaliba architecture

(a) Diagram of 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 size of

the ancestral mounds increasing from youngest to oldest.

Here only one such array is shown, but typically there

are several in the same houschokl.

(a, from Blier 1987.)

Blier's diagram indicates that the size of the ancestral mounds increases from

youngest to oldest, and she notes that this reflects the Batammaliba's idea of a

spiritual power in proportion to age. So far it would appear that there are only

two scaling cascades-one to shrink houses to soul mounds, and another to

divide soul mounds into cylinder rows--and no iterative loop. But if the largest

mound represents the oldest, then recent mounds would be increasingly

threatened by vanishing scale. How would the first descendant have known how

large to make the first mound? Blier notes that many of the symbolic features

of the architecture are replastered with additional layers of wet clay on ritual

occasions, and we can surmise that this applies to the ancestral mounds as

well. Thus an iterative loop, in which each new ancestor adds power to the older

ones by increasing their mound's size, would be at work in the scaling sequence

we see accumulating around the central tower.

Ricision

The Mirsogho society of Gabon includes several religious associations that

(are housed in the same temple (ebanda). Figure 8. ga shows the central post of.

an ebandza featuring scaling pairs of human figures. As in the chi wara figure, there

is only one iteration; the significance lies in this figure as the seed transformation

for a recursive process. The use of a cross shape may be due to Christian influence,

but rhe bilareral scaling is quite indigenous, as we see in the classic Bakwele sculp-

ture (fig. 8.gb) elsewhere in Gabon. Most important, the ebandza post provides

a visualization for the iterative concept of descent that is widely used in this cul-

cure area. This is beautifully described by Fernandez (1982) in a detailed ethnog-

raphy of the Mitsogho's neighbors and cultural relatives, the Fang.

Although the Fang are patrilineal, they believe that the active principle

of birth--a tiny human (what was called a "homunculus" in early European med-

ical theory)—is contained in the female blood. The idea of the new existing within

the old, and vice versa, is a strong cultural theme. For example, in one ritual the

mother places a newborn child on the back of her oldest sibling to symbolize

continuiry of the lineage. Fernandez (1982, 254) notes that the rebirth con-

cept is so strong that "Fang fathers often called their infant sons ata, the

familiar form of father." 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 become 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 (fig. 8.0c). Even in Christian-animist syncretism, biblical characters

are reinterpreted as cyclic rebirths: the African gods Zame and Nyingian

become Adam and Eve, who become Cain and Abel (understood as male and

female), who become Christ and the Virgin Mary. Fernandez notes that these

cycles are not mere repetition, but rather iterative transformations: "The

spiritual-fraternal relation of Zame and his sister is converted into the carnal

relation of Adam and Eve which degenerates into the materialistic and divisive

relation of Cain and Abel which then is regenerated as the immaculate and

filial relationship of Mary and Jesus" (p. 339). According to Fernandez, these

degeneration/ regeneration differences are visualized as horizontal versus

vertical,? which could explain the alternation in the ebandza posts. In apply-

ing this cyclic conception to the ebandza structure (fig. 8.gdl), we can see the

descent model in its full fractal expansion.

The Tabwa, who occupy the eastern section of the Democratic Republic

of Congo (Zaire), have also developed several geometric figures to serve as mod-

els for their conceptions of kinship and descent. Maurer and Roberts (3987, 25)

explain that in the Tabwa origin story, an aardvark's winding tunnel results in

727

FIGURE 8.9

Recursive kinship in Gabon

(a) The central post of the ebandza temple in western Gabon suggests an iterative descent

concept. This is actually a museum reproduction. (b) Bakwele masks from eastern Gabon show

similar bilateral scaling.

(a, from Perrois 1986; b: left, from Perrois 1986; right, Metropolitan Museum of Art; from Zaslavsky

1973 .)

(figure continues)

les)

Recursion

I29

FIGURE 8.9 (continued)

Recursive descent in Gabon

spirit is explicity 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. The implication of infinite regress is discussed in

chapter 9.

(c, from Fernandez 1982.)

Nganga dance

a "bottomless spring" from which emerges the first human, Kyomba, whose

descendants spread in all directions from this central point. This spread is visu-

alized by the mpande, a disk cut from the end of a cone snail, which is worn as

a chest pendant (fig. 8. 10a). The central point is drilled out, representing the emer-

gence of Kyomba from the deep spring, and the logarithmic spiral of the shell

end symbolizes the expansion of kin groups from this origin.8

One way to represent these expanding iterations through time is to take a

series of portraits as the structure changes: projections at different points along

the time axis. Figure 8. 1ob shows the first step toward this design: a more linear

version of the mpance disk, in which an Archimedean spiral fits between a series

130

African fractal mathematics

of triangles (which represent the wives of the guardian of the ancestors). In

figure 8.1oc we see that the linear spiral has become concentric squares, but

they are now portrayed in a scaling sequence, suggesting a series of portraits of

the kinship spiral as it expands through time. Similar scaling square sequences,

carried out to a great number of iterations, can be seen in the staffs of their

northern neighbors, the Baluba (fig. 8.1od).

FIGURE 8.10

Tabwa kinship representations

(a) The mpande shell worn by Chief Manda Kaseke Joseph. (b) A more linear version of the

mpande disk, in which an Archimedean spiral fits between a series of triangles (which represent

the wives of the guardian ancestors). (c) The linear spiral has become concentric squares, but they

are now portrayed in a scaling sequence, suggesting a series of portraits of the kinship spiral as it

expands through time. (d) Sinilar scaling of square sequences can he seen in the staffs of their

northern neighbors, the Baluba.

(a-c, from Roberts and Maurer 1985; d, Museum für Volkerkunde, Frankfurt.)

Recursion

I3I

Recursive cosmology

In all the descent representations we have examined, kinship groups trace them-

selves to a mythological ancestor at the beginning of the world, and thus we move

from the origins of humanity to the origins"of the cosmos. African creation con-

cepts are often based on a recursive nesting. The best-known example is that of

the Dogon, as described by French ethnographer Marcel Griaule (1965). His work

began during the 1930 Dakar-Djibouti expedition, where he first made contact

with the Dogon of Sanga in what is now Mali. In 1947 his studies took a dra-

matic turn of events when one of the Dogon elders, Ogotemmêli, agreed to intro-

duce Griaule to their elaborate knowledge system. Clifford (1983) provides a

detailed review of the strong.

reactions to Griaule's resulting ethnography.

While many of the critiques were really about the failings of modernist anthro-

pology in general--the tendency to prefer a static past over the present, or a

singular "tradition" over individual invention-there were also those who

simply did not believe that such elaborate abstractions could be indigenous.

For the Dogon the human shape is not only a biological form, but maps

meaning at all levels: "The fact that the universe is projected in the same

manner on a series of different scales—-the cosmos, the village, the house, the

individual-provides a profoundly unifying element in Dogon life" (Duly 1979).

The Dogon house is physically structured on a model of the human form, with

a large rectangle for the body, smaller rectangles on each side for arms, a door

for the mouth, and so on. The Dogon village, however, represents the human

form with a symbolic structure rather than a geometric structure: it is not phys-

ically arranged as a human shape, but various buildings are assigned meaning

according to their social function (the smithy stancs for the head, the menstrual

..- lodges as hands, and so on): Flie ase of to chifferent systems of representation

prevents self-similarity in the physical structure of the architecture, but some

of the Dogon's religious icons do show human forms made out of human forms

(fig. 8.11a).

A threefold scaling appears in several aspects of the Dogon religion, and

it is here that we find an indication that the Dogon are using more than just

cascade. Griaule (1965, 138) summarizes Ogotemmêli's creation story:

"God... had three times reorganized the world by means of three successive

Words, each more explicit and more widespread in its range than the one

before it." But these reorganizations are not merely layering one on top of the

other; rather the output of each reorganization becomes the input for the next.

The earth gives birth to the first spirits; these "Nummo" regenerate ancestral

beings into humanlike reptiles; the reptile-ancestors are again reborn as the first

true humans. Within rebirth, the threefold iteration is again enacted. In the first

(a) In the Dogon cosmology, the structure of the human form is

created from human form.

(h) The symbolism of the stacked pots,

représenting the breath of life, within the,

feteus, within the womb. We can use an

iterative drawing procedure to better

understand how this kind of scaling can

result from a recursive loop. Suppose we

have a routine that can draw the circle of

the pot given a diameter, and one that can

draw a lid.

While diameter ≥ minimum do:

Draw a circle of size diameter

If size = minimum, draw a lid

Shrink diameter by 4/3

End of "while" loop.

This procedure first checks to see if we are

past the smallest diameter possible. If not,

it draws a pot, shrinks the diameter value

by ⅔s, and then goes back to the start of

the while loop. In other words, the output

of one iteration—a given diameter-

becomes the input for the next iteration.

child.

FIGURE 8.11

Scaling in Dogon religious icons

(a, from Laude 1973; courtesy Lester Wunderman; c, from Carnegie Institute 1970; courtesy of

Jay C. Leff.)

Recursion

regeneration, for example, each ancestral being enters the earth's womb, which

turns each of them into a fetus, which allows the breath of life (nummo) to enter.

The cosmological narrative suggests that in the Dogon view the birthing

processes at all scales are, in some sense, iterations through the same transfor

mation, and that these iterations are actually nested loops.

Why should the Dogon require such deep iterative nesting? I suspect that

there are two motivations First.)there is an insight into modeling the world:

recursion is an important feature in biological morphogenesis, as well as in

environmental and social change. The second is thé cultural context of this

knowledge: elders need to ensure that the younger generation respects their

authority, which can only be done by giving them gradual access to the source

of this power, which is knowledge. A knowledge system in which endless exe-

gesis is possible makes the initiation process a lifetime activity. But having so

much explanatory elbow room also presents a problem with translating such

narratives into machematics.' We had to be careful with translations for more

formal

practices, such as interpreting the Bamana divination system as a binary

code, or adire cloth as a geometric algorithm. A narrative is not a quantitative

or geometric pattern, and its ambiguity requires all the more care in produc-

ing a mathematical translation that does not embellish indigenous concepts.

Pirst, we have to distinguish between modeling the narrative-something a

structural anthropologist like Claude Lévi-Strauss would do-and the narra-

tive as an indigenous model, such as the Dogon's system for representing their

own abstract ideas. The best way to limit our translation to ideas that the Dogon

themselves are trying to convey is to compare these abstractions of the narra-

tive with other, more formal Dogon systems. This means missing some ideas

that do not have such formal counterparts, but it is better to err on the safe side...

in this context.

The material designs of the Dogon are more restricted than the narrative

in terms of their iterative depth. The best case is probably in the iconography

uf the granary, where Ogotemmêli explains a stack of three pots: the largest rep-

resents the womb; the one on top of it, creating its lid, represents the fetus; and

the lid of that por is the smallest pot, containing a perfume that represents the

breath of life (Griaule 1065, 39). The smallest por is capped by a normal lid; at

this point the recursion "bottoms out." This is not merely a stack of different sizes;

in the Dogon view the womb creates the preconditions that give rise to the fetus,

which is the precondition for the entry of the breath of life. The recursion is empha-

sized in the way that each new pot begins before the previous pot ends (fig. 8.11b),

that is, one pot's lid is the next pot's body (Griaule 1965, 199). In the sculpture

in figure 8. 11c the mother's breasts become the child's head- again, a new one

I33

134

African fractal mathematics

begins before the previous one ends. As we saw in the chi wara sculpture of the

Dogon's Bamana neighbors, reproduction is modeled as recursion.

The Dogon view of a cosmos structured as nested human form is quite

similar to certain ancient Egyptian representations. Figute 8.12 shows a relief

from a tomb in which the cosmos encloses the sky, which encloses the earth.

It is interesting to note that there are again three iterations of scale. A three-

(iteration numeric loop is indicated for the Egyptian god of wisdom, Thoth. He!

is referred to as Hermes Trismegestus, which means "thrice great Hermes," but

he is also referred to as "eight times great Hermes." Why both three and eight?

It makes sense if we think in terms of those common elements of African numeric

systems, recursion and base-two arithmetic. Thrice great because while an

ordinary human may rise as high as the master of masters, Hermes Trismegestus

is the master of masters of masters (three iterations); thus we can surmise "eight

times great" refers to 23 = 8.

FIGURE 8.12

Recursion in the cosmology of ancient Egypt

Geb, the Earth, enclosee by Shu, space, enclosed by Nut, the stellar canopy.

(From Fourier 1821.)

Recursion

Many of the processional crosses of Ethopia also indicate a threefold iter-

ation (fg. 8.13). Although the crosses are now used in Christian church pro-

ceedings, Perczel (1981) reports that related designs can be found on ornaments

excavated from the city of Axum in northern Ethopia in the second half of the

frst millennium B.C.e., so we should not assume that the threefold iteration was

originally related to the Christian trinity, although a connection may have

occurred later (fg. 8.13b). Could there be a common history behind all these occur-

rences of triple iterations in the religious icons of the Sudan and North Africa?

I think the common use of recursion itself is due to a mutual influence, but the

occurrence of triple iteration may be only due to the similarity of circumstances

rather than diffusion. For one thing, given the materials the artisans are work-

ing with, minute scales are difficult, so that the tendency to be limited to three

iterations may simply be a practical consequence of the craft methods. It may also

be that if one wishes to get the concept of iteration across, two is too few, while

more than three is unnecessary (which is why modern mathematicians often rep-

resent an infinite series by the first three elements, e.g., "1,2,3 ..."). On the other

hand, there are cases where many such "unnecessary" iterations are made in the

most difficult of craft materials. Figure 8.14 shows an ancient Egyptian design,

carved in stone, representing the origin myth in which the lotus flower (its petals-

within-petals illustrated by a multitude of scaling lines) begins the self-generating

creation of the material world.

I35

Self-reference

Self-reference is the most powerful type of recursion. The ability of a system ro

reflect on itself is at the heart of both the limits of mathematical computation

as well as our subjective experience of consciousness. But there are relatively

trivial applications of self-reference as well (one can always use a blowtorch to

light a candle). Self-reference first came to the artention of mathematicians in

simple examples of logical paradox; for example, the "liar's paradox" we exam-

ined earlier. To see how self-reference can be more than just a logician's joke,

let's examine how it works in programming. Recall that a simple cascade could

not be.used. if. we did-not.know.how.many.transformations were needed.ahead

of time. The same problem occurred for the Batammaliba ancestral mounds; since

the first descendant did not know how many would be needed, the system has

to allow for iterative resizing. We also saw the possibility of nested iterative loops,

illustrated by the two-loop drawing program for Mokoulek architecture. But stip-

pose we didn't know how many nested loops we were going to need? In the same

way that the recursive cascade could not deal with an unknown number of iter-

Sced shape

(all lines are

active lines)

Second iteration

Third iteration

FIGURE 8.13

Fractals in Ethiopian

processional crosses

(a) Fractal simulations for Ethiopian

processional crosses through three iterations.

(b) Ethiopia converted to Christianity in

333 c.e., and in the thirteenth century King

Lalibela directed the construction of churches

to be cut from massive rocks in one of the

mountain regions. The church of Sr. George

(at right) shows a triple iteration of nested

(a, all Ethiopian processional crosses from Portland

Museum in Oregon; photos courtesy of Csilla

Perczel, b, photo by Georg Gerster.)

Recursion

137

FIGURE 8.14

The lotus icon in ancient Egyptian cosmology

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 Fourier 1821.)

ations, nested iteration has trouble with an unknown number of loops. 10 Here

is where self-reference can help out. An example of self-reference in program-

ming is illustrated for the Dogon pot stack in figure 8.15.

We know that the Dogon pot stack can be drawn with a single iterative

loop-it does not require self-reference. But the task can be accomplished by

self-reference, and we might similarly ask if there are cases of scaling in African

designs in which self-reference plays a role, regardless of whether it is required.

In. European history, self-reference begins with the story of Epimenides of

Crete, the "liar's paradox." Similar utilizations of narrative self-reference to cre-

ate uncertainty can be found in certain African trickster stories. For example,

in an Ashanti story of Ananse (who became "Aunt Nancy" in African Ameri-

can folklore), a man named "Hates-to-be-contradicted" is tricked into con-

tradicting himself. Pelton (1980, 51) notes that the application of such

self-referential paradox is a theme in many Ananse stories: "Thus Ananse

rejects truth in favor of lying, but only for the sake of speech; temperance in

favor of gluttony for the sake of eating; chastity in favor of lasciviousness for

the sake of sex." The following tale is not nearly as sparse but carries the fla-

vor of self-referential paradox quite well:

One of the most common of all stories in Africa describes the encounter of a

man and a human skull in the bush. Among the Nupe of Nigeria, for instance,

they tell of the hunter who trips over a skull while in pursuit of game and

exclaims in wonderment, "What is this? How did it get here?" "Talking

138

African fractal mathematics

FIGURE 8.15

Drawing the Dogon pot stack by self-reference

The symbolism of the stacked pots represents the breath of life,

within the fetus, within the womb: We have aready seen how this

can he drawn using an iterative loop; now let's see how it can be

drawn using self-reference.

Suppose we have a routine that can draw the semicircle of the

pot given a diameter.

Procedure DRAW-POT

If size = minimum, draw a lid.

Else

Draw a circle of size diameter

Shrink diameter by ⅔3

DRAW-POT

End of "else" clause

End of procedure

Notice that this procedure first checks to see if we are at the

smallest diameter possible. If not, it draws a pot, shrinks the

diameter value it by ⅔s, and then calls itself-an application of

self-reference. Now the program has to execute a DRAW-POT

procedure again. The recursion will "bottom-out" when it finally

draws a lid. The program then skips to the "End of procedure" line

and can finally pop back up to the place it left off after executing

the previous DRAW-POT call.

brought me here," the skull replies. Naturally the hunter is amazed and

quickly runs back to his village, exclaiming about what he has found. Even-

tually the king hears about this wonder and demands that the hunter take him

to see it. They return to the place in the bush where the skull is sitting, and

the hunter points it out to his king, who naturally wants to hear the skull's

message. The hunter repeats the question: "How did you get here?" but the

skull says nothing. The king, angry now, accuses the hunter of deception, and

orders his head cut off on the spot. When the royal party departs, the skull

speaks out, asking the hunter "What is this? How did you get here?" The head

replies, "Talking brought me here!"

(Abrahams 1983, 1)

Self-reference is also visually portrayed in some African designs. Figure 8.16a

shows another abbia carving from Cameroon, seen also in the nested fish earlier

in this chapter. But this abbia carving is an icon for itself--it is an abbia of abbia.

According to the Cameroon Cultural Review (inside cover, June 1979), its mean-

ing is "reproduction." Another example of self-reference from Cameroon is

shown in figure 8.16b, a life-size bronze statue of the king of Foumban. Here we

see the king smoking his pipe, the bowl of which is a figure of the king smok-

ing his pipe, the howl of which is a figure of the king smoking his pipe. Like the

Kellogg's cornflakes hox described earlier, the visual self-reference instantly

leads to infinite regress. But it could be more than just humor in the bronze sculp-

Recursion

ture. Since the pipe is a well-known symbol of royal prestige in Foumban, it may

be that the artisans were making purposeful use of the infinite regress: "The king's

power is never-ending."

Figure 8.16c shows a Bamana headdress, that is, a sculpture worn on the

head during ceremonies. Fagg (1967) suggests that this enacts self-reference:

a headdress of a person wearing a headdress of a person wearing a headdress.

Others (cf. Arnoldi 1977) have described this as a symbol of fertility spirits, but

the two interpretations may not be mutually exclusive. Returning to the

139

FIGURE 8.16

Self-reference in African icons

(a) The abbia carvings from Cameroon show a wide variety of images, but chis abbia carving is

an icon for itself-it is an abbia of abbia. (b) A life-size bronze statue of the king of Foumban.

Here we see the king smoking his pipe, the bowl of which is a figure of the king smoking his pipe.

(e) Bamana headdress.

&, drawing based on abbia pictured on the cover of Cameroon Cultural Review, 1979; c, photo courtesy

Fana University Museum of African Art.)

6: H

140

African fractal mathematics

Bamana's close cultural relatives the Dogon, we see self-reference suggested by

Ogotemmêlli's description of how the eighth ancestor, "who was Word itself," was

able to use Word (that is, the breath of life) to self-generate into the next iter-

ation of humanity. In examining the self-similar iterations of the Dogon mother

and child in figure 8.1 1c, we noted a structural characteristic that can be

expressed in the phrase "a new one begins before the old one ends." This would

also describe the structure of the pipe in the statue of the king of Foumban, which

we know to be explicitly self-referential. Perhaps the self-referential version of

the Dogon pot stack was the correct one after all.

Iconic representations of recursion

The abbia of abbia, as a symbol of " reproduction, is more than just an appli-

cation of self-reference; it represents the conceps itself. If recursion is really a

conscious (that is, self-conscious!) aspect of African knowledge systems, then

we should expect such representations, rather than just instances in which the

concept is applied. Figure 8.1 7a shows the application of recursion in the tra-

Chaguet sal

FIGURE 8.17

Reflux

(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) Ancient 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, but it also symbolized the alchemists' goal of refinement of the

human soul.

(a, photo courtesy IFAN, Dakar; b, drawing hased on Taylor 1930.)

Recursion

ditional distillation of palm wine into liquor in the Casamance region of

Senegal. Such distillation techniques were developed to sophisticated levels

in ancient Egypt, where the process became an iterative loop which modern

chemists call a "reflux" apparatus. Figure 8.1 7b shows the iconic representation

of the reflux system in the oldest known alchemical writings (first century c.E.),

which are attributed to Maria (who wrote under the name of Miriam, sister of

Moses), Cleopatra (not the famous queen), Comarius, and the mythic figure

of Hermes Trismegestus (Thoth). Taylor (1930) notes that although these

were written in Greek, "the religious element ... links them to Egypt rather

than to Greece," and he suggests that the most likely origin is from the tradi-

tions of the ancient Egyptian priesthood."' In these writings we find the reflux

icon associated with the aphorism "as above, so below," recalling the self-

similar scaling cosmology we have seen in sub-Saharan Africa, as well as its links

to the recursion of self-fertilization. 12

Of course, one can go too far in attributing links between ancient Egypt

and sub-Saharan Africa (see Oritz de Montellano 1993; Martel 1994; Lefkowitz

1996). There is good evidence for the origins of the Egyptian base-two arithmetic

system from sub-Saharan Africa, and for the persistent use of recursion in knowl-

edge systems across the African continent. But it would be unwise to assume that

one can attribute more specific features to diffusion. In particular, it is highly

unlikely that the same figure of a serpent biting its tail, appearing as an icon

for the god Dan in the vodun religion of Benin (fig. 8.18a) could have derived

from the Egyptian image, or vice versa. As we shall see, the meaning of the

vodun icon has nothing to do with the Egyptian reflux concept.

In August 1994, thanks to the aid of Martine de Sousa (one of the African

descendants of the famed Francisco de Souza), I was granted an interview with

the chief of the Dan temple in Ouidah, Benin. Both the chief and his wife were

quite responsive to my interest in the geometric features of Dan's representations

and identified the sinusoidal icon in iron (fig. 8.18b) as "Dan at work in the world,"

pointing out that he creates order in wind and water. The cyclic Dan was more

abstract, existing in a domain where he was in communication with other gods

of vodun. Maupoil (1981, 79) also found that Dan (Dangbe) was there "to

assure the regularization of the forces," and Blier (1995) summarizes his role as

"powers of movement through life, and nature's blessings." Regular phenomena

in nature--the periodic aspects of weather, water waves, biological cycles,

etc.---are attributed to the action of Dan.

The relation between the undulatory Dan "at work in the world" and the

circular form of Dan as a more abstract spiritual force maps neatly on to the dif-

ference between the sinusoidal waves we see in space and time—-waves in

141

noise (external temperature changes)

input (desired

temperature)

If > 0 switch on furnace

1f < 0 switch off furnace

output (new room

temperature)

The thermostat that regulates temperature in a house is a negative feedback loop. The word "negative"

is used because we subtract the current room temperature from the desired temperature set by the

thermostar control: Over time this will tend to produce cycles of heat and cold.

noise (road bumps)

input (desired

position on road)

If > 0 steer right

*I'< 0 steer left

* output (new road position)

Driving a car can aiso be modeled by a negative feedback loop. The driver attempts to stay in the center

of the lane, and will correct to adjust for bumps. Again, given enough humps; we will tend to see cycles

of swerving to ger back to the center.

FIGURE 8.18

The vodun god Dan

In the vodun religion of Benin, the snake god Dan represents the cyclic order of nature. Dan's

shape reflects this idea in two ways. As an abstract force, he is represented as a feedback loop (a).

As a concrete manifestation, his body is always oscillating in a periodic wave (b). This same idea

of a periodic time series from cyclic feedhack is also used in Western models of nature (c).

(a, photo courtesy IFAN, Dakar.)

lion)

inter

cles

(a).

dea

Recursion

water and cirrus clouds, daily fluccuations in heat and light, the biannual rainy

seasons, etc.— and the abstract idea of an iterative loop that generates these wave-

forms. The association can be derived from the kind of empirical observation

one. gets in everyday occurrences. A-lopsided wheel will produce undulatory

tracks in sand; friends who periodically give gifts are in a "cycle of exchange,"

and so forth. What did take great insight and intellectual labor, however, was

the religious practitioners'

generalization of such observations into specific,

abstract, universally applicable categories, represented by icons with the appro-

priate geometric structure.,

The mathematical equivalents in nonlinear dynamics are limit cycles and

point attractors- the results of what engineers call a "negarive feedback loop."

We have already seen such characterizations in cellular automata and owari,

where spatial patterns remain bounded within a cycle or frozen in a static pat-

tern. Figure 8.18c shows some commonplace examples of negative feedback loops,

and how they act to keep the behavior of systems bounded or stabilized, even

in the presence of noise. But the vodun system would not be complete if it could

only account for regularity-what causes deviation in the first place? Hence

the role of Legba, god of chaos. Figure 8.Iga shows another iron icon, the forked

path of Legba, "god of the crossroads." As explained to me by Kake S. Alfred,

a divination priest of vodun in Cotonou, Benin, Legba is represented by the

fork because "the answer could be yes or no; you don't know which path he will

take." For divination, in which a "path" (question) is often pursued for further

questions, the image becomes one of endless bifurcations. At the Palais Royal

in Porto Novo, Benin, I was told that the shrine to Legba was placed at the

threshold because his force was so disruptive that it would undo both good and

evil, creating a purification at the entrarce: Kake also explained that while the

music of Dan was slow and regular, the music of Legba was both fast and slow—

signifying his unpredictable nature- an observation I was able to confirm by

recording the drumming that was used to call each god at the temple of Dan in

Ouidah. 13 As the converse to Dan, the bifurcating uncertainties of Legba are

like a positive feedback loop, amplifying deviation and noise (fig. 8.19b).

Contrasts between a negative feedback loop, creating stability, and the pos-

itive feedback of uncontrolled disorder are also featured in the iconic carvings

of the Baule. Vogel (1977, 53) notes that the Baule chief is chosen by consen-

sus, and that in all important decisions he serves as mediator in public meetings

rather than as an autocrat. The Baule carving in figure 8.20a shows two caimans

(relatives of the alligator) biting each other's tails. It is said to represent the chief

and the people in balance--if one bites, the other will bite back. It nicely

recalls the kinds of negative feedback loop models that are often proposed in West-

143

FIGURE 8.19

Legba

(a) The vodun god Legba represents the forces of disorder.

Vodun divination priests explain this icon as the path to the

future: with Legba there is no way to know which path will be

caken. Since one crossroad leads to another, the resulting image

is one of bifurcating unknowns, the uncertainty multiplying with

each crossroad.

noise (road bumps)

input (desired

position on road)

- Il > 0 steer too far right

i < O steer too tar lett

output (new road positior

I dontres do ver, five adie, which sill ele cabilize me me enterine easier large destabilize it.

eventually running off the road.

Nation A

buys more arms

Nation B sees arms

increase and

becomes worried

Nation A sees arms

increase and

becomes worried

Nation B

buys more arms

Here we see positive feedback in an arms race.

Recursion

1 45

FIGURE 8.20

Feedback loops in Baule iconography

(a) This Baule carving shows two crocodiles biting

each other's tails. It is a symbol showing the chief and

she 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 circuit fermé of fécondité (closed

circuit of fecundity); Soppelasa (1974) and Odica

(1971) identify these animals as symbols of "increase."

(ascal bo, photo courtesy of IFAN, Dakar.)

emn political theory, but this flowchart is a purely indigenous invention. So, too,

is the Baule positive feedback loop of figure 8.zob, showing that "power creates

the appetite for more power"-little fish are eaten by bigger fish, who then

become even bigger fish. The fish-within-fish abbia from Cameroon we saw

carlier may have had similar connotations.

Conclusion

Recursion can be.found in almost every corner of African material culture and

design, from construction rechniques to esthetic design, and in cultural repre-

senfations from kinship to cosmology. Most of these are specific enough to

allow us to distinguish berween the first two types of recursion--cascade versus

I46

African fractal mathematics

iteration-and in some cases the third type, self-reference, is also made explicit

by the indigenous knowledge system. We have seen several cases in which the

iterative loops are nested, but these are rarely more than two loops deep, so it

would not appear that the application of self-reference is motivated by the com-

plexity of the computation. The only potential exception is the cosmological nar-

rative of the Dogon, and this narrative is too vague to serve as a mathematical

foundation. There is, however, another route to the limits of computation. As

we will find in chapter 1o, the combination of negative and positive feedback

indicated by certain recursion icons provides another path to the helghts.of com-

putational complexity, one we will explore in detail. But first, we need to take

a short detour through infinity.

CHAPTER

-Infinity

- - The first time I submitted a journal article on African fractals, one reviewer replied

that Africans could not have "true" fractal geometry because they lacked the

advanced mathematical concept of infinity. On the one hand, that reviewer was

wrong about fractals at a pragmatic level. If he or she saw a fractal on a computer

screen it would be taken as a "true" example, and in fact no physically existing

fractal is infinite in its scales; at best it will have to bottom out into subatomic

particles. On the other hand, it raises an interesting question. Infinity has been

an important part of fractal mathematics in Europe; so how does that compare

to the use of infinity in Africa?

To the ancient Greeks, infinity was associated with what they thought of

as the horrors of infinite regress. Aristotle tamed this problem by redefining

infinity: it was a limit that one could tend toward, but it was not considered to

be a legitimate object of mathematical inquiry in itself. Most European mathe-

maticians kept to this definition until the Cantor set, Europe's first fractal, cre-

ated the proper defnition of an infinite set, thus allowing infinity itself to be

considered. We will discuss this in more detail in chapter 13, but for now it is

sufficient to note that this distinction does not shape African concepts of infin-

ity. Many African knowiesige systems using infity in the sense of a progression

without limit do nor hesitate to represent it with iconic symbols suggesting

147

148

African fractal mathematics

"the infinite" in its Cantorian sense as a completed whole. This is by no means

a more sophisticated or elaborated definition than that of pre-Cantorian Euro-

pean mathematics; it is rarely linked to much more than either a narrative or a

geometric visualization. But far from being nonexistent, these culturally specific

representations show a strong engagement with the same concepts that coupled

infinity and fractals in contemporary Western mathematics.

The most common African visualizations for infinity are snail shells. The

Baluba) for example, use spiral land snails (fig. g.r), and the Joln lse the spiral

end of a sea snail, which forms a drinking cup that can only be used by the chief.

Unlike the ancient Greek associations with troubling paradox and pathology, the

African infinite is typically a positive association, in this case to invoke prosperity

without end. If these infinity icons were only meant to communicate this desire

they would fit Aristotle's definition: a process without end. But the spiritual ele-

ment of these icons.adds another requirement: the icons need to convey the sense

that they are drawing on the power of infinity itself, Snail shells are used because

of the scaling properties of their logarithmic spirals; one can clearly see the poten-

tial for the spiral to continue without end despite its containment in a finite space—

indeed, it is only because of its containment in a finite space that there is a sense

of having gained access to or grasped at the infinite,

We have already seen another example of an infinity icon in the Nankani

architecture discussed in chapter 2. There the coils of a serpent of infinite

FIGURE 9.1

Baluba use of snail shells

to symbolize infinity

Davidson (1971, 120) descrihes this as a fertilit

figure, and notes that the snail shells represent

endless growth.

(Collection Tristan Tzera, Paris; photo by Eliot

Elisofon.)

Infinity

length, sculpted into the house walls, made use of the same association between

prosperity without end, and a geometric length without end. The conscious

creation of this infinity concept is more clear than in the case of the snail shells,

because one cannot actually see the infinite coils of the snake. And unlike the

naturally occurring shells, the packing of this infnite length into a finite space

(the Nankani describe it as "coiling back on itself indefinitely") cannot be mis-

taken for mere mimicry of nature; it is rather the artifice of fractals. This snake

icon does not exist in isolation; we saw that-the Nankani map out a scaling pro-

gression that passes through their architecture, the zalanga and the kumpio,

which provides a recursive pathway to this concept of infinity.

In chapter 8 we discussed the Mitsogho and Fang iterative model of

descent. Fernandez (1982, 338) notes the contrast to Christian theology: "The

question as to whether God was one or many may have bothered the mission-

aries in their contacts with Fang more than the Fang themselves. Holding Chris-

tian beliefs in the 'Uncreated Creator' and 'Unmoved Mover,' missionaries were

challenged by the 'infinite regress' of the geñealogical model employed by the

Fang- their belief that the God of this world is one of a long line of gods and

like man has his own genealogy."

The Fang theory of infinite regress is a complete, coherent view; it does not

need further amendment, for the Christian theory of uncreated creator is no more

free of contradiction-and perhaps less so. Of course, as Fernandez himself

warns, one cannot simply proclaim that a particúlar African narrative is just another

work of theology or philosophy-or, for that matter, mathematics. Recent works

such as Mudimbe's Invention of Africa ( 1988) have shown that such translations

to specife European disciplines are always partial, highly interpretive, and in dan-

ger of misrepresenting the indigenous view. Yet Mudimbe is also respectful of the

work that has been done. Of particular relevance here are his citations of African

theologian Engelbert Mveng

Mveng included several notes on infinity in his studies of the relation

berween the African and Christian views. His beautiful text, L'Art d'Afrique Noire

(1964), contains diagrams (pp. 100-103) showing what he termed "radiation

omplifcative," scaling patterns in African art and music that he interpreted as

representations of a transcendental path to infinity. "Une fois de plus, nous

découvrons que le mouvement rythmique, dans notre art, n'est autre chose

qu'une course vers l'infini" (Once again, we discover that the rhythmic move-

ment in our art is none other than the path toward infinity) (p. 102). Father

Mveng was a wonderful inspiration during my research in Cameroon, both for

his deep cultural knowledge as well as for his courageous work as a cross-cultural

mediator. During our last meeting we discussed Mudimbe's book, and I promised

149

150

African fractal mathematics

to send him a copy. Shortly after doing so a reply came from the American

Cultural Center in Yaoundé: Mveng had been murdered "under suspicious ? !

circumstances"— apparently the result of opposition to his cross-cultural

activism. He has finally taken the course vers l'infini.

CHAPTER

Complexity

- In ordinary speech, "complex" just means that there is a lor going on. But for

mathematicians the term is precisely defined, and it gives us a new way to

approach mathematics in African material culture. In chapter 7 we saw how cer-

tain African symbolic systems, like the Bamana divination code, could be

generated by a recursive loop. Such numeric systems clearly translate into the

Western definitions of what it means to "compute." But the translation was less

clear for some of the physically recursive structures in African material culture.

Can a system of physical dynamics be said to "compute"? Mathematical com-

plexity theory, which is based on fractal geometry, provides a way to measure

the computation embedded in physical structures, rather than just symbol sys-

rems. By looking at African material culture in the framework of complexity.

rheory, we can better understand the presence of fractal geometry as an African ) !

knowledge system.

Analog computing

By the misk rodos it was clear to many researchers that digital computers would

be the wave of the future. But before then, analog computers held their own, and

they may yet make a comeback. In digital systems, information is represented by

I51

152

African fractal mathematics

physically arbitrary symbols. As Bateson (1972) said, "There is nothing sevenish

about the numeral seven." The geometric structure of a digital symbol has little

or nothing to do with its meaning, which is simply assigned to it ay social con-/

vention. In analog systems, the physical structure of the 3ignal changes in pro

( portion to changes in the information it represents.' Rather than being arbitrary,

the physical structure is a direct reflection of its information. Loudness in human

speech is a good example of analog representation. As I get more excited, I speak

louder: the physical parameter changes in proportion to the semantic parame-

ter. This is not true for the digital parts of speech, such as the average pitch ("fomat

frequency") of each word. In English the word "cat" has a higher pitch than the

word "dog," but that does not infer a relation in meaning-in fact, the difference

is reversed in Spanish, since "gato" has a lower average pitch than "perro." This

same analog/digital distinction occurs in neural signals. In the frog retina, for

example, some neurons have a firing rate in proportion to the speed of small mov-

ing images (Grusser and Grusser-Cornehls 1976). That is, the faster a fly moves

across the eye, the faster the pulses of the neuron: an analog system. A digital!

example can be found in the muter neurons that lins open the crayish claw. Here 3

a specific firing pattern (off-on-on-off) switches the claw to this defense teflex

(Wikson and Davis 1965).

So far we have only examined how analog systems can represent infor-

mation; figure yo.1 shows a simple example of how analog computing works.

Although most computer scientists eventually settled on digital systems, ana-

log computers were quite popular up until the igoos. Even when they began to

die out as practical machines, there was an increasing awareness that much of

our own brain operates by analog computing, and this led some scientists

toward the development of what are now called "neural nets"--computing

devices that mimic the analog operations of natural neurons (fig. 10.2). By the

raid-108 os neural nets and related analog devices had achieved enough success

(and digital computers had run into enough barriers) to begin to compare the

two. There was an odd moment of analog optimism, when a few brash claims

were made about the potential superiority of analog computing (see Dewdney

1985; Vergis et al. 1985), but these assertions were eventually proved incor-

rect (Blum, Shub, and Smale 1989; Rubel 198g). As it turns out, analog sys-

•tems have the same theoretical limits to computing as digital systems.?

Although the studies did not result in releasing the known limitations, they

did produce a new framework for thinking about computing in physical dynam-

ics: complexity theory.

Before this time, mathematicians had defined complexity in terms of

randomness, primarily based on the work of Soviet mathematician A. N.

lis

ior

ves

tal

cre

Rex

for-

rks.

h of

rists.:

ning.

y the

ccess

e the

laims

dney

ncor-

g sys-

ems.?

, they

y nam:

jis of

J. N.

Complexity

x53

FIGURE 10.I

Analog computation

Dewdney (s985) shows a great variety of simple physical devices that demonstrate analog

computing. This device, created by J. H. Luerh of the U.S. Metals Rehning 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 distances of these

sources are o, c, and i, then the refinery should be located at the point where 00 + cC + IL is at

a minimum. The holes through which the strings pass are at the source locations, and the weights

un the ends of the strings are proportionate to O, C, and L. The brass ring attached to the strings

quickly moves to the optimal location on the map.

(Contesy A. K. Dewdney.)

Kolmogorov and Americans Gregory Chaitin and Ray Solomonoff. In this def-

inition, the complexity of a signal (either analog or digital) is measured by the

length of the shortest algorithm required to produce it (fig. 10.3). This means

that periodic numbers (such as .2727272...) will have a low algorithmic com-

plexity. Even if the number is infinitely long, the algorithm can simply say,

"Write a decimal point followed by endless reperitions of '22" or even shorter:

"3/ux." Truly random numbers (e.g, a string of numbers produced by rolling

*dice) will have the highest algorithmic complexity possible, since their only

algorithm is the.number.itself-for.an infinite lengrh, you get infinite com-

plexity In analog systems a periodic signal such as the vibration from a single

guitar string or the repetitive swings of a pendulum would have the lowest algo-

rithmic complexity, and random noise such as static from a radio that has lost

154

African fractal mathematics

its station (what is often called "white noise") would have the highest algo-

rithmic complexity.

One problem with defining complexity in terms of randomness is that it does

not match our intuition. While it's true that the periodic signal of a ticking

metronome is so simple that it becomes hypnotically boring, the same could be

said for white noise—in fact, l sometimes tune my radio between stations if l

want to fall asleep. But if I want to stay awake I listen to music. Music some-

how satisfies our intuitive concept of complexity: it is predictable enough to fol-

low along, but surprising enough to keep us pleasantly attentive. Mathematicians

eventually caught up with their intuition and developed a new measure in

which the most complex signals are neither completely ordered nor completely

disordered, but rather are halfway in between. These patterns (which include

almost every type of instrumental music) also happen to be fractals-in fact, as

we will see, the new complexity measure exactly coincides with the measure.of

fractal dimension.

The first step in this direction was through studies of cellular automata. Recall

from chapter 7 that computer scientists in the early 198os had started to think

input

output

input

output

FIGURE 10.2

Neural nets

(a) Suppose we balance a ball on a teeter-totter. Unless the ball is at the precise center, the

teeter-totter will start to stope toward one side, which will cause the ball to roll even farther

toward that side. In other words, there are two stable states, and anything in herween (except fo

tiny neutral point) will get caught up in the positive-feedback 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 outpur. Since the invert

output is connected to the input of the other amplifer in each, they will balance out like the b.

at the exact center of the teeter-totter, but rapicily lip to one of the two stable states in which t

amplifier is at its maximum ("saturated"). That means that this circuit can solve a simple task:

which of two numbers is larger? By putting an initial charge proportionate to one of the two

numbers at each inpur, the system rapidly flips to the sarurated stable state favored by the large

number. Linking thousands of these simple amplifers togerher allows computer scientists to ma

sophisticated machines for pattern recognition and other artificial intelligence tasks.

Complexity

about cellular automata as the simulation of complicated physical dynamics, such

as that seen in living organisms. Physicist Stephen Wolfram began to wonder:

just how complicated is it? Clearly, living systems are more complex than ran-

dom noise, so he knew that the old complexity measure of Kolmogorov would

nor do. But Wolfram had studied a good deal of computer science, and he real-

ized that the way in which different types of recursions are used to measure com-

puting power could also be applied to physical dynamics. Recall from chapter 8

that we divided recursion into three types: cascades, iterations, and self-reference.

155

time

frequency

amplicude

power

time

frequency

FIGURE 10.3

Koinogorov-Chuitin compiexity incasure

(a) Whether it is in digital or analog signals, complexity can be

mensured in terms of the information content. The first such

measure was that of Kolmogorov and Chaitin, who thought of

complexity in terms of randomness. The sine wave is about as

nonrandom as we can get. Here it is given as a time-varying

signal, although the same would apply to a spatial pattern, such

complexity

as waves in water or sand (in which case we could measure it as

wavelength, which is simply the reciprocal of frequency).

• randomness

(b) The same signal in a spectral density plot. This tells you

how much power is at each frequency. In the case of the sine wave, all the signal power is at one

Itequency. (c) White noise is a completely random signal, such as that produced by the sound of

bacon frying. By the Kolmogorov-Chaitin definition, white noise is the most complex signal.

Again, this would also apply to a spatial pattern, such as dust sprinkled on a cable. (d) Spectral

density plot for white noise. Because it is completely random, there is an equal likelihood of any

wivelength occurring at any time, so the signal's power is equally distributed across the spectrum.

(e) In summary, the Kolmogorov-Chaitin complexity measure is simply a measure of randomness.

(c, courtesy R. F. Voss.)

input tape

A read only

....

156

African fractal mathematics

These correspond approximately to the three formal categories of recursion

used in computer science, which we will now examine in detail

Three types of recursion: the Chomsky hierarchy

In a recursive system, present behavior depends on past behavior. It is the capa-

bility of this access to memory that defnes the relative difference in recursive

power. The scaling cascade, for example, could not produce the Fibonacci

sequence, because it could not recall previous members of the sequence. Simi-

lar distinctions are used in computer science to rank computational power into

three types of abstract machines, referred to as "Chomsky's hierarchy." These

abstract machines are compared by their ability to recognize certain categories

of character strings. A machine that can recognize periodic character strings

such as "ababa.

i."." occurs at the lowest level of the hierarchy: the Finite State

Automaton (FSA). An example of the FSA is shown in figure 10.4

What would it be like to be an FSA? Since the FSA has no memory stor-

age, the experience would be somewhat analogous to neurosurgery patients, who

have had bilateral hippocampal lesions (Milner 1966). These patients are fully a

aware and intelligent but have lost the capacity to transfer knowledge to long

term memory. The hippocampal surgery patient who finds herself at the end of

a book can deduce that she has read its contents, although she does nor know

what the previous chapters were about. An(FSA has only an implici memory,

fact that it must have passed through one of the sequences of states that termi-

because its present scare cannot reveal anything about its past, other thin the f

By nite bertise present state.

FRA

Current state

Transition table

Current symbol

on input tape

New state

FIGURE 10.4

The finite state

automaton

The finite state automaton

(FSA) has a list of transition

rules that tell it how to change

from one state to the next,

depending on its current state

and the symbol it is reading on

the input tape. It has no men-

ory, other than that implied by

its curent state. This FSA will

end up in the "accept" state S,

if the tape ends after an even

number of h's.

Complexity

157

The set of palindromic strings (e.g., aabbaa) is a good example of the lim-

itation of the FSA: it lacks the ability to memorize the first half of the string

and therefore cannot compare it with the second. The least powerful machine

capable of this memory storage is the Push*Down Automaton (PDA), illus- T-ba

trated in figure 10.5. The stack memory of the PDA is usually compared to the

spring-loaded tray stack often used in cafeterias; once a symbol is read from mem-/

ory it is gone. As a knowledge analogy, we might think of a reader who accu-

mulates stacks of books but gets rid of each book after it is read. This is a

remporary explicit memory, since the PDA. can.make.two different transitions

given, the same state and input, depending on its past. It is important to under-

stand that greater recursive capability does not necessarily.imply larger mem-

ory storage; it means an improved ability to interact with memory. Size only matters

insofar as it restricts the interaction.

Although the PDA can recognize all sets of strings recognized by an FSA,

as well as many others, there are still (infinitely) many sets of strings that it can-

not recognize (For example, it cannot recognize the set of all strings of the form

aNbNeN (where we have N repetitions of a, followed by the same for b and c),

because it has to wipe out its memory in the process of comparing the number

of a's and b's, leaving no information for checking the number of c's.

At the top of the hierarchy (fig. 10.6), the Turing Machiné (TM) can TM

recognize all computable functions. It is simply a PDA with unrestricted mem-

ory, but because of this capability it can achieve full self-reference: the abil.

ity to analyze.its own program. Again, it is not a difference in memory size,

but in memory access--unlike the PDA stack, the TM memory interactions

can occur over any past sequences of any length, and it does not lose memory

input tape

read only

FIGURE 10.5

The push-down

read/write

Transition table

"Stack" memory. This allows new symbols

to be pushed down on top of the stack, but

symbols can be read only by popping them

off the top, and each one popped is lost.

158

African fractal mathematics

input tape

FIGURE 10.6

The Turing machine

read/write (inoves in both directions)

Transition table

after it is read. To continue the text analogy, if the FSA is a person who accom-

plishes tasks with no books, and the PDA is a person whose simple tasks are

limited to books that are removed after they are read, then the TM would be

able to collect and recall all books, in any order. Unfortunately this does not

solve all of our problems, because the unbounded nature of the TM means

that it foolishly accepts some tasks that require an infinite library. This is called

the "halting problem," and Turing himself proved that it is unavoidable.

Mathematician Rózsa Péter showed that one can define a restricted set of pro-

grams that are haltable (which she called the set of "primitive recursive

functions"), but in doing so we would always sacrifice some of the TM's

computing power.

These three machines, FSA, PDA, and TM, illustrate the ascent up the

Chomsky hierarchy. They differ in having implicit memory, temporary explicit

memory, and permanent explicit memory. By looking at memory as the basis for

the recursive loop in these systems— that is, as the element that governs the abil-

ity of the system to perform interactions between its present input and past behav-

ior-we can see that the differences in computational power for these machines

depends on the differences in recursive power.

Measuring analog complexity with digital computation

Now let's return to Wolfram and his cellular automata. After running thousands

of trials, Wolfram found that all cellular automata generally divided into four spe-

cific classes. Classes 1 and 2 were those that either died out, or went into a peri

odic cycle. Class 3 was just the opposite: it was uncontrolled growth that led to

apparently random behavior, like white noise. But class 4, which included the

"game of life" cellular automaton, had something that Wolfram described as "com-

plex" behavior: not as random as white noise, but not as boring as a periodic cycle.

Wolfram found that this highest complexity also demanded the highest com-

Complexity

putability: while pure order and pure disorder could be recognized by an FSA,

the pacterns of the complex behavior required a Turing machine.

Mathematical physicist James Crutchfield (1989) found an even

simpler example of recursive computation in a physical system. Crutchfield

used the population equation made famous by biologist Robert May (1976):

Pn+| = PnR(1 - Pn) (where P is a population number, scaled so that it is between

0 and 1, and R is the birth rate). May found that when R is low, the popu-

lation is simply a periodic cycle, switching back and forth between the

same sequence of levels. As you increase R, the length of the cycle (that is,

the number of different population levels you pass through before returning

to the first one) increases extremely fast. At R = 3.1, the population is in a

two-level cycle, at R = 3.4 in a four-level cycle, and at R = 4.0 the cycle length

is ar infinity: deterministic chaos. Crutchfield was able to measure the com-

putability of these chaotic fluctuations and found results similar to those of

Wolfram: both completely periodic waves and completely disordered waves

were computationally quite simple, but those in berween, with a mix of

order and disorder, had a high degree of computational complexity. The

simple equation examined by Crutchfield required only a PDA, but other

researchers (Blum, Shub, and Smale 1989) demonstrated that more complex

analog feedback systems would be capable of signal complexity equivalent to

TM computability.

Figure 10.7 shows how these complex waveforms, called "1/F noise," com-

pare to periodic and white noise waveforms. This is easiest to see in the spec-

tral density plots. A periodic signal has all its power at one wavelength, while

a white-noise signal has the same power at all wavelengths. 1/F noise is a com-

promise beiwecirthe ewor-biased so that it has the greatest amount of power'

at the longest wavelength, and the least at the shortest. For this reason, 1/F noise,

is fractal; it has fluctuations within fluctuations within fluctuations. When we

think of the length of these waveforms in terms of memory, we can begin to

see a connection to computational power. If a system had the same behavior over

and over again, it would be too fixed on memory. If it randomly picked a new

behavior every time, then it would be too free from memory. But useful behav-)

ior is generally a mixture between the two. For example, think of something)

unusual you did today-moving socks to a new side of the drawer, or eating pret;

zels instead of crackers. Whatever it was, chances are it was pretty trivial. If we

took the same whimsical approach to major life-events each day—"today I think

I'll move to Spain, or get pregnant, or become a podiatrist"-we would be in

trouble. Our life is typically arranged as 1/F noise: high-power events should be

long-term changes, and low-power events should be short-term changes." In fact,

159

power

frequency

power

rime

frequency

power

time

frequency

FIGURE I0.7

Crutchfield-Smale complexity measure

(a-b) Periodic noise: A simple signal. (c-d) White noise:

From the viewpoint of the Crutchfekd-Smale 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

complexity

signals in the Crutchfield-Sinale measure are "scaling

noises" in which there are fluctuations within fluctu-

ations. These signals have the greatest amount of their

periodic

noise

fractal

thise

random

noise

power in the lowest frequencies (longest wavelength).

Since power is the reciprocal of frequency, it is often referred to as 1/F noise. (g) In summary, the

Crutchfield-Smale complexity measure is a reflection of the fractal dimension. The "most fractal"

(e.g., dimension of t.g) will be the most complex, and the function decreases with both higher and

lower dimensions.

(c and e, courtesy R. F. Voss.)

ise

ind

Complexity

many of the analog waveforms produced by intelligent human behavior appear

to be 1/F signals (Voss 1988; Eglash 1993).

As more scientists began to think of complexity in terms of computation

and 1/F noise, they began to accumulate examples that suggested that this was

what it meant to have a "self-organizing" system. In the evolution of life, for

instance, most of the genetic information stores long-term events, such as the

physiology that underwent change in life's evolution from water to land. More

short-term adaptations, such as skin color, take up very little of the genetic mate-

rial. Here again, we have something like 1/F noise, with long-term events tak-

ing up the buik of the system, and short-term events taking up proportionately

less. Physicists Per Bak and Chao Tang (Bak and Chen 1991) found several

examples of simple physical self-organizing systems that produced 1/F noise. In

forest fires, for example, very dry woods would spread fire in an orderly circle,

while fires in wet wood would be too sporadic or random, and thus die out. But

in-between fires spread in a fractal pattern, with most of the fire in long-length

patches, less of the fire in medium patches, even less in smaller patches, and so

on. In water we have orderly crystals and disorderly liquids, but in between we

can get the fractal patterns of snowflakes.

Since we are familiar with our own recursive interactions with memory,

we have a good intuitive sense for why 1/F noise should accompany complex

behavior, and clearly it can characterize many varieties of self-organizing sys-

tems -perhaps all of them if we use the proper definition. But how does this hap-

pen? What is the mechanism that makes it work? Complexity theorists have not

hesitated to suggest implications of their work for culture; here I would like to

suggest the reverse: that certain aspects of African culture can provide impor-

tant implications for complexity theory. More so than.any of the previous ethno

mathematics models we have seen, this part of my research was much more of

a collaboration, much closer to my sense of the "participant simulation"

method—although if truth be known I had to be dragged kicking and scream-

ing much of the way.

Christian Sina Diatta: an African physicist looks at culture

"Rhab." "Phantom." "Rhab!" "Phantom!!" A strange dialog flew across the com-

puter lab at the Institut de Technologie Nucleare Appliquée at Senegal's Uni-

versity of Dakar. I was seated with Professor Christian Sina Diatra, director of

the lab, watching the pulsating forms of cellular automata flow about the screen.

Dr; Diatta was the local sponsor for research under the United States' Fulbright

Fellöwship program, and was eager to discuss his own ideas. His physics lab was

I 62

African fractal mathematics

an inspiring place to be. I had already been able to sit in on a graduate student's

presentation; after having witnessed the same ritual in the physics department

at the University of California at Santa Cruz, it made for a fascinating bit of cross-

cultural comparison. I tried to make myself useful by setting up a demo of an elec-

trical circuit that produced deterministic chaos ("Chua's circuit") and installing

various types of software for simulations of nonlinear dynamics. It was one of these

software demos, Rudy Rucker's calaB, that caused our multilingual exchange.

As noted in chapter 7, some of Rucker's most interesting programs are those

he calls "Zhabotinsky CAs," which can produce paired lug spirals. In acklition

to the two states of live cell and dead cell, these cellular automata require at

least one "ghost state." Since someone had previously mentioned the indige-

nous term for ghost, rhab, it seemed like an opportunity for creative transla-

tion. I explained (in French, the official language of Senegal) that after l'état mort

(the dead state) the cell went to l'état rhab. To my surprise, Diatta corrected

thab back to the French: "phantom." We went back and forth a couple of times

before I realized that it was not just my poor pronunciation. Only later did

I discover my blunder: Diatta was not from the Islamic Wolof majority (in whose

language rhab occurs) but from one of the animist minority groups, the Jola.

Using Wolof was no more of a cultural translation for him than it would have

been to use English.

This was only the start of my mistranslations. Although Dr. Diatta was

greatly enthusiastic about my work on fractals in African architecture, he

seemed disinterested in the fractal generation software. But he persistently

brought up African architecture during the cellular automata demos. I found this

entirely too frustrating; the whole point of my research on African fractals was..

to explore the intentional side of these designs. Cellular automata create pat-

terns not by preplanned design, but rather by the interactions of its aggregate

cells. From my point of view, having fractal architecture as the result of aggre-

gate self-organization destroyed the possibility of intentionality. By focusing on

cellular automata as an architectural model, Diatta seemed to he undoing all

my carefully prepared research. His enthusiasin was unbeatable, however, and

I began to study aerial photos of his place of origin, the Jola settlements south

of the Casamance River. Figure 10.8 shows the settlement of Mlomp, not far from

Diatta's hometown; its paired log spiral structure could have come right out of

Rucker's Zhabotinsky CAs.

A trip to the Casamance was clearly called for. I was fortunate in finding

Nfally Badiane, a Jola graduate student who had done his master's thesis on indige-

nous architecture of the southern Casamance, as a guide. Nfally's background is

ideal for an anthropologist: raised among the Islamic majority in Dakar, he is both

FIGURE 10.8

The Jola settlement of Mlomp, Senegal

(4 ts) (6 Mloey phlel se senely combisg of ego sdo hnageieand recursive process.

I64

African fractal mathematics

stranger to and member of the Jola society. As we traveled the delta area of the

Casamance River, using cars, trucks, canoes, and anything else that moved, his

warnings about the secrecy of Jola religious knowledge were repeatedly confirmed.

Secular information about technical methods of house construction, precolonial

and postcolonial social changes, kinship. groups, and many other aspects of

Jola society were readily forthcoming (Eglash et al. 1994). We were told that the

circular building complexes were not preplanned, nor were the broad curves of

these complexes in each neighborhood, but that they could not tell us anything

about the sequence of construction because, unlike the Wolof, "we do not have

a griot (oral historian) in Jola society." The spiral structure visible in the photo

was mainly due to the carefully maintained sacred forest surrounding each local

neighborhood. But the mechanisms for creating such coherent structures over

such an enormous range of scales remained hidden. A tantalizing glimpse of the

Jolas sacred geometry, however, led us to suspect that there was a conscious ele-

ment to the CA-like settlement structure. First, there was the symbolisin of the

chief's drinking vessel: a spiral shell. Second, Nfally had seen the interior of one

of the settlement altars, and said that it consisted of a spiral passage.

The best clue we found was from Diatta himself, who described a log spi?

ral path in certain rituals that took place in the sacred forest. But how to rec-

oncile this self-conscious modeling with what appeared to be the emergence

of the settlement structure through aggregate self-organization? I finally con-

fessed my disturbance to Diatta, and asked him how I might understand the appar-

ent contradiction. He suggested yet another simulation: the Jola funeral ritual

(fig. 10.ga). We had been alerted to this ceremony as a result of a suspicious death

during our visit, bur were not allowed to attend. Diatta described the ritual in

detail. The body of the deceased was placed on a platform, and posts at each of

the four corners are held aloft by palibearers. If critical knowledge is thought

to have been held by the deceased (e.g., as in the case of a murder), a priest asks

questions. The pallbearers, reacting to the force of the deceased, move the plat-

form to the right for yes, left for no, and forward for "unknown."

The simulation for this ritual (fig. 1o.gb) is based on an analog feedback

network. We don't need to make any assumptions about whether the pallbear-

ers are exerting force due to conscious opinions or subconscious beliefs; it is only

necessary to assume that they exert force in proportion to this motivation.

Since they can both exert force and sense it from others, this would theoretically

allow the summation of knowledge among the participants to be expressed in the

most effective way possible. In fact, the technique is more effective than a vote,

since voting can lead to the paradox of a minority opinion win if there are more

than two options. * The inforination emerged froin the bottom-up interaction of

Complexity

the parts, yet it was also intentional in the sense that this mechanism for aggre-

gate self-organization of knowledge had been consciously designed. This was not

intentionality as I knew it; it sounded more like the description of a neural net-

work in computer science:

",,.

If a programmer has a neural network model of vision, for example, he or she

can simulate the pattern of light and dark falling on the retina by activat-

ing certain input nodes, and then letting the activation spread through the

165

Unknown

(a) In the Jola funeral ritual four

pallbearers hold a platform aloft and

move it in response to questions. Since

the information (whether one believes it

to be of spiritual or mundane origin) is

held by the pallbearers, we can model the

force of each corner as having direction

and magnitude (a vector) determined by

the pallbearer's conviction. Decision

making based on a continuous range

rather than on yes/no is called "fuzzy

logic" in mathematics.

Yes

yes

padlocarers:

(imput)

(b) We can think of the information

processing in the Jola funeral as the

equivalent of a neural net (similar to that

in hig. 10.2) in which the sum of the force

vectors of all four pallbearers are inpurs to

shrer amplifers, with each inverted output

connected as negative feedback to the

other two. This would require pallbearers

to hoth exert force as well as sense it, but

such force-feedback is actually quite

common in motor tasks.

unknown

FIGURE 10.9

Neural net model for the Jola funeral ritual

IVU

connections into the rest of the network. The effect is a bit like sending

shiploads of goods into a few port cities along the seacoast, and then letting

a zillion trucks cart the stuff along the highways among the inland cities. But

if the connections have been properly arranged, the network will soon settle

into a self-consistent pattern of activation that corresponds to a classifica-

tion of the scene. "That's a cat!"

(Waldrop 1992, 289-90)

The tricky part is "if the connections have been properly arranged."

Clearly it could be arranged for four people, but could it for this"city of Mlomp,

with dozens of local neighborhoods and hundreds of people in each? And

Mlomp is not an anomaly. While we saw a more explicit formal system in the

construction of several fractal settlement architectures in chapter 2, there are also

many African settlements that have a large, diffuse fractal structure (see Denyer

1978, 144). Self-organizing mechanisms that arrange such vast aggregations

into coherent patterns would have to be more global and less explicit.

One key mechanism in complexity theory is memory; the theory predicts

that self-organizing systems will utilize 1/F distributions in memory length: The

lukasa, a visual "memory board" developed by the Baluba of Congo (Zaire), shows

just such fractal scaling (fig. 10.10). The memory system of the lukasa is partly

based on digital (that is, physically arbitrary) coding, such as color, but Roberts

(1996) notes that much of the lukasa is a "geometry of ideas," mapping the beaded

spatial structure to analogous historical events. Although there is considerable

interpretive and coding variation, there is a tendency to have single beads rep-

resenting individuals, groups of beads representing royal courts, and larger bead

arrangements showing the sacred forests that have been growing over many

generations. This visualization of a l/F-like distribution of memory suggests at

least the possibility of indigenous awareness of scaling properties in maintain-

ing self-organized complexity.

The strongest candidate for a mechanism underlying self-organization is

the complementary pair of indigenous feedback concepts. we examined in

chapter 8. In the vodun religion of Benin, we found Dan representing the sta-

bilizing force of negative feedback, and Legba representing the disruptive

force of positive feedback. Similar feedback pairs were found in the Baule

door carvings; the caimans biting each other's tails are a symbol of negative

feedback, and the fish eating ever larger fish represent positive feedback. This

combination of opposing feedback loops also appears to be at the heart of the

conditions that sustain self-organizing structures. Of course, mnost.self-organizing

systems will have more than two loops; but in every case I have examined, at

'least one of each is present, and it is through this interaction that sustained

complexity can arise.

FIGURE 10.10

Lukasa

(From Roberts and Roberts 1996; photo by Dick Beaulieux.)

I68

African fractal mathematics

Returning to the most basic example of complex behavior, May's popula-

tion equation, we have two components. On the one hand, there is population

growth: Pn+1 = PnR. Next year's population will be this year's population times

the growth rate. As long as R is a positive number, this will be a positive feed-

back loop. But the other part of the equation, multiplying by (1 - Pn), was a neg i?,

ative feedback loop, acting like an epidemic that kills more people with larger

population size. Together they create deterministic chaos: the positive feedback

keeps expanding the population, and the negative feedback keeps it within

bounds. This works for other chaos equations as well. Figure 10.11 shows a

chaos equation called the "Rossler attractor" modeling a car with two drivers.

One is drunk and overcompensates by steering too far with each correction; the

other is sober and pulls it back on the road when the drunken oscillations get

too large. Because it always steers back to a slightly different position, the oscil-

/ lations never repeat--deterministic chaos.6

We can see the same combination of negative and positive feedback cre-

ating self-organization in aggregate systems. The "game of life" cellular automa-

ton offers a particularly clear illustration of this phenomenon. If we give a rule

set that makes birth too easy (e.g., the cell goes to the "live" state if there is one

or more nearest neighbors alive), then there is too much positive feedback and

we get a rapidly spreading disk. If we make death too easy (e.g., the cell goes to

the "dead" state if there is one or more nearest neighbors alive), the screen goes

FIGURE 10.11

Rössler attractor as feedback in automobile driving

The Rössier attractor is a set of three simple equations whose output is deterministic chaos, that_is.

a signal with variable oscillations which remain bounded but never repeat the exact same pattern.

How can such a simple system produce infinite variation? An automobile driving model can help

us see what these equations are doing.

(a) Positive feedback. 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. Note 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) Negative feedback. The 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. "Caution" is represented by

the third variable, Z.

work. The "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 back 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 back toward the center,

it would appear to he completely random, with no predictable pattern. Yet the patrern is entirely

deterministic (chat is, determined only hy this set of equations); it could be predicted if you knew

the initial conditions with infinite precision.

t is,

rh.

s not

ving

gle of

ely

new

Driver

noise

5000

posillon

Driver observation:

has car been devtaling?

YES

new steering angle (Y)

Move steering ongle in

the wrong direction

a Positive feedback

Driver ooservatton.

(2)

d/dt

noise

sotpoint

(x)

Driver observation:

Is cor position › setpoint ?

YES

new position on road (x)

Incroose

coutionl

(2)

Decrease

stoering

ongie

Negative feedback

road position X

c Combination

of negative

and positive

feedback

tune

(Y)

time

Facial expression = Z

Total system: x' = - (y + z)

= x + 0.15y

= 0.2 + 2(x - 10)

I70

African fractal mathematics

blank in a few generations. The "classic" life rule set. (found by John Horton Con-

way in 1970) is often referted to as "3-4" life because it takes 3 nearest neigh-

bors to give birth, but 4 results in death. Conway discovered that this combination

of negative and positive feedback maximized the complexity of behavior. Sim-

ilarly, when Per Bak found empirical data for self-organization in physical sys-

tems-forest fires, earthquakes, avalanches, etc.—he noted that it occurred only

at a "critical state" in which there was a balance between noise-suppressing mech-

anisms-which would correspond to negative feedback-—and the positive feed-

back of noise-amplifying loops.

It is unfortunate that so much of the classic research on African social mech-

anisms came from functionalist anthropology, since they made an almost exclu-

sive emphasis on the role of negative feedback in achieving equilibrium. When

it comes to conscious knowledge systems, African societies do not exclusively

focus on balance, harmony, and stasis. The complimentary roles of Dan and Legba,

of order and disorder, are much more common, as we see in this passage: "In the

mind of the Bambaras the air, wind and fire ... are indispensable elements of

the world's onward movement. But as these principles may be active in an

uncontrolled, that is, unruly and often excessive manner, Nyalé is considered

to be a profuse and extravagant being.... So by her very nature Nyalé is, to a

certain extent, a factor of disorder. That is why it is said that Bemba... took

away her 'double' to entrust it to Faro... whose essential attribute is equilib-

rium" (Zahan 1974, 3).

A similar pairing occurs in the Dogon religion, where Amma, the high god,

creates the Nummo to enact order, and accidentally creates the disorderly

Ogo; together the two generate life as we know it. In the repertoire of dynam-.

ical concepts occurring in several African knowledge systems, there is recognition

of the useful tension between equilibrium and disequilibrium, the dance between

order and chance that results in self-organized complexity. And just as Stuart

Kauffman has shown a bias toward order in evolution's "edge of chaos," the high)

god ensures that the trickster can act only sporadically, thus creating more power)

toward long-term order in these African cosmologies.

Although fractals resulting from geometric algorithms are usually seen as

static structures, they too can be viewed as the combination of feedback loops.

A seed shape with a huge number of tiny line segments (fig. 10.12a) will tend

to be shape-preserving under self-replacement iterations; here deviations due to

replacement are camped-fthe difference between a line segment and the seed

shape is usually not important (and the resulting graph will have a low frac-

tal dimension, i.e., tending toward 1.0). But for seed shapes made up of only

a few large lines (fig. ro.12b), the difference hetween a line segment and its

Small line segments:

negative feedback

Large line segments:

positive feedback

Medium line segments:

a feedback combination

Fractal dimension = 1.3

Fractal dimension = 2.0

FIGURE 10.12

Fractal graphics as feedback

Fractal dimension = 1.6

172

African fractal mathematics

replacement shape will be very important. Large deviations tend to be ampli-

fied in a quick positive feedback, sometimes explosively growing out of bounds

in only a few iterations. Figure xo. 12b has been scaled down to fit on the page,

but the actual fractal graph will quickly grow out of bounds and blacken the

screen entirely (i.e., a fractal dimension close to 2.0). Figure 10.12c shows a

fractal dimension close to t.s; the "most fractal" measure, which results from

a balance between the negative feedback of small segment shape preservation

and the positive feedback of large segment replacement deviation.

There is no quantitative measure of fractal dimension in precolonial

African knowledge systems. But the idea of a spectrum progressing from more

orderly to less orderly is vividly portrayed in certain material designs. The best

examples are in the raffia palm textiles of the Bakuba (fig. 1o.13a). These tend to.

show periodic tiling along one axis, and aperiodic tiling--often moving from order

to disorder--along the other. Similar geometric visualizations of the spectrum

FIGURE I0.13

From order to disorder in a Bakuba cloth

(a) The Bakuba often create cloth designs that stay fairly constant along the vertical axis, but

gradually change along the horizontal axis. In many cases, the horizontal transformation suggests

an order-disorder range. (b) Computer scientist Clifford Pickover created this pattern to show

how a spectrum from order to disorder could be visualized by allowing a random variable to have

increasing influence on the graph's equation. Thus it, too, makes use of periodic tiling along the

vertical axis and aperiodic along the horizontal.

(a, from Meurant 1986, by permission of the author; b, from Pickover 1990, by permission of the author.)

Complexity

from order to disorder have been used in computer science (fg. 10.13b). As far

as I can tell, the Bakuba weavings never reach more than halfway across the spec-

trum-they are typically moving between i and 1-5, that is, from periodic to frac-

tal, rather than stretching all the way to, pure disorder.?

I know of only one African textile that takes this last step, and that is the

block print shown in figure 10.14. This pattern is reminiscent of the title of Niger-

ian author Chinua Achebe's famous novel, Things Fall Apart. Given the anti-

colonial context of Achebe's writing, it might be tempting to read it as an

indication that white noise only comes with white people, but at least in terms

I73

SAT

FIGURE 10.14

Block print textile

This print from West Africa suggests the full spectrum from order to disorder.

(From Sieber 1972.)

174

African fractal mathematics

of the indigenous knowledge system such assuiptions are unfounded.® There

is, for example, a form of music indigenous to Nigeria that has something like

a white noise distribution of sounds. Akpabot (1975) describes "the random music

of the Birom," a flute ensemble designed to allow each musician to express indi-

vidual feelings through whatever idiosyncratic noise (or even silence) he or she

chooses, resulting in "an indeterminate process (in which] the sounds produced

by the players are not obstructed by a conscious attempt to organize the rhythms

and harmonies" (p. 46). Pelton (1980) refers to the Nigerian (Yoruba) trickster

Eshu as the "lord of random," and notes that there is a coupling between the

orderly work of Olirun and this unpredictable spirit, similar to the negative

feedback/positive feedback combinations we noted earlier. The characteriza-

tion of extreme disorder might well be applied to the experience of colonial

rule, but we should not assume that the concept was unknown before then. A

summary of selected African complexity concepts is shown in figure 10.15; note

that the central peak of spiritual power is analogous to the central peak of com-

putational power in the Crutchfield-Smale complexity measure.

Conclusion

This chapter is only the bare outline of what I hope will be future areas of

research, examining the relations between technical, cultural, and political

systems through the new frameworks offered by complexity theory. For the

moment, we will have to limit ourselves to the few fragments that my Senegalese

colleagues pointed out so diligently First, this does not negate the previous

examples of explicit algorithmic design in African fractals,? but it does suggest

that at least in the case of settlement architecture they can arise from another

source as well. The creation of fractal settlement patterns through aggregate self-

organization, while unlike the planned structures we saw in chapter 2, do not seem

to be the result of unconscious social dynamics (as we saw.for the urban spraw!

of European cities in chapter 4). This may be due to a difference between African

concepts of intention, which can apply to a group project created over several

generations, versus the Western focus on an individual performing immediate

action in defining intentionality. Most important, there are indications that this

pattern creation through group activity is supported by conscious mechanisms

specific to self-organization as defined in complexity theory. Both the scaling

distribution of interactions with memory and the spectrum from order to dis-

order have at least some graphic counterparts in African designs. The best can-

didate for a conscious mechanism is the combination of negative and positive

feedback. We did not examine every possible case of deterministic chaos and

spiritual/cultural power

order

fractal

disorder

Akin

(Ghana):

Ananse

the trickster

Icon for "calm waters"

Nyame's power of life;

turbulent waters of

Tanu

(Benin,

Nigeria,

African

diaspora):

Legba, Eshu

the tricksters

Dan

Mawu (acts through

lower gods, c.g., the

bifurcating doublings

of Shango)

Dogon

(Niali):

Ogo

the trickster

Nummo (drawing based on

photo of ritual staff in Imperato

1978)

FIGURE 10.15

African complexity concepts in religion

176

African fractal mathematics

aggregate self-organization, but it would appear that the combination of neg-

ative and positive feedback loops, which form the basis of several African knowl-

edge systems, also formi a key mechanism of general self-organizing systems.

As noted in the first chapter, it is just as important to find what is miss-

ing as it is to find what is present. While four of the five basic concepts of frac-

tal geometry-scaling, self-similarity, recursion, and infinity- ate all potent

aspects of African mathematics, a quantitative measure of dimension (the Hausdorf-

Besicovitch measure) is completely absent. There is a weak sense of a complexity

spectrum of order-disorder, which would covary with the Hausdorf-Besicovirch

measure, but that spectrum is neither quantitative nor (to my knowledge) ever

compared to a concept of dimension in any indigenous African system. This is

an enormous gap in the African knowledge of fractal geometry, especially since

the dimensional measure is often considered the most valuable component by

contemporary researchers in the field.,

On the other hand, we also need to appreciate all knowledge systems in

their own right, and African fractals have a surprisingly strong utilization of

recursion. Indeed, in Mandelbrot's seminal text, The Fractal Geometry of Nature

(1977), the index lists "recursion" only twice, and the terms iteration, self.

reference, self-organization, and feedback are entirely absent. As we will see,

this absence is no accident; it reflects a European historical trend. But why have

Europeans traditionally placed such little importance on recursion, and why was

it so strongly emphasized in African fractals? In part iu of this book we will take

up such cross-cultural comparisons in detail.

- Implications

PART

IMI

CHAPTER

Theoretical

frameworks-

"in-

-cultural studies

of knowledge-

• Parts 1 and i1 of this book emphasized the geometric, symbolic, and quantita-

tive aspects of African fractals. Some cases were more speculative than others-

a difference that l hope was clearly indicated--but even in the use of mythic

narrative, I generally restrained conclusions to those that had geometric or quan-

titative counterparts. In other words, the claims made in parts 1 and i1 should

be falsifiable in the sense of Karl Popper; the data either supports the hypothe-

I sis or refures it.' But the chapters in this last section will switch to topics in cul-

tural politics and other humanities. These issues are too complex and

multidimensional to be reduced.

to formal representations; they can only be

approached by exploring their interpretative depths, Poetry çan reveal as much

truth about the world as any science; we only need to keep in mind that it is a

different way of going about it. While the philosophy, politics, and poetics of

culture are not strictly falsifiable, they can often approach the areas of life chat

Popperian positivisin cannot--areas we cannot live without.

Given that one can make a good case for at least four of the five basic ele- 1.

ments of fractal geometry in African mathematics, what should we make of it

in terms of culture? To ask this question effectively we need to avoid two pit-

falls. The first is the possibility of "overdetermined" explanations for African

Tractals, explanations that seem to be waiting for us before we've even begun

179

I80

Implications

to examine the evidence. The second is the difficulty of sustaining skepticism

in a racially charged environment, the possibility that we might shy away from

critique over fears that expressing a negative view could be taken as having an

ethnocentric or racist motivation, Both failings are equally damaging. Recently,

researchers have drawn attention to the ways that theories of knowledge

(epistemology) can sneak unexamined into cultural portraits. If we are to avoid

the trap of seeing African fractals as an indication that Africans are "closer to

nature," or concrete rather than abstract thinkers, or unified in a single homo-

geneous culture, then we need to know a bit about the origin of these mis-

conceptions. The first step in that process is to examine the epistemological

frameworks that are applied to the study of culture.

The unity/diversity debate and thin description

According to Muclimbe $1988), the concept of a unitary, traditional "African cul-

ture" is an invention created frst by colonialists, who sought to rationalize their

conquest with the myth of the primitive, and subsequently by anticolonialists seek-

ing to consolidate their opposition. A similar critique is provided by Appial (1992),

'who suggests that the differences among various African societies were much too

broad to allow any generalizations (p. 25): "Surely differences in religious ontol-

ogy and ritual, in the organization of politics and the family, in relations between

the sexes and in art, in styles of warfare and cuisine, in language-surely all these

are fundamental kinds of differences?"

Appiah and Mudimbe promote various kinds of solidarity in contemporary

Africa (as well as internationally in the diaspora); they only caution that this

cultural unity is of relatively recent origin, and that attempts to see an African

"essence" or a unified African culture preceding major European intervention

(i.e., previous to the First World War) will eventually have to fall back on racially

defined categories, which is certainly a self-defenting basis for antiracist

movements. From Appiah's antiessentialist point of view one cannot discuss

precolonial "African culture," only "African cultures."

On the other extreme of the unity versus diversity debate lies the Afro-

centric position. While its proponents also agree that there was no single,

homogeneous African culture, they emphasize the shared elements. Asante

and Asante's African Culture: Rhythms of Unity (1985), for example, begins by

stating that while black unity cannot be based on genetic grounds, broadly shared

cultural undercurrents were found throughout the diverse societies of pre-

7 colonial Africa:

Theoretical frameworks in cultural studies

Although the precise actions and ideas may differ within the acceptable range

and still remain squarely in the category of African culture, there are some behav-

iors among some African ethnic groups which may have the opposite mean-

ing among others. Twinness is commonly considered a positive characteristic

in African societies, yet there are some ethnic groups which accept twinness

as a negative characteristic.... Yet this particularistic emphasis would not make

the echnic group uncelated to the others. Patterned behaviors by African erh-

nic groups are cultural, not rigid or fixed, but related to history and experience.

Culture can vary over time, but in the case of African culture, it will always

be articulated in the same way.

There is a lot going on in this paragraph, but the crucial point for my analy-

sis is Asante and Asante's distinction between the surface particularities of

various ethnic groups, which may differ, and deeper cultural sensibilities or pat-

terns of articulation (which they later illustrate with "the three traditional

values: harmony with nature, humaneness, and rhythm" (p. 7l). In this Afro-

centrism, it is only at the deep level in which we find important cultural attrib-

utes held in common.

Appiah also makes this distinction between trivial surface and the "fun-

damental" depths. The only disagreement between him and the Asantes is

whether or not the depths reveal differences. One way around this question is

in the "thick description"

proposed by anthropologist Clifford Geertz (1973).

Geertz was motivated in part by his dissatisfaction with the ways that Claude

Lévi-Strauss's structuralism seemed to reduce symbolic culture to a flat, mecha-

nistic syntax. For Geert?, cultural symbols should be in a kind of dynamic play,

and the ethnographer should show their turbulent expansion through layers of

meaning, not their reduction to a single fixed structure. Geertz defined these deep

elements, which tend to be more subjective and literary, as specifc to a partic-

ular community. For him, it would be extremely difficult to compare deep ele-

ments from one location to the next, because the deep elements are the result

of local interpretations. Taken to the extreme, Geertz's thick description would

simply reply that the question Appiah and the Asantes are asking cannot be

answered

The framework I have used in parts 1 and in of this book, which is that of

ethnomathematics in general, might be referred to as thin description: a study

of the surface particularities, such as material designs and symbolic formulas. As

the Asantes point out, a mathematical element like doubling ("twinness" in their

quotation) is just a surface feature. Whether or not it has deeper meanings--and

thus the entire Afrocentrism/antiessentialism debate--is a question outside of

thin description. For this reason, the thin description use of African icons to

181

182

Implications

represent specific mathematical concepts or structures (e.g., the trickster =

disorder) is not necessarily in conflict with the thick description of these sym-

bols in their deep semiotic dynamics. Pelton (1980) sets up just such a conflict,

and perhaps rightly so-there has indeed been a tendency for structuralists to

claim that they had reduced culture to irs true essence. Their error was to

insist that these bare-bones structures were the truly deep mechanisms of cul-

ture, and that the discursive play of meaning should be disregarded as shallow

distraction. As long as we keep the thick stuff as the deep, and the pared-

down structures as the surface, there is no conflict.

While the lack of African unity in "twinness" is not a problem for those

concerned only with deeper meanings, wouldn't it present a problem for thin

description? That is, if doubling is supposed to be an important feature of African

mathematics, then how does one explain the African societies that do not use

it? Indeed, how is fractal geometry supposed to be an African knowledge system

if the examples of its use are so disparately scattered across the continent? To answer

this question, we need to consider what Wittgenstein called a "family resemblance."

When we look at the photograph of a large family we can see that everyone is

related, even if there is no single characteristic that they all share (some have

big noses and some small, some light hair and some dark, etc.). In the same way,

it is not uncommon for a group of mathematical ideas to share many common-

alities without a singular essence. In James Gleick's (1987) history of chaos

theory, for example, he shows that the emergence of nonlinear dynamics as a dis-

cipline was due to a slow gathering of many different strands of mathematics—

strange attractors, fractal geometry, cellular automata, and so on. In order for

scientists to collaborate on this development, there was a long period in which

several researchers worked hard to point out the family resemblance of these dis-

parate mathematical tools, and many aspects of their relationships are still

uncertain today. Similarly, African fractal geometry is not a singular body of knowl-

edge, but rather a pattérn of resemblance that can be seen when we describe a

wide variety of African mathematical ideas and practices. And as we saw in the

case of Banneker's quincunx, it is not the only pattern possible.

Participant simulation

Whether one believes in Geertz's thick description or in some other method for

researching the deeper meanings of a local culture, anthropologists generally agree

that it requires long-tern local ethnographic study. My thin description fieldwork

lasted only a year and moved through Senegal, Mali, Burkina Faso, The Gam-

bia, Cameroon, Benin, and Ghana. This dispersed investigation is quite unlike

Theoretical frameworks in cultural studies

what is undertaken by most anthropologists, who often spend a couple of years

in one village alone, using "participant observation" to traverse the depths of the

local culture by actively living it. There is, however, an important difference:

I was not trying to understand how the Yoruba experience grief, or to determine

the inner meaning of cominunal spirit among the Baka. My interest was primarily

in the formal properties of design, in methods of construction, and in other

rechnical questions that could often be answered in a direct and simple fashion.

Many of the Africans I spoke with were clearly relieved to hear that I was a

mathematician. Of course I was still faced with several of the same problems

involving ethnographic accuracy and authority (see Clifford 1983). But even

these were somerimes differently posed. In particular, I began to think of my

methodology not as participant observation, but rather as participant simulation,

seeking to collaborate in mathematical analysis and virtual reconstruction

With my African colleagues.

Participant simulation was carried out to conclusion only in the research

with Christian Sina Diatta, but I tried to maintain the practice at some level with

everyone I had the opportunity to work with. That meant hauling diagrams of

fractal graphics with me into the equatorial rain forest and across the savannah,

and disrupting research time with math lectures, but in the end it was well

worth it. There was the potential problem that someone who knew what l was

after might fabricate what I wanted to hear (as in St. Louis, Senegal, when one

of the local children heard me talking about Benjamin Banneker and claimed

to know him personally). A more pressing problem was my resistance to their

suggestions, as occurred in my initial disappointment with the lack of place value

notation in the Bamana divination code, or hearing the description of the oscil-

latory snake as "Dan at work" (all I could think of at the time was a road con.

struction sign). Of course, there are always the aftereffects-Senegalese

sociologist Fatou Sow said "if there are not fractals in Africa now, there surely

will he by the time you leave"— but then that is a feature of all ethnography; and

participant simulation is about turning that into an advantage.

The reason collaborative approaches like participant simulation were not

enditionally used in erhnography comes from concerns over accuracy-the

desire to obtain an objective account-and concerns over authority, a suspicious

motive in the colonial context of most traditional anthropology. Clifford (1983)

Jescribes the move roward collaborative rechniques as both the anthropologists' /

own self-critique of authority and as a growing recognition that since the ethno-

grapher has as much motivation as the informant does, accuracy and objectiv.

ity can be better approached by sharing aurhority with indigenous voices than

by using them in a kind of ventriloquist act. Simply proclaiming a collaborative

183

184

Implications

approach is of course no guarantee that you will have one, and participant sim-

ulation is perhaps even more susceptible to manipulation due to the role of tech-

nological expertise,

On the other hand, since the creation of virtual worlds—simulations--is

in some ways the production of something fake, participant simulation does have

the advantage of avoiding some old-fashioned concepts of authenticity. It was,

after all, the creation of an "authentic native" (see Appadurai 1995) that helped

colonists to jail rebels among black South Africans and Native Americans; and

one could even hear the occasional guilt-ridden lament among the colonial rulers

that they themselves were to blame for having accidentally polluted the natural

purity of these "children of the forest" with their own troubling artifice (see the

apartheid culture comedy, The Gods Must Be Crazy). Locating indigenous activ-

ity in virtual worlds can, if done properly, counter chis habitual tendency to place,

artificiat on the Western side and natural on the indigenous side..

Doing it properly relies on the other root, which comes from the old-

fashioned-and, I think, still crucial--method of participant observation.

Participant observation recruits a kind of responsibility that can be sadly lack-

ing in virtual ethnographies. Take, for example, the growing field of cyber-

ethnography, in which anthropologists study the virtual communities of the

Internet. Since "lurking" (observing the electronic exchanges without partici-

pating) is so easy, there have been a number of studies in which the ethnog-

rapher is reduced to eavesdropper or spy, with no attempt to work with the

community in either off-line or on-line lives. On the other hand, recruits can

include both draftees, who have little real interest in working collaboratively,

and fanatics, who aie all too interested in what Gayatri Spivak (1987) calls the

"benevolence of the western gaze."

Thus participant simulation is an attempt to take the best of both approaches,

and to use them in a kind of checks-and-balances system. By insisting on par-

ticipation we can help avoid glib irresponsibility; and by using simulation we can

strive to avert the policing of boundaries around constructions of authenticity

and realism. From this point of view we do not need to emphasize tradition over

invention; the mathematical creations of a single individual are still examples

of indigenous mathematics, even if she is the only one who knows they exist.

Intentionality and ethnomathematics

There are clear advantages to a methodology that can credit the inventions of

a single individual, but what about those creations that do not have a single inven-

tor? As we saw in the case of complexity in chapter 10, it is possible to err on

.........I

Theoretical frameworks in cultural studies

the other side by insisting that conscious creations can only come from singu-

lar inventors. A better understanding of this problem can be gained through the

contrast between ethnomathematics and mathematical anthropology. Mathe-

matical anthropology is generally focused on revealing patterns that are not con-

sciously detected by its subjects of study.'In part this is due to a conviction that

many of the underpinnings of society are forces unnoticed by its members- not

only because such forces operated at levels beyond individual awareness, bur also

because regulatory mechanisins would have to be covert, obscured, or otherwise

protected from manipulation and conscious reflection. For these reasons, mathe-

matical anthropology makes good sense, and it has indeed produced wonderful

insights. But its emphasis on unconscious process also arose from imitation of

the researcher-object relation in the natural sciences: if anthropologists were

simply reporting indigenous discourse, then they would not count as scientists.

This problem of mere reporting is indeed the case for "non- Western mathematics,"

which is mainly focused on direct translations for Chinese, Hindu, and Muslim

mathematics and thus considered a subject for historians. Hence mathematical

anthropology's rendency to avoid intentionality can be problematic.

The intentionalicy problem in mathematical anthropology can be seen in

Koloseike's-(x974) model for mud terrace construction in the low hills of

Ecuador. Koloseike began with two hypotheses: either the Indians learned from

the Inca stone terraces in the high mountains above, or they were unintentional

by-produces of cultivation on hillsides. He then made a list of nine observations

that were relevant to deciding berween the two. Of particular interest are the

following:

. 3. The same hillside soil is used in rammed-dirt houses and fence walls, and

these stand for years.

4. But I never saw a terrace being constructed, nor did people talk about such

a project.

5. Small caves are often dug into the terrace face for shelter during rain-

storms. That this potentially weakens the terrace face does not seem to con-

cern people.

(1974,20-30)

Koloseike concludes that these terraces are the unintentional result of

an accretion process from the combination of cultivation and erosion, and

then proceeds to develop a mathematical model for the rate of terrace

growth. My point is not in questioning the accuracy of the model, but rather

the way that indigenous intentionality is positioned as an obstacle that must

be overcome before mathematics can be applied. Even a small degree of

awareness-being aware that a cave dug into a terrace face might weaken it—

must be eliminated.

185

186

FT:

Implications

In addition, it reveals a particular cultural construction of the supposed uni-

versal attribute of "intention." As a Westerner, Koloseike is used to a society in

a hurry. Projects to be done must get done, and always with someone in charge.

The idea of a long-term intentional project, perhaps extending over several

generations, or the constitution of collective intentionality rather than individual

intent, is not brought under consideration. It may well be that the mathe-

matical model Koloseike offered was not only accurate, but also had an indige-

nous counterpart.

Ethnomathematics, in contrast, has emphasized the possibilities for indige-

nous intentionality in mathematical patterns. For example, Gerdes (1991) used

the Lusona sand drawings of the Chokwe people of northeastern Angola to

demonstrate indigenous mathematical knowledge. While it would have been

possible to attribute this practice to an unconscious social process, such as the reg-

ulation of authority, Gerdes chose to focus on their properties as conscious indige-

nous inventions. Ascher (1997) notes the same cype of Eulerian path drawings

in the South Pacific, and shows them to be primarily motivated by symbolic nar-

ratives, in particular their use by the Malekula islanders as an abstract mapping

of kinship relations. Again, this is in strong contrast to the tradition of mathe-

matical anthropology, where kinship algebra was considered a triumph of West-

ern analysis (and even a source of mathematical self-critique; Kay (1971] harshly

notes the anthropologists' tendency to invent a new "pseudo-algebra" for various

kinship systems rather than apply one universal standard).

Ascher's description of the Native American game of Dish shows this

contrast in a more subtle form. In the Cayuga version of the game, six peach stones,

biackened on one side, are tossed, and the numbers landing black side or brown

side up were recorded. The traditional Cayuga point scores for each outcome are

(to the nearest integer value) inversely proportional to the probability. Ascher

does not posit an individual Cayuga genius who discovered probability theory,

nor does she explain the pattern as merely an unintentional epinhenomenon of

repeated activity. Rather, her description (p. 93) is focused on how the game is

embedded in community ceremonials, spiritual beliefs, and healing rituals,

specifically chrough the concept of "communal playing" in which winnings are

attributed to the group rather than to the individual player. Juxtaposing this con-

rext with detäiled attention to abstract concepts of randomness and predictability

in association with the game-in particular the idea of "expected values" asso-

ciated with successive tosses--has the effect of attributing the invention of,

probability assignments to collective intent.

At the skeptical extreme in ethnomathematics, Donald Crowe has refrained

from making any inferences about intentionality and insists that his studies of

Theoretical frameworks in cultural studies

symmetry in indigenous pattern creations (see Washburn and Crowe 1988) are

simply examples of applied mathematics. But since Crowe has restricted his

work to only those patterns which could be attributed to conscious design (paint-

ing, carving and weaving), it creates the opposite effect of mathematical anthro-

pology's attempt to elimínate indigenous intent. This is evidenced by Crowe's

dedication to the use of these patterns in mathematics education, particularly his-

teaching experience in Nigeria during the late 196os, which greatly contributed

I to Zaslavsky's (1973) seminal text, Africa Counts.

While non-Western mathematics is exclusively focused on direct trans-

lations (such as Hindu algebra or Muslim geometry), ethnomathematics can be

open to any systematic pattern discernable to the researcher. In fact, even that

description is too restrictive: before Gerdes's study there was no Western cate-

gory of "recursively generated Eulerian paths"; it was only in the act of their par-?

ticipant simulation that Gerdes-and the Chokwe—created that hybrid. And

unlike mathematical anthropology, ethnomathematics puts an emphasis on

the attribution of conscious intent to these patterns. At the same time, it

demands quantitative or geometric confirmation that is lacking in the purely

interpretive approach of New Age mysticism, such as that of Fritjof Capra's Tao

of Physics (see critiques in Restivo 1985). Claims that ancient knowledge sys-

tems reveal the structure of the atom or the equivalence of matter and energy

do more harm than good—first because they are wrong, and second because there

is no means by which such knowledge could be obtained. Such mystification dam-

ages credible research in indigenous knowledge systems, and removes the attri-

bution of intentionality and intellectual labor from the putative knowers.

Evolution is a bush and not a ladder:

the cultural location of African fractals

We are increasingly surrounded by explanations based on biological determin-

isin, and there is none more virulent than racism. Even in the supposed liberal

climate of U.S. academia, my lectures on fractals in Africa are frequently followed

by a question about neuroscience-Typically this is an innocent remark concerning

Noam Chomsky's ideas on universal cognitive structure, but even so, it is quite

telling thará lecture on European fractals invokes questions about the genius of

individuals, while African fractals are compulsively attached to biology.

The mythology of race is too complex to recount here (see note 6), but it

is useful to distinguish between two categories of racism Primitivist racism

operates by making a group of people too concrete, and thus. "closer to nature"—

not really a culture at all, but rather beings of uncontrolled emotion and direct

187

I88

Implications

bodily sensation, rooted in an edenic ecology. Orientialist racism operates by mak-

ing a group of people too abstract, and thus "arabesque" —not really a natural

human, but one who is devoid of emotion, caring only for money oran inscrutable

spiritual transcendence.

The alternative to biogenetic explanation is sociocultural, and here the ,

categories of primitive and oriental can be much more complex. Historically,

many researchers who strongly opposed both racism and ethnocentracism have

been located in institutions with titles like "Museum of Primitive Arts" or

"Department of Orientalist Studies," and it would be unwise to simply sneer at!

their work, particularly considering the antiracist contributions by black anthro-

pologists such as Zora Neale Hurston or Jomo Kenyetta. There is value to be found

in even the weakest of these oppositional theories, and problems in even the l

strongest.

In general these theories can be grouped into two strategies: sameness and

difference. Sameness can usually avoid orientalis and primitivism, since it

argues that what is important about a non- Western culture are those things held

in common with the Euro-Americans, and what is different is (in this context)

trivial. Claude Lévi-Strauss, for example, argued that the "savage mind" is based

on systems of symbolic structures, just like the European mind, so that am

African working with a system of mythological symbols is performing the same

cognitive operations as a European working with a system of computer code syn-

bols. One drawback of sameness is that we become players in a game created by

someone else: "I am worthwhile only insofar as I am the same as you." Difference

can avoid this trap, although it has more trouble avoiding primitivism and ori-

entalism. For example, Aime Céstire's neologism "negritude" began as a way of

speaking about the difference of African culture in open-ended, dynamic, cre-

ative terns, but later (in the hands of others) the comparison was frozen into a

set of binary oppositions (infuitive vs. analytic, concrete vs. abstract, etc.).5 In

other words, both sameness and difference have moments of failure as well as

moments of success.

The recent focus on ancient Egyptin certain circles of African studies has

certainly seen both moments. Motivated by considerable scholarly work (e.g., Drake

1984), it has also become attached to some disreputable and questionable claims

(see critiques in Oritz de Montellano 1993; Martel 1994; Lefkowitz 1996). It is

worth noting, however, that some of the critiques have been equally lacking in

their restraint. In his review of the Portland Baseline Essays, for example, Rowe

(1995) — while rightly pointing to a number of unsupported assertions implied

that claims for an ancient Egyptian glider should he dismissed because the

author was merely an aerodynamics technician rather than a Ph.D. Rowe was

Theoretical frameworks in cultural studies

quite right in objecting to the wild leap from empirical tests of a small wooden

carving to the authoritative claims for ancient Egyptians flying from pyramids;

but to imply that simple experiments are automatically suspect because they were

macle by, rechnician is nothing but classis, prejudice. On the other hand, the

fact that this researcher was a rechnician rather than a PhD speaks to the under-

lying cause for these problems: the lack of institutional resources and precarious

economics among many black educational communities.

Appeals to ancient Egypt can also encounter problems as a strategy of same-

ness. On the one hand, ancient Egypt's status as a state empire directly opposes

primitivist assumptions that Africa consists of nothing but tribal villages. On

the other hand, it reinforces the view that the knowledge systems of nonstate

indigenous societies are not comparable to those of state societies. This view comes

from the old idea of cultural evolution as a ladder, a unilineal progression from

"primitive" to "advanced." In the ladder model the small-scale decentralized

("band") societies would be on the bottom rung, the more hierarchical ("tribal"")

societies would be on the next rung, and the most hierarchical ("state") societies

would be on the top rung. Of course, simply positing that the societies with com-

plex social organization (e.g., labor specialization and political hierarchy) have

greater technological complexity is not inherently demeaning; but it is not

entirely accurare. Anthropological research has persistently shown that neitherg

social structures nor their knowledge systems can be consistently ranked in a

unilineal sequence; for example, monotheistic religions tend to occur in band'

and state societies more than in tribal. Just as biological evolution has been

revised from Lovejoy's "great chain of being" to Gould's "copiously branching

bush, "ó so too cultural evolution is now typically portrayed as a branching diver-

sity of forms. There is no reason to focus on state societies over nonstate soci-

eties in the pursuit of antiprimitivist portraits.

The difficulties of theoretical frameworks in the epistemology of nonstare

societies have been much more mixed. Appiah (1992) provides an extensive dis-

cussion of this intersection, starting with ethnophilosophy. His analysis weaves

between the positions of Wiredy (1979), who critiques the focus on comparison

to Western science rather than religion (noting that it leaves the superstitions

and folk philosophies of the West unexamined), and Hountondfi (1983), whol

argues against any mimetic comparison, suggesting that ethiophilosophy and its

allies are dressing European motivations in autochthonous garb. Both critiques

'could certainly be applied to African fractals. But like Mudimbe's (1988) Fou-

caultian analysis of African epistemology, and Gilroy's (1993) fractal history (which

we will examine in the following chapter), Appiah's dialectical contour maps

African epistemology as an historical process rather than an object of strictly

180

I90

Implications

pre-or post-Western presence. The cautions of Wiredu, Hountondji, and oth-

ers are serious reminders that African fractals can only succeed as participant sim-

ulation, not as Indiana Jones discovering another lost temple.

( Given those precautions, it makes sense to see African fractals as just

another moment in

a historical sequence. One could, for example, place

them in Mudimbe's history of ethnophilosophy, or Zaslavsky's (1973) history

of research on African math. But there are other researchers who have pointed

out some of the fractal characteristics of African designs and practices, and

it is useful to examine them as a group, even if they lack the clear historical

trajectory of other categories. We have already mentioned the observation of

nonlinear scaling by British art historian William Fags chapter 6), and the

interpretation of scaling designs as signifiers ofinfinitx.inthe.work.of

Cameroonian theologian, Engelbert Mveng (chapter 9). Léopold Senghor,

the distinguished man of letters who became Senegal's first president, also had

an eye for African fractals. His term was "dynamic symmetry," which he took

from art historians. But Senghor's motivation was primarily ideology; defin-

ing a "negritude" that would encompass the kind of cultural politics he saw

as necessary to independence. Most recently, Henry Louis Gates (1988)

explored the doubling practices of vodun divination in terms of a literary veri

sion of deterministic chaos; here the recursion generates a cultural uncertainty

that frees gender identity from static boundaries: "The Fon and Yoruba escape

the Western version of discursive sexism through the action of doubling the

double; the number 4 and its multiples are sacred in Yoruba metaphysics. Esu's

two sides 'disclose a hidden wholeness,' rather than closing off unity, through

the opposition, they signify the passage from one to the other as sections of

a subsumed whole."

While all four have hit ipon mathematical aspects of African fractals, none

of these authors have focused on representations of mathematical knowledge.

Mveng, the theologian, provides a theological interpretation. Fagg, the artist,

concludes with a comparison to D'Arcy Thompson's famous nature drawings.

Senghor, the statesman, sees his dynamic symmetry as a sign of cultural-and

thus national-identity. And Gates, as a literary critic, sees it as discursive tech-

nique. Surely my insistence on indigenous mathematics is no less an imposition

of seeing the world though my own lenses, but since that is no different from

the other explanations, why does ethnomathematics appear to be so much

more controversial? It is because a portrait of mathematical sophistication in

nonstate societies creates a strong conflict with the old ladder model of cultural

evolution, a model that is itself overdue for extinction.

Theoretical frameworks in cultural studies

I9I

Conclusion

So far we have outlined several theoretical frameworks that could raise prob

lems for African fractals. On the one hand, there are theories in which the

designs could be dismissed as unconscious biological or social process. On the

other hand, great care must be taken to avoid either inflated claims or a

mathematical version of negritude. With the exception of biological deter-

minism, none of the frameworks reviewed here are necessarily good or bad

There are cases in which mathematical anthropology is more appropriate

than the ethnomathematical approach, or when sameness is a better strategy

than difference, or when attention to ancient Egypt needs to supersede atten-

tion to sub-Saharan Africa-just as there are cases in which the opposite is

true. Our goal is not to find the one true final framework—-it does not exist--

but to keep a well-stocked toolbox and know how to pick the right tool for

the right job. Now that we are well prepared for constructive tasks, it is time

to move to politics.

CHAPTER

The

politics of-

African-

fractals-

- Given the possible dangers in misinterpreting African fractals, how can we put

them to good use? Social theorists from many different disciplines have used two

mathematical concepts we have discussed, recursion and the analog-digital

dichotomy, in constructing their ideologies. Many theorics of communication have

assumed that there is some kind of universal ethical or social difference between

using analog signals and using digital symbols. Other theories have maintained

that recursion has some kind of universal ethical or social value. Both are ulti-

mately failures in the sense that ethics and values do not lie within mathe-'

matical distinctions. Yet they are also on the right track in that such associations

can be locally formed-it is just that different locations will result in different social

meanings. Such locally specified social attachments to fractals can be useful for

understanding cultural politics in Africa and beyond.

The politics of the analog-digital distinction

Jean Jacques Rousseau is often credited as a founder of "organic romanticism,"

the theory that the Natural is inherently hetter than the Artificial. Whether or

not this is deserved, Jacques Derrida (1974) takes him to task for proposing that

a natural/artificial difference can be found between different languages. Jean Jacques

I92

The politics of African fractals

Rousseau proposed that the "natural crys of animals," music, and "accentuation"

(that is, pitch intonation in the human voice) are all a similar type of com-

munication. In this I would tend to agree, since, instrumental music and human

pitch intonation are for the most part apalog representations and since he was

probably thinking of analog examples of animal communication (although

many animals, for example vervet monkeys, use digital communication as well).

Rousseau contrasted this to "articulation" in the human voice, by which he

meant the linguistic (and hence digital) parts of speech. But instead of seeing

the distinction as two different types of representation, one analog and the other

digital, Rousseau claimed that analog signals were not a form of representation

at all. In his view, digital versus analog was representation versus The Real. Music,

animal cries, and emotional intonation were somehow more natural and

authentic. Worse yet, he inflated this into a cultural difference, maintaining

that while European languages were largely based on (digital) articulation, the

language of the nobel savage was closer to nature.

One might hope that Derrida would correct the matter and point out that

analog signals are just as much a representation- just as much fakes, just as easy

to lie or tell truth with, and just as artificial-as digital symbols are. But he 100

failed to produce a balanced portrait. Derrida did insist that all human linguis-

tics is fundamentally digital (quite true), but he did not bother to say a word about

other modes of vocal representation. This error is due to Derrida's concern over

the authoritarian ideology that organic romanticism can produce. For example,

history is full of dictators who claimed that their ethnic group was the real or

natural one, and that others were artificial pollutants in their Eden. Rousseau

himself did not have such fascistic tendencies, but Derrida is right in pointing

out that organic romanticism-can alwäys be used in that-way, no matter who it

is coming from.' One need not panic so much, however, and banish analog sig-

nals from existence; it is enough to give them the same epistemological status

as digital symbols—no more and no less.

I have found this egalitarian view of the analog/digital distinction very dif-

ficult to promote; it seems that everyone has their own favorite view. When I spoke

to chaos theorist Ralph Abraham, for example, he explained that analog systems

were in his view the realm of spirit, the vibrations of Atman. Postmodern theory

maven James Clifford, to the contrary, insisted that only digital representation

is capable of the flexible rearrangements that constitute human thought. This

same battle has been played out in the history of African cultural studies. Dur-

ing the 196os, realism was in vogue, and what could have been a wonderful explo-

ration of the analog representation techniques in African culture was often

reduced to romantic portraits of the "real" and "natural," while African symbol

193

194

Implications

systems suffered from neglect. During the late 197os, this began to reverse

itself--with the advent of postmodernism, African cultural portraits became

increasingly focused on discourse and symbol systems, even at the expense of ignor-

ing analog representations.

It is important, however, to see how these restrictions have been contested,

particularly in black intellectual communities. Hooks (1991, 29) summarizes her

own reaction to romantic organicism: "This discourse created the idea of 'prim-

itive' and promoted the notion of an 'authentic' experience, seeing as 'natural'

those expressions of black life which conformed to a pre-existing pattern or stereo-

type." Rose (1993) describes the history of rap music, also arising in the mid-197os,

as not just a resistance to organic romanticism, but as a technocultural rebellion

that makes Derrida look like Gutenberg. Cornel West, Houston Baker, Hortense

Spillers, and Hazel Carby have made interventions in African American intel-

lectual discourse in similar ways, as have works of black science fiction such as

George Schyler's Black No More, Ralph Ellison's Invisible Man, Toni Cade Bam-

bara's The Salt Eaters, Samuel R. Delany's Dhalgren, and Octavia Butler's Xeno-

genesis trilogy. An egalitarian view of the natural/artificial dichotomy can be seen

in black intellectual history running from George Washington Carver's concept

of "God's Kingdom of the Synthetic" to Mudimbe's "Invention of Africa."3

Indeed, Carver and Mudimbe's concepts are quite similar; it is not Mudimbe's con-

tention that African unity lacks a spiritual bond, but rather a celebration of the

spirit of invention, which requires resistance to the European claim that spirit

can exist only in categories of the natural. African animism is marked by an extra-

ordinary acceptance of the religious significance of artifice,' from gris-gris to the

mojo hand, and its techniques for passing information through the physical

dynamics of sound and movement show that this faith in the power of analog rep-

resentation is not misplaced

The politics of recursion

While Derrida was trashing organic romanticism, Michel Foucault was attempt-

ing to do the same for humanism. His historical studies demonstrate that human-

ist goals of recursion-—to be self-governed, self-controlling individuals-are not

innocent; but rather develop historically in combination with various tech-

niques of social control. In an era where "self-management" usually means that

the corporation you work for has developed improved techniques for self.

exploitation, it is not hard to see what Foucault is getting at. As in the case of

Derrida's warnings against claims that analog representation will automatically

lead to more ethical living, Foucault warns against seeing recursion as a moral'

The politics of African fractals

formula. While African analog systems raise the problem of someone making

claims about what is more real or more natural, African recursion-especially the

recursive architecture of African settlements-raises the problem of humanist

claims. •

To see how this can be a problem, consider the following two case studies

of African architecture. Caplan (1981) studied the relation between housing and

women's autonomy in Tanzania. She described how the flexibility of housing

allowed women to create new homes if they wanted a divorce, or to extend old

homes if they wanted to shift the family structure. As in many African settle-

ments, this self-organized housing created a self-similar structure-fractals-which

allowed greater social self-control for women. When socialism brought mod-

ernization programs, this autonomy was threatened by the "improved" housing

design, which sometimes resembled concrete army barracks. Here one would con-

clude that fractal is better.

Stoller (1984) described a Songhai town in which a caste system ensured

• that the best land was voluntarily given over to the highest caste members. It

was not a matter of forcing people against their will, but simply unquestioned com-

mon sense that one should want to be located in their proper place. This frac-

tal, self-organized architecture was a form of self-exploitation. Eventually several

members of the community decided to break out of this oppressive structure by

building houses along the new highway. Thus liberation in this case meant

leaving the fractal geometry, and lining up in straight Euclidean formation— exactly

the opposite of the Tanzanian village studied by Caplan. Scoller's work ificely illus-

trates Michel Foucault's warning against simplistic humanist formulas: self-

determination is not necessarily liberating; it can serve to support social control

rather than resisi ic. Neither fractal nor Euclidean geometries have any inher-

ent ethical content; such meanings arise from the people who use them.

195

Colonialism and architectural fractals

René Descartes was not much of a humanist; in his view self-organized architecture

is junk. He makes this clear in his famous Discourse on Methodology:

(There is less perfection in works made of several pieces and in works made

by the hands of several masters than in those works on which but one master

has worked. Thus one sees that buildings undertaken and completed by a

single architect are commonly more beautiful and better ordered than those

that several architects have tried to patch up. ... Thus I imagined that people

who, having once been half savages and having been civilized only gradually,

have made their laws only to the extent that the inconvenience caused by crimes

196

Implications

and quarrels forced them to do so, would not be as well ordered as those who,

from the very beginning of their coming together, have followed the fundamental

precepts of some prudent legislator.

(1673, 12)

For Descartes, "self-organized" is synonymous with savages, the imperfec:

tion of both material and social structure. Lack of complete Euclidean regular-

ity means randomness: for "streets crooked and uneven, one will say that it is

chance more than the will of some men using their reason that has arranged them

thus" (p. 12). The lack of Cartesian coordinates in many African settlements would

thus evidence their need for the guidance of colonial reason. As Hull (1976) notes,

huge centers of urban life in Africa were indeed disregarded by Europeans as

"unstructured bush communities" on just these principles. While Timbuktu was

granted cityhood due to its grid pattern of streets, the Yoruba cities of equal pop-

ulation size and economic, technical, and labor specialization have been disre-

garded as merely giant villages due to their lack of Cartesian regularity.? Thus

fractal architecture was used as colonial proof of primitivism. This debate over

the urban status of non-Euclidean settlements continues in the postcolonial era

(see Schwab 1965; Lloyd 1973).

The occasional Cartesian linearity in African architecture threw a hitch

into this colonial justification. In 1871 the German geologist Carl Mauch "dis-

covered" the ruins of Great Zimbabwe. Stunned by the evidence of precise stone

cutting on a massive scale, he proposed that the buildings were not of African

design, but were instead due to the Queen of Sheba's visit to Solomon. The .

Rhodesian government used this explanation as a part of its propaganda against

Black rule (Macintosh and Macintosh, 1989). Actually, they had much less to

fear in the truth: the stone was not cut, but it naturally broke into linear

sheets (after heating) due to its geologic properties. Moreover, most of the out-

side walls were originally covered with smooth clay, creating a nonlinear set

of scaling shapes (which Connah [1987) refers to as "random curved forms").

This is not to diminish the remarkable technological skill of the construction,

but to point out that one culture's sign for "artificial" can he another's sign for

"natural." Euclidean versus fractal does not necessarily mean artificial versus nat-

ural; that, too, is culturally influenced.

During the development of colonial cities, the chaos of African architec-

ture was used as both symbol and symptom of European fears over social chaos.

Pennant (1983) provides an example of this concern about proper settlement geom-

etry in his examination of colonial development in Malawi: "The language of this

1930s policy discourse is significant. Medical experts wrote of 'investigations' show-

ing 'unquestionably' and of 'abundant proof.' ... Lay Europeans showed 'con-

cern,' 'alarm,'

and 'hortor.' Africans, with their 'primitive habits,' of "promiscuous

The politics of African fractals

defecation' formed a 'floating' or 'scattered' population in need of 'control' and

"supervision' in a 'properly laid-out village or location.'"

In the above case where "primitive" mixeswith modern, the fractal tradi-

tion was a threat. But kept in what colonialists thought of as its natural role, it

could make fractal settlements appear to benefit the colonial enterprise. The nov-

elist Karen Blixen (Isak Dinesen), in Out of Africa (1988), described her attempts

to lay out grids for African workers' houses on her ranch. They refused to follow

these linear instructions and fit their houses in patterns matching the irregular

configuration of the land. That such ecological fit could be quite efficient was not,

however, lost on the colonists. "The squatters' land was more intensely alive than

the rest of the farm, and was changing with the seasons" (p. 9). Architectural

fractals could be part of colonial romanticism as long as they ensured a supply.

of self-supporting workers.

Even in the case of social control, indigenous fractals could be utilized. British

coloniai policy, for example, at first failed in cases where there was a decentral-

ized network rather than a large hierarchy. This was approached in the case of

the Ibo with a system of "indirect rule" based on "warrent chiefs" (Isichei 1976).

The Ibo autonomy of self-organization was turned against them; in a sense it was

grass-roots colonialism. The architectural equivalent of this system can be seen,

in a manual for colonial-era housing designs from the Agency for International

Development (Hinchcliff 1946, 31). Here the Ibos' fractal settlement pattern (radial

houses around a center in each village, radial villages around the settlement cen-

ter) is tidied up to suit European conceptions of symmetry while retaining the

overall indigenous fractal structure.

I97

Fractals and racial redistricting

In the introduction to his seminal Fractal Geometry of Nature, Benoit Mandelbrot

examines some of the disparaging comments that were made about the early

fractal forms of Georg Cantor, Helge von Koch, and others. Rejected as "bizarre"

and "torturous," these "dragons" were consigned to the oddities section at the

end of the few math texts that would even consider them. Strikingly similar

language has been used to reject the outlines of voting districts that were

altered to include larger African American populations, and these do indeed

appear to be fractals (fig. 12.1).& Were the courts as mistakenly hasty to disre-

gard fractals as mathematicians were?

• The Euclidean shape of voting districts is not an arbitrary sampling—-this

could only be done by randomly selecting voters from everywhere in the state.

According to the 1993 Supreme Court ruling in Shaw v. Reno, it is meant to

198

Implications

BARROW

WALTON

NEWTOM

MONRCE

JONES

TALBOT

EMANAJEL

TAYLOR

EACIF

LAURENS

BIX.LOCK

TREUTLEN

MACON

CANDLER,

EFFINOH

tone

WHEELER

TOOMBS

BRYAN

FICURE

12.1-

Georgia congressional district xI in 1992

Similarity of irregular redistricting pattern can be seen at multiple scales.

(Original maps courtesy of che Carl Vinson Institute, graphic highlighes ly the author.)

designate a geographic locale in which "shared interests" inform the vote. The

objection to creating a district in which contours are "predominantly motivatel"

by race is that it creates a bias in the sampling of the geographic location. This

would certainly be the case if we were to take a random sampling, separated vor-

ers by ethnicity, and then designated those ethnic groups as the voting districts.

However, if some ethnic groups are distributed in Euclidean settlement patterns,

and others in fractal settlement patterns, then why consider Euclidean district

shapes to be unbiased, and fractal district shapes to be biased? I do not know if

African American housing distributions are more fractal than orhers-and

even if they are, I would not necessarily assume a cultural connection to African

The politics of African fractals

fractals -but the fact that we now know of societies in which fractal settlement

parterns are beautiful fusions of form and function suggests that we might

reconsider their potential role in American politics.

African fractals from cultural visionaries

Fractals and chaos theory have been increasingly mentioned in the humanities

as either a tool or an object of cultural analysis, but too often the approach of

these studies has left the impression of mathematical ink blots allowing writers

to see whatever they please. Lyotard (1984) saw fractal geometry as contribut-

ing to a "postmodern condition" whose contradictory nature would disrupt

auchoritarian certainty; a more cautious version of this thesis is floated in Deleuze

and Guattari (1987). At least two authors (Steenburg 1991; Argyros 1991)

have argued that fractals and other branches of chaos theory have created a direct

challenge to postmodernism, integrating the disruptions it created. Porush

(1991) and others insist that "deterministic chaos" is attempting to substitute

a feeling of free will for fatality? Sobchack (1990) suggests that it implies "an

embrace of irresponsibility in a world already beyond control." When Sobchack

cites Peitgen and Freeman in her condemnation of chaos theory as a denial of

"the specificity of human embodiment and historical situation," I can't help but

think of Peitgen's fractal geometry course at the University of California at Santa

Cruz, where he commented on German mathematicians who altered their

careers to oppose Nazi anti-Semitism or support peace efforts; or of Freeman's

(1981) use of Martin Luther King in his discussion of chaos in neurophysiology.

How can we critique the work of chaos theorists as lacking historical specificity

and embodiment if we ignore their own histories and bodies?

Hayles's Chaos Bound (1990) took a more subtle approach. Like Porush and

others, Hayles's literary method allows her to glide far too easily between un-

related ideas; by the time she has tossed together quantum theory, entropy, and

Gödel's theorem with deconstruction and "holism," one can only conclude that

any complicated idea can be a metaphor for any other complicated idea. But her

derailed analysis of literary works, showing deep parallels berween self-reflexive

writing and self-referential mathematics, suggests that when grounded in specific

locations the fusion of fractal geometry and cultural interpretation can be pro-

foundly rewarding.

Paul Gilroy makes explicit use of fractals in his portrait of the diversity and

lynamicism with which both traditional Africa and the African diaspora have

organized their cross-cultural flows. The recursive construction of his Black

Atlantic can be seen, for example, in this quote from James Brown on a visit to

199

200

Implications

hear Fela Kuti in Nigeria: "[Bly this time he was developing Afro-beat out of

African music and funk. His band had a strong rhythm; I think Clyde picked up

on it in his drumming, and Bootsy dug it too. Some of the ideas my band was get-

ting from that band had come from me in the first place, but that was okay with

me. It made the music that much stronger" (1993, 199).

Gilroy cites the impact of the Virginia Jubilee Singers on tour in South

Africa in 1890, the return of slaves from Brazil to Nigeria, the Rastafari culture:

in Zimbabwe, and other examples of "mutations produced during its contingent

loops and fractal trajectories." Perhaps his most radical move is a claim for dias-

poric mixing with jewish culture-- W.E.B. Du Bois passing for a jew to main-

tain safety in Eastern Europe, the use of the Exodus theme in Martin Luther King,

Jr., and Marcus Garvey, and E. W. Blyden's childhood in a Jewish commu-

nity. !O The fractal imagery works in many different ways for Gilroy-from the

turbulent metaphor of hybridity to the concrete description of ships' paths

and travelers' routes (or "roots/routes" as he puts it). 11 While music is, with-

out doubt, Gilroy's strongest example, he does slip into the problematic labels

of representational versus "nonrepresentational" rather than digital versus

analog, 12 but he makes it clear that the music reverberating across his Black

Atlantic is neither pure nor natural.

While Gilroy is primarily focused on fractals as spatial representations of

blurred boundaries, he also briefly mentions their potential for "a striking

image of the scope of agency within restricted conditions" (1993, 237028); that

is, the ability for geometric expansion within bounded space becomes an anal-

ogy for oppositional political expansion in human bondage. The metaphor is car-

ried to a more exacting relationship in Gary Van Wyk's study of Sotho-Tswana -

murals under the apartheid system of South Africa. Van Wyk (1993) found that

the litema, or the house painting patterns of the Socho-Tswana women, utilize

alternations of irregularity and regularity at several scales, sometimes resulting

in a resemblance to fractal patterns. Noting that the scaling is associated with

the geometric structure of flowers, and flowers with the regenerative power of

women (both spiritually and in social struggles), Van Wyk's ethnography con-

cludes that the murals expressed political opposition to apartheid by providing

a visual analog in which "a woman can be secretive while at the same time hid-

ing nothing" (Deleuze and Guattari 1987, 289-290).

Although the word "fractal"

is nowhere mentioned in his text, Anton

Shammas's novel Arabesques (1988) is an exemplar of nonlinear insight and recur-

sive cultural commentary. Heaver (1987) analyzes the novel through the North

African artistic form of the "arabesque," and shows how Shammas has used this

fractal to sustain the cyclic time and multiple identities required to articulate a

118

RAMLES

MOSQUE

PASKA

Mosque

GOGUE

i25

128

3.8

MasQue

130

TOMOR

"MOUSH

НафЕ

133

132

Pos,

OFFICE

MaSqUE

134

ASQUE

135

136

(CH. SIKKET

137

138

FIGURE 12.2

Religious institutions in the map of Cairo, 1898

139

202

Implications

political flexibility crucial to the survival of cultural diversity. "As an 'Israeli Arab,'

Shammas is a member of a minority group-but as a Christian, he falls outside

the Islamic mainstream of the minority.... On the other hand, he writes in

Hebrew, the language of the dominantly Jewish culture, which is itself a minor-

ity within the predominantly Arab Middle East" (p. 49).

Such recursive nesting is emphasized throughout the temporal flow,.

narrative structure, and conceptual dynamics of the novel. Heaver suggests that

the "nonmimetic geometrical abstractions of the arabesque" are a spatial model

for Shammas. He notes that in part these cyclic reentries act to negate one

another; undermining, for example, the fruitless argument of "I was here first."

But negation is not the only meaning behind the arabesque, as Heaver points

out in a passage that ties Islamic social structure to analog representation,

recursion, and the scaling properties of fractals.

The arabesque does not serve only a negative, critical function; it also bears a

positive, utopian message. It acts as an analogue, in the area of visual arts, to

the position of Islamic "contractualism" in the social sphere.... In contrast

to western corporativism, with its preference for hierarchical structures in

which a limited number of conclusions are drawn from a limited number of

premises (on the model of geometry), the cyclical rhythms of the arabesque could

well be said to characterize an "indefinitely expandable" structure. The

arabesque provides a framework within which it becomes possible to reduce the

apparently "chaotic variety of life's reality" to manageable proportions, yet with-

out "arbitrarily setting bounds to it."

(Heaver 1987, 61)

Clearly, when Heaver refers to the limiting dangers of a "model of geom-

etry" he is thinking of Euclidean structures; it is the fractal geometry.of.the

arabesque which conveys the hopeful message of Shammas. In chapter 2 we

examined the arabesque branches of streets that appear in a map of Cairo, Egypt.

In another section of this map (fig. 12.2), a wide diversity of religious insti-

tutions flower at the ends of these branches, attesting to the positive poten-

tial of fractals in cultural politics.

CHAPTER

Fractals-

-in the

-European

-history of

mathematics

- Anthropologists have recently taken an increasing interest in the cultural analy-

sis of Euro-American societies. In part this is a reaction to the many decades of

focus on indigenous societies, as if their behavior required explanation while that

of Europeans was self-evident. At frst this "reflexive ethnography" sounded like

an ingenious way to turn tables on some very troubling aspects of anthropolog-

ical authority, but it too has drawbacks. Occasionally one suspects a hidden sigh

of relief from anthropologises who decide they can place themselves on the cut-

ring edge by "studying their own tribe" (just as cyberethnography sometimes seems

suspiciously convenient). Nevertheless, there is an important place for anthro-

pological studies of Euro-Americans. It would be an unbalanced portrait if we were

to see African fractals in need of cultural analysis, and Western fractals as merely

self-evident mathematics.

A cultural history of European fractals

Ancient Greek philosophy is often remembered for Plato's rational realm of

unchanging, static forms. But in the history of mathematics, it is important to

consider other intellectual currents in that society, in particular the paradoxes

of the philosopher Zeno of Elea and the discovery of irrational numbers by the

Pythagoreans.

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204

Implications

According to ancient historians, Pythagoras of Samos gathered knowledge

in Egypt and Babylon in the sixth century B.c.e. and established a secret soci-

ety in Magna Graecia (what is now southeastern Italy). His disciples. includ-

ing one of the first recorded women mathematicians, Theana, swore an oath

to maintain strict dietary regulations, secrecy, and a religious faith in numbers.

The Pythagorean cosmology was a harmonious unity based on whole numbers

(1, 2, 3 ...) and their ratios (fractions such as 2/3, 5/2, etc.). From the motion

of heavenly bodies to the laws of music, they found increasing evidence for their

arithmetic religion. But at some point-and much ink has been spilled in the

date debate-came the discovery of what they termed alogos, the "irrational"

numbers (a name that we have kept to this day). Unlike whole number ratios,

which either terminate (5/2 = 2.50000 ...) or repeat (13/11 = 1.181818...),

irrational numbers, such as the square root of two (1.41421356...), continue

to change forever. They cannot be expressed as the ratio of two finite integers;

as geometric magnitudes they are "incommensurable lines." The most plausible

origin for the Pythagorean knowledge of irrationals is in an attempt to deter-

mine the diagonal of a pentagon. If you wish to determine the ratio of diago-

nal to sides for a regular hexagon, it is quite easy, because all diagonals intersect

in the center. But the diagonals of a pentagon just form a smaller pentagon. Since

the same operation can be repeated again and again, an irrational number is

exposed.' This "irrationality" in the heart of their spiritual practice was too much,

and members of the group agreed not to reveal this secret on pain of death.

Zeno of Elea (fl. ca. 450 в.c.E.), a disciple of Parmenides, provided a

series of paradoxes that also conflicted with the numerical faith of the day. His

most famous example is a race between Achilles, the fleetest of runners, and a

tortoise. Allowing the tortoise a sporting chance, Achilles gives it a consider-

able lead (let's say 100 feet). But by the time he caught up to the place where

the tortoise began, it had already advanced 10 feet. By the time he gained that

distance, the tortoise has crept forward one foot. Zeno concluded that although

experience proves otherwise, logically the tortoise should win the race. Back in

450 B.C.E., these paradoxes of infinity (and infinity's flip side, the infinitesimal)

were unnerving, even shocking to philosophers who depended on rationality as

the gateway to religious perfection.

In Plato's philosophic cosmology, spiritual perfection was seen as the higher

level of transcendent stasis, and illusion and ignorance were the result of life in

our lower realm of changing dynamics ("flux," which in ancient Greek also

means "diarrhea"). Several of Plato's students attempted to improve the match

between the characteristics of mathematics and the requirements of the static

realm. Eudoxus proposed to eliminate irrationals by redefining "ratio," and

Fractals in the European history of mathematics

Xenocrates introduced a doctrine of indivisibles to oppose Zeno's paradoxes. Aris-

totle, noting that infnity + infnity = infinity, suggested that this "self-annihi-

lating" characteristic could be eliminated by restricting reference to infinity as

a limit to be approached, rather than as a thing itself, a proper object of mathe-

matical inquiry?

The Platonic reform was quite successful, and as a result inathematicians

in the following centuries paid littie attention to the kinds of recursion that led

to Zeno's troubling infinite regress. One early exception was that of Leonardo

Fibonacci in the twelfth century. He introduced the first recursive series shown

to be of use in modeling the natural world. In chapter 7 we saw that the Fibonacci

series appears to have been utilized in the temple architecture and weight bal-

ances of ancient Egypt. There may actually be a connection between the two.

While little biographical material is available, Gies and Gies (1969) and other

sources have put together a good account of what life was probably like for

young Leonardo of Pisa. Following schooling in Pisa, in which arithmetic was

largely based on the Latin writings of Boethius (circa 500 c.E.), Leonardo's

father sent for him from the North African city of Bugia (Bougie). There he learned

the Indian place-value notation (probably through Arabic sources). He was

inspired by this innovation and traveled along the Mediterranean to Constan-

rinople, Egypt, Syria, Sicily, and Provence, collecting mathematical knowledge

from both scholars and ordinary merchants.

The resulting text, Liber Abaci (Book of the Abacus), has a strong Islamic

influence. Levey (1966), for example, shows that many of abu Kamil's sixty-nine

problems can be found in Leonardo's text. But the Fibonacci series, introduced

unobtrusively as the solution to a problem in rabbit population growth, does not

have a known Islamic counterpart. Perhaps it is simply an independent inven-

tion, but if the weight balance system was in use at that time, Leonardo could

have easily picked it up from a merchant during his travels in Egypt. And it is

possible that through its religious use in ancient Egypt the series had retained

some significance as an item of sacred or mystical knowledge and was thus trans-

mitted through scholarly contact.

Gies and Gies (1960, 61) nore that Leonardo's practice of reducing all frac-

tions to 1 in the numerator "went back to ancient Egypt, and perhaps derived

from the fact that fractions were regarded less as numbers in their own right than

as signs of division." Boyer (1068, 281) suggests that the Liber Abaci problem with

recursive nesting of sevens ("Seven old women went to Rome, each woman had

seven mules; each mule carried seven sacks..") originated in its ancient Egypt-

ian counterpart (Rhind Mathematical Papyrus problem #79). And Fibonacci does

provide a narrative statement of the recursive construction, highlighting the

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Implications

same self-generating aspect of the series that would be emphasized by the ancient

Egyptian belief system.

If this influence (whether merely contextual or direct) does in fact exist,

it should not detract from the genius of Leonardo's work. His"general solution

for finding "congruent numbers" for squares has been hailed as "the finest piece

of reasoning in number theory of which we have any record before the time of

Fermat." But when it comes to the use of the Fibonacci series in the contem-

porary history of mathematics (cf. Brooke 1964), there is actually no evidence

of a direct contribution from Fibonacci himself. By all accounts, German

astronomer Johannes Kepler rediscovered the series independently in 1611, and

it was only in the mid-r8oos, with the formal publication of Liber Abaci, that'

French mathematician Edouard Lucas found the Pisan historical predecessor and

named it accordingly. This fact has received little attention, and most texts pres-

ent Fibonacci's discovery as if it were in a direct intellectual line of descent

rather than an honorary title given to a well-deserving but disconnected ante-

cedent. Fibonacci himself seemed unhesitant about the multicultural contri-

butions to his work; the first sentence of Liber Abaci states, "The nine Indian

figures are ...." No doubt he would have been quite content attributing the

series to originators of any heritage.

Fibonacci's series was simply unbounded growth; there was no introduc

tion of the infinite except in ways that Aristotle would have approved. The sev-

enteenth century brought attention to the concept of the "infinitesimal"

(revived from its Greek banishment in Kepler's Stereometria (r615)), and con-

vergence to a limit as infinity is approached (e.g., the algorithms for generat-

ing pi); bui infinity would still exist only as a never-reached orientation rather

than a legitimate object of study. The Aristotelian voice could still be heard in

1831, when mathematician Carl Friedrich Gauss (1277-1855) cautioned his friend

Schumacher against infinity: "I must protest most vehemently against your

use of the infinite as something consummated, as this is never permitted in mathe-

matics. The infinite is but a façon de parler, meaning a limit to which certain

ratios may approach as closely as desired when others are permitted to increase

indefinitely." But Gauss's distinction was short-lived. As we saw in chapter 1,

the work of Georg Cantor, which had produced the first fractal, the Cantor set,

ended the Aristotelian view on infinity. Like Fibonacci, Cantor too may have

had some non-European influence in his work.

The Cantor set (fig. 13.1a) was his visualization of transfinite number

theory. It shows the interval of zero to one on the real number line, and indi-

cates that the number of points is not denumerable--that is, greater than infin-

ity. But at the time, pure mathematics was only one of Cantor's concerns. His

Fractals in the European history of mathematics

real fascination was in the theological implications; the increasing classes of infin-

icy he discovered seemed to point toward a religious transcendental. Cantor's biog-

raphers differ greatly on the cultural signifcance of this point. E. T. Bell felt that

Cancor's Jewish ethnic origin ruled his life, and he made several remarks about

the inheritance of personality traits-particularly disturbing in light of his

207

| |l

|I H

FIGURE 13-1

The Cantor set

(a) The frst fractal, created by Georg Cantor in 1877. (b) This design is found on the top of

culumos in the temples of ancient Egypt. Georg Cantor's Rosicrucian beliefs and his cousin Mortiz

Cantor, an expert on the geometry of Egyptian art, may have put him in contact with this Egyptian

design.

(bi from Fourier 1821 .)

208

Implications

remarks on Cantor's arch rival, the Jewish mathematician Leopold Kronecker:

"There is no more vicious academic hatred than that of one Jew for another when

they disagree on purely scientific matters. When two intellectual Jews fall out they

disagree all over, throw reserve to the dogs, and do everything in their power to

cut one another's throat or stab one another in the back" (Bell 1939, 562-563).

Another Cantor biographer, J. W. Dauben, says that since Cantor's mother

was Roman-Catholic "she was by definition non-Jewish, thus it follows that Georg

Cantor was not Jewish, contrary to the view which has prevailed in print for many

years" (Dauben 1979). Nazi scholars solved their worries by spreading a story

that Cantor was found abandoned on a ship bound for St. Petersberg (Grattan-

Guiness 1971, 352).

Actually Cantor's Jewish identity was quite complex. His family had indeed

converted to Christianity, but he was well aware of his heritage. He referred to

his grandmother as "the Israelite" and wrote a religious tract that attempted to

show that there was no virgin birth, and that the real father of Jesus Christ was

Joseph of Arimathea. Cantor eventually joined the Rosicrucians, whose mysti-

cal/scientific approach to a supposed Egyptian origin for all religions probably

appealed not only to his intellectual interests, but also to his syncretic ethnic-

ity. Cantor chose a Hebrew letter as his new symbol: the aleph, beginning of the

alphabet, was used to represent the beginning of the nondenumerable sets.

While his biographers argued Jew or not-Jew, off or on, zero or one, Cantor him-

self proved that the concinuum from zero to one cannot be delimited by any sub-

division process, no matter how long its arguments.

Given Cantor's Rosicrucian theology and the proximity of his cousin

Moritz Cantor—at that time a leading expert in the geometry of Egyptian art

(Cantor 1880)—it may be that Georg Cantor saw the ancient Egyptian repre-

sentation of the lotus creation myth (fig. 13.1b), and derived inspiration from

this African fractal for the Cantor set. We may never know for certain, but the

geometric resemblance is quite strong.

As noted in chapter 1, Cantor's mathematics was considered utterly imprac-

tical; it was not until Benoit Mandelbrot that fractal geometry became useful to

science. Mandelbrot reports that his inspiration came from a study of long-term

river fluctuations by British civil servant H. E. Hurst. Hurst examined the flood

variations over several centuries and concluded that it could be characterized in

terms of a scaling exponent. Later, Mandelbrot realized that this was the same

scaling mathematics that the "remarkable curves" of Cantor and others described.

But where did Hurst find a reliable source for several centuries of flood data? Hurst

lived in Egypt for 62 years and was able to demonstrate long-term scaling in Nile

flood records because of the accurate "nilometer" readings going back fifteen cen-

Fractals in the European history of mathematics

turies. Artempts to find patterns in the floods are quite ancient in the Nile val-

ley; in some ways, Hurst and Mandelbrot were simply the latest-and most suc-

cessful--participants in that search.

209

*>=

Recursion and sex: a cross-cultural comparison

Throughout the exploration of African fractals, we failed to find any one cultural

feature that was persistently associated with these forms. They ranged from

practical construction techniques to abstract theological icons, from wind-

screens to kinship structures, from esthetic patterns to divination rechniques. There

is no singular "reason" why Africans use fractals, any more than a singular rea-

son why Americans like rock music. Such enormous cultural practices just cover

too much social terrain. At best we can make a lower-dimensional projection of

such high-dimensional dynamics, the silhouerte that appears given one partic-

ular axis of illumination. This section will focus on the relation between recur-

sion in mathematics and sexuality in culture. Sex is convenient in that other

researchers have developed African-European comparisons, and that sexual

reproduction is often connected to recursive concepts.

Taylor (1990) describes sexuality in Rwanda as based on the concept of a

"fractal person" in which society is perceived "not in terms of monadic individ-

uals but in terms of ... structures of meaning whose patterns repeat themselves

in slightly varying forms like the contours of a fractal topography" (p. 1025). His

analysis on expressions of this sociality in terms of the circulation of fluids is used

to examine the failure of programs to encourage condom use. Carolyn Martin Shaw

(1980, 1905) analyzes Kikuyu sexuality in related ways and provides an illumi-

nating contrast to European sexuality. Using Foucault's critique of humanism, Shaw.

challenges the usual portrait of European sexual repression and African sexual

license. She demonstrates that in both cases, the social system controls sexual

behavior, but while the European locus of control is in the privatization of plea-

sure, the Kikuyus's sexual regulation occurs through public expressions of plea-

sure and "sociality of individual conscience." For example, she highlights the

practice of ngweko, in which teenagers wrap themselves with a few leather strips,

oil their bodies, and engage in a public display of sexual behavior. From a Euro-

pean point of view this sounds like an unregulated orgy, but Shaw found that the

practice was a method of preventing teenage pregnancies and channeling the teens'

sexual desire into socially approved forms.4

When we look at many African fractals we can see an emphasis on sexu-

ality in terms of reproduction. The self-similarity of the Bamana chi wara ante-

lope headdress and merunkun fertility puppet, the self-generating Dogon

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Implications

cosmology, the cyclic kinship iconography of the Mitsogho, Fang, and Baluba,

the iterations of birthing in the Nankani architecture, and many other cases of

recursion are closely ried to sexual reproduction. Thus one contributing factor

to the African mathematical emphasis on recursion could be this African con-

struction of sexuality through positive public domain expressions.

The European counterpart of Shaw's theory would predict the opposite, and

indeed we find that the banishing of infinite regress in the Platonic reform was

closely tied to a kind of sexual prohibition. In Plato's Symposium, Socrates

explains that there is a hierarchy of reproduction. Love between a man and a

woman will only result in a flesh child, a creature of flux who will eventually die,

at best producing more flux. Love between a man and a boy (lover and beloved)

is higher, because it can result in raising the boy to a higher plane--that of a

philosopher. And a philosopher can have a "brain child," a perfect idea that never

changes or dies. The Platonic ideal of static, eternal perfection conflicts with the

ever-changing dynamic of sexual reproduction. The Greek preference for the sta-

tic shape of the Archimedean spiral suggests this Platonic ideal, just as the

growing shape of the logarithmic spiral suggests the African emphasis on fertil-

ity and reproduction. Of course, this is a gross generalization; there are, for

example, plenty of Archimedean spirals in African designs. Conversely, European

mathematician Jacobo Bernoulli was utterly dedicated to the logarithmic spiral

and specified that one would be engraved on his tombstone. But the stone cut-

ter did not go against the grain of his culture; Bernoulli's grave is still marked with

an Archmedean spiral (fig. 13.2).

It would be dangerous to suggest that there is an ethical difference at

stake here, as so many organic romanticists have maintained. Again, there is no

historical evidence for a consistent relationship between mathematical distinc-

PHUY DESIDERATISS

HEM. P

FIGURE 13.2

Bernoulli's tombstone

Although Bernoulli asked for a logarithmic spiral to be

inscribed on his tombstone, the engraver was apparently

only familiar with the linear spiral.

(From Maor 1987, courtesy Birkhäuser Verlag AG, Basel,

Switzerland.)

itly

Fractals in the European history of mathematics

tions and the ethics of their users. Some strictly linear, logical thinkers like Bertrand

Russell and Noam Chomsky have been famous for their progressive ethical

standpoints, just as some holistic organicists have been prone to fascism. And

of course vice versa. What does count for ethics is how people are able to use mathe-

matics in the particular events and ideas that surrounded their life. With that

in mind, let's look at three of the innovators who brought recursion into Euro-

pean computational mathematics.

The story of Ada Lovelace is well known in computing science history. Her

fame stems from her writings in x843 on the mathematical possibilities of

Charles Babbage's proposed "analytical engine"— a plan for a mechanical digi-

tal computer. Lovelace is often promoted as a recovered feminist ancestor, a posi-

tion that tends to overestimate her achievements and obscure her own thinking.

Against these reductive portraits, Stein (1985) has written a detailed, critical exam-

ination of Lovelace that reveals a much more interesting and complex story than

the popularizations have allowed

Lovelace's mother was always worried that she might have inherited the

notorious sexual proclivities of her father, Lord Byron. Her childhood revolved

around strictly prescribed educational activities, and at times she was forced to

lie perfectly still, with bags over her hands to ward off any "wildness.". This repressed

upbringing eventually inspired rebellion in the form of an attempted elope-

ment, but the failed affair left her humiliated and repentant. She wrote to a fam-

ily friend, William King, requesting mathematical instruction as a cure for her

sinful impulses. King agreed, sending her both mathematical and religious texts.

But despite her declarations to apply her mathematical imagination "to the

greater glory of God," she turned away from the moralizing of King to the more

glamorous social company of Babbage and his famous "thinking machines."

Babbage's motivations were far removed from King's religious intellectu-

alism. He was primarily concerned with economic and scientific progress. This

switch from King to Babbage was an act of independence, and Lovelace began

to turn her imagination loose. While pursuing a much more intense area of mathe-

matical study, her religious thinking also took an expanded turn. She began to

describe herself and her work in terms of magical imagery: the mechanisms of sym-

bol manipulation were "mathematical sprites," and she advised Babbage to allow

himself to be "unresistingly bewitched" by "the High Priestess of Baggage's

Engine."

Stein also notes that it was actually Babbage who first drew up the "table

of steps" constituting the first computer programs. Babbage was having diffi-

culty obtaining funding for his work, however, and realized that Lovelace's social

• position and notoriety-both as the daughter of Byron as well as a "Lady of

21I

2I2

Implications

Mathematics" —could be put to his advantage. The reputation of Lovelace as

the originator of programming stems from this public relations ploy of Babbage.

There was, however, one table for which Ada was wholly responsible: the

recursive generation of a sequence known as the Bernoulli-numbers. Moore

(1977) states that this table used recursive programming. Huskey and Huskey

(1984), apparently referring to this claim, suggest that this is a confusion with

Lovelace's description of mathematical "recurrence groups" and note that the

term "recursive programming" generally refers to a procedure that calls itself"

(i.e., self-reference) impossible for Lovelace since her code had no procedures.

But they also note that Lovelace introduced a new code notation to describe

what she referred to as "a cycle of a cycle," which would be equivalent to the

recursive structure of nested iteration in use today.

Significantly, this iterative recursion was the one program for which Bab-

bage claimed credit: "We discussed various illustrations that might be introduced:

I suggested several, but the selection was entirely her own. So also was the alge-

braic working out of the different problems, except, indeed, that relating to the

numbers of Bernouilli (sic], which I had offered to do to save Lady Lovelace the

trouble" (quoted in Stein 1985, 89).

The appropriation may have been anticipated by Lovelace: Stein notes that

• in the letters concerning this program, Lovelace is atypically vague--she had

always been overdependent on Babbage for mathematical specifics—-and spec-

ulates that the vagueness was a deliberate move to protect her iterative inno-

vation. Many feminists have written about male envy of women's reproductive

capacity, and there might well be a parallel in Babbage's appropriation of

Lovelace's recursive achievement. But the organicist versions of such analyses

portray the conflict in terms of women being more natural or embodied, and men

being more artificial or abstract. In this story of male womb envy and the pro-

tective mother, it is the digital abstraction of recursion, not concrete embodi-

ment, over which the struggle is fought. The birthing metaphor was mentioned

by Lovelace herself; the finished programming study was "her first child." Con-

trary to Plato, sexual reproduction is not in opposition to the abstract realm of

mathematics; Lovelace used her mathematics to rebel against attempts to limit

her to a repressive femininity and used this artificial sexuality-a bewitching

high priestess, jealously guarding her programming progeny—to develop the first

computational recursion.

In the discussion of the mathematical theory of computability in chap-

ter 1o, we noted that the set of "primitive recursive functions," developed by

Rozsa Pèter, had the greatest computing power short of a Turing machine.

Unlike Lovelace, Pèter's capability as a mathematician is uncontested; in fact,

Fractals in the European history of mathematics

she is widely regarded as "the mother of recursive function theory" (Morris and

Harkleroad 1990). But she, too, implied that parallels existed between her

gender identity and mathematics; maintaining that women could provide a spe-

cial insight that men could not (Andréka 1974, 173). Since we know that, as

a mathematician, she would not be thinking of this special insight as being more

concrete or less logical, it may be that Pèter also made connections between sex-

ual reproduction and recursion.

Following Pèter's class of primitive recursive functions, one reaches the upper

limit of recursive power in the Turing machine. Alan Turing's contributions were

not only in the mathematical abstractions of computing, but in its application

to artificial intelligence as well. In his classic paper titled "Computing Machin-

ery and Intelligence," he proposed what is now called the Turing test. At first,

most definitions of machine intelligence were based on a particular task or

behavior (e.g., chess playing). But as the field of artificial intelligence (AI) has

developed, these have shown to be increasingly inadequate, and the Turing test

is widely regarded as the most reliable definition for Al (in fact, yearly Turing

tests are now held, with no machine winners thus far).

Turing begins by describing a game in which a man and a woman are

behind a door and answer questions from an interrogator by written replies. The

interrogator must determine who is the man and who is the woman; both must

try to deceive him in their answers. Turing then suggested replacing one person

with an Al machine; the Turing test holds that if the interrogator cannot dis-

tinguish person from machine, then one has created true machine intelligence.

Turing's biographer, Andrew Hodges, suggests that this "imitation game" was

inspired by Turing's own life: struggling to define his identity as a homosexual

in a homophobic society. Both the Turing machine's ability to imitate other

machines and this game of cognitive imitation echo she social experience of gays

who live in a community where they must pretend to be someone they are not.

And to some extent, the endless self-reference of metamathematics was Turing's

hiding place from the antigay world surrounding him. But the sexual guessing

game on which the Turing test was based worked against such normative

gender restrictions: it suggested gender as something more fluid, less fixed—a

feature which the virtual communities on the Internet have started to demon-

strate (ef. Stone 1995; Turkle 1995). Douglas Hofstadter (1985, 136-167), a mod-

ern master of recursion, has also written about the potential for a more fluid gender

identity in digital dynamics.

Mathematics had a double meaning for Turing. It was both an emotional

shield, a closed world of endless interior self-reference, as well as an opening into

consciousness and community. In the end, this desire for opening killed Turing:

213

214

Implications

during a robbery investigation he admitted his homosexuality to police detec-

tives and was arrested and forced to submit to hormone treatments. This even-

tually drove him to suicide. It was a tragic fairy-tale ending: he killed timself by

eating an apple dipped in poison. Hodges writes about this death in terms of the

double meaning that mathematics had in his life. "Lonely consciousness of self-

consciousness was at the center of his ideas. But that self-consciousness went

beyond Gödelian self-reference, abstract mind turning upon its abstract self. There

was in his life a mathematical serpent, biting its own tail forever, hut there was

another one that had bid him eat from the tree of knowledge."

In Africa these two serpents are one; sexual reproduction exists in the same

public realm as social intercourse. That is one possible reason why we see recur-

sion--the snake that bites its own tail--so prominently emphasized in African

fractals, and a possible explanation for why these pioneers of recursion in Europe

happened to be people who took issue with sexual repression. That's not to say

there is a deterministic link between the two. In analog feedback theory, for

example, we see both anti-authoritarian feminists, like Norbert Wiener (Heims

1984), as well as authoritarian prudes like Howard Odum (Taylor 1988). Mathe-

matics is not a mere reflection of personal interests, nor is it an abstraction that

is entirely divorced from our lives. We make meaning for ourselves out of what-

ever metaphors—-technical or otherwise—-we find useful; conversely, personal

meanings can often inspire new technical ideas.

While recursion is prominent in African fractals, it has been less so in Euro-

pean fractal geometry.® In the historical appendix to The Fractal Geometry of

Nature, Mandelbrot provides an erudite history of mathematical developments

that led to his work; recursion is never mentioned. Even when recursion does... -..

come up in the fractal geometry literature, the treatment is typically informal

or cursory. For example, Saupe (1988, 72) merely notes that "in some cases the

procedure can be formulated as a recursion."? Similarly, the fractal time series

produced by deterministic chaos is rarely regarded as the product of feedback

loops, and in one of the few studies that is focused on this relationship, Mees

(1984, 101) merely states that "chaos is certainly possible in feedback systems."

On the contrary, it is not that chaos is possible with feedback, but that chaos

is impossible without it.

It would be inaccurate to say that European mathematics has disregarded

recursion in general, and perhaps the observation I am making is simply due to

disciplinary specialization; there is no reason why someone studying applications

of graphics to analysis and mensural theory should necessarily be thinking about

Turing machines or recursive functions. But it is precisely this lack of necessity

in mathematics that is so easily forgotten in a discipline where certainty goes

Fractals in the European history of mathematics

beyond that of any empirical science imaginable. Mathematics is both an inven-

tion and a discovery. We discover the constraints inherent in the fabric of space

and time, constraints that are the stuff of which our universe is composed. But

mathematics does not stop there. The constraints are not just negations, but rather

the building blocks with which further mathematics is constructed. And like any

construction, there are choices to be made, decisions about how these building

blocks are to be connected, interrogated, and deployed in further discovery.

This is where the human side of mathematics enters the picture, especially that

most human of endeavors, culture. Conversely, culture is not mere whim, a

purely subjective matter of choosing favored social practices. This is where the

mathematical side of humanity enters the picture, for we are only free to con-

struct culture within the constraints of the universe in which we live. Neither

mathematics nor culture should be viewed as firmly fixed on the universal/local

divide; there are divisions within divisions never ending.

215

CHAPTER

-Futures-

for

African

fractals

- Most anchropologists have long abandoned the tendency to create a frozen

"ancient tradition" in defining indigenous society; change and synthesis are

now integral parts of the cultural portrait. So, too, with African fractals; they are

necessarily as much of the future as they are of the past.

Fractals in African contemporary arts

There are many works of modern African professional art which incorporate

aspects of fractals, spanning a wide range of cultural viewpoints. At the National

Museum in Yaoundé, Cameroon, one can see organic romanticism in Nyame's

paintings of logarithmic spirals morphing into people. The double-sided post-

modern metal sculptures of Legba in Benin, by artists such as Kouass, show a

cyborg' trickster whose traditional bifurcating abilities are rendy for the

binary codes of new technologies. In East Africa, painter Gebre Kristos Desta

produces nonlinear scaling circles he describes as pure abstractionism (Mount

1973, 118). African fractals continue to evolve. Besides being present in pro-

fessional studio art, fractals have also appeared in large-scale public art works,

such as on the facade of the University of Dakar library (fg. 14.1). This scaling

design, in which the alternation of painted rectangles at the small scale

216

Futures for African fractals

217

07020000

FIGURE 14.1

The library of the University of Dakar

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).

matches the alternation of the building walls at the large scale, is reminiscent

of certain African fabrics.

One of the most active areas of today's African art comes not from pro-

fessional studios, but rather from the undistinguished sellers of tourist art.

Tourist art was formally disregarded in the professional art world, but cultural

studies have increasingly shown chat this is a dubious position. First, neither

the "traditional artist" creating royal works for a king, nor art students trying

to please their instructors, nor even professional studio artists who must also

be concerned with sales are completely free to create whatever they wish, so

there is no reason to single out the creators of tourist art for being constrained.

Second, opportunities for professional studio artists are few, and the tourist

market creates a large number of economic opportunities; it seems suspicious

to disregard this vibrant activity in favor of a tiny elite. And finally, as Cullers

(1981) notes, tourism is not the opposite of authentic culture, rather tourism

creates authenticity.

Cullers's observation was repeated to me by Max (he did not want his last

name to be used), a Senegalese artist in Dakar who sells to the tourist trade. Max

complained that his most creative work--the designs which came to him in

dreams-was difficult to sell because of the tourist conception of tradicion and

authenticity. Like many creators of tourist art in Dakar, he produces imitations

of the kora, the Senegalese stringed instrument that features a single fret run-

ning down the center and a hand grip on both sides. Figure 14.2 shows the usual

kora model, along with Max's innovation, the recursive kora. The recursive

kora makes use of each hand grip as the fret of two smaller koras. I asked Max

218

Implications

FIGURE 14.2

The recursive kora

At right, a typical kora; at left, the innovative

recursive kora created by Senegalese artist Max.

if he had ever considered continuing to smaller scales, and he said that he had

once done so, but that it was impossible to sell such innovative work; tourists

did not want anything that smacked of originality.

Fractals in African contemporary architecture

Many indigenous African designs have been incorporated into modern archi-

•tectural projects in Africa, and some of these have been fractals. For example,

the Sierpinski-like iterative triangles from Mauritania were used in an institu-

tional building in Senegal, and the circles of circles in the architecture of West

African villages became the basis of a design for a building complex in down-

rown Bamako, the capital of Mali (fig. 14-3).

One of the most potent visions of an African fractal future has come from

the architectural studies of Dr. David Hughes at Kent State University in

Ohio. Working as a Fulbright scholar in several African countries, Hughes (1994)

put together a portrait of what he termed "Afrocentric architecture," which

embodies several aspects of the fractal model. First, Hughes combined a char-

acterization of the self-organizing properties of African building design (an

"organic architecture" which "grows from its site") with its self-similar prop-

erties (what he termed "the outside/inside relationship," a mutual shaping of

units, clusters of units, and communal spaces formed by the surrounding clus-

ters). Second, he explicitly rejected primitivist or naturalizing portraits. While

.....

Futures for African fractals

noting its environmental harmony, Hughes also emphasized that African

architecture is always an intentional act of design and semiotics, not merely

an unconscious adaptation to the ecosystem. In his framework, "tradition"

includes the tradition of innovation, or as Gates (1988) puts it, the African

theme of "repetition with revision."

2I9

FIGURE 14.3

Indigenous fractals in modern architecture

(a) Here a traditional Mauritanian fractal design is used in a modern building in the Casamance,

Senegal. (b) The DPC huikling 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

Implications

Given this combination of self-organized structure and intentional design,

it is not surprising that Hughes's work led him to a beautiful example of the poten-

tial fractal future. Figure 14.4 shows a design by Alex Nyangula, one of Hughes's

students at the Copperbelt University in Zambia (Hughes 1994, 165-166). This

architecture provides a powerful syncretic fusion of indigenous and modern

forms. The figure traced by the walkway shown in the ground plan is a classic

FIGURE 14.4

Design for Kitwe

Community Clinic

(a) Kitwe Community Clinic

in Zambia; design by David

Hughes and Alex Nyangula.

(b) Kitwe Community Clinic

ground plan.

(Photos courtesy David Hughes.)

First iteration

Second iteration

Third iteration

Fourth iteration

Fifth iteratión

FIGURE 14.5

Fractal iterations of Nyangula's community clinic design

Fractal based on Nyangula's architecrural design. The "active lines" of the generation process hav

been removed, as have any self-intersecting hexagons.

222

Implications

example of the fractal branching pattern referred to as a Cayley tree (see

Schroeder 1991, 87-88; Peitgen et al. 1991, 19-20), and can be extended from

the two iterations given by Nyangula to infinity. Adding the hexagons Ta syn-

cretism between the cylinder of Zambian indigenous architecture and the rec-

tilinear forms of modern materials) violates the Cayley requirement that the graph

is self-avoiding (that is, that the branches do not intersect). Since I was inter-

ested in exploring the fractal structure by taking Nyangula's design to higher iter-

ations, I made two adjustments for this problem. One is suggested by the approach

elevation sketch (Hughes 1994, 167), where it is clear that the central unit is

slightly larger than the others. This means that self-intersection will be forestalled

to higher iterations.? The other is simply the elimination of units whenever they

overlap. With these two qualifications, Nyangula's design makes for an infinitely

expandable (yet bounded) architecture, as shown in figure 14.5. Such flexibil-

ity could contribute to the efforts to encourage a more participatory approach

to African architectural design (Fathy 1973; Ozkan 1997).

If we take an aerial view of the modern European settlement of Paris,

France, we would see linear concentric circles surrounding its center of social

power. The difference between this linear, radially symmetric series of circles

and Africa's nonlinear, decentralized architecture is perhaps subtle, but impor-

tant. The term "Afrocentric" is misleading in that "centric" is much more the

geometry of Paris than of Logone-Birni, Mokoulek, Labbezanga, and the other

African architectures we have explored. Hughes's call for a "multidimensional

Afrocentrism" is both an affirmation of "Afro" and a challenge to "centrisin";

it is a call for cultural portraits that do not reduce to a single one-dimensional

center but rather combine the boundaries of tradition with the infinite expan-

sion of innovation.

African fractals in math education

Several researchers have independently explored fractal aspects of African

mathematics. Chonat Getz of the University of the Witwatersrand has created

Iterated Function System simulations of Zulu basket weaving. John Sims, nathe-

matician and artist at the Ringling School of Design in Florida, has developed

fractal patterns based on Bakuba rafia cloth (and inspired by his African heritage).

In chapter 5 we encountered the lusona analysis of Paulus Gerds, a professor at

Universidade Pedagogica of Maputo, Mozambique, whose prolific writings have

recently ranged from the ethnomathematics of wonen's art in southern Africa

(Gerdes 1998a) to the use of Mozambique basket weaving geometry in model-

ing fullerene molecules (Gerdes 1998b).

Futures for African fractals

While there are clearly benefits to utilizing indigenous knowledge for

development and education in Africa, African fractals might also be of use in the

United States. Despite the low mathematics participation of African American

students as an ethnic group, it has been demonstrated that changes in the learn-

ing environment can improve their mathematics proficiency to levels equal to

the majority population. Evidence suggests that although direct institutional bar-

riers in economically disadvantaged schools, such as the emphasis of vocational

over academic subjects (Davis 1986) and lack of computer access (Anderson,

Welch, and Harris, 1984) can account for some of this difference, more subtle

curricular changes can play a key role in retention and achievement. For example,

Baratz et al. (x985) found that African American students are more likely co

use computers for routine drill; hence, the problem is not simply the availabil-

ity of computers, bur also their style of utilization. The National Assessment of

Educational Progress (1983) study of math performance in seventeen-year-old

African Americans reported the greatest deficiencies at the applications level,

and several researchers (Usiskin 1985; Davis 1980; Malcom 1983) have recom-

mended revision of courses to emphasize more interdisciplinary and "real-

world" mathematics instruction as well as "action-oriented" pedagogy.

Computer-based learning has demonstrated the capability for both interactive

and interdisciplinary mathematics instruction (Keitel and Ruthven 1993), and

Stiff et al. (1093) specifically point to computer-based learning as a promising

forum for bringing these changes to African American students. These needs

could be directly addressed by applying African fractals to the classroom.

In addition to changes in structural aspects of mathematics teaching,

several researchers and instructors have initiated culturally enriched curricula.

The rationale for this approach comes from a variety of perspectives (e.g.,

Vygotskian learning cheory). Powell (1990) notes that pervasive mainstream

stereotypes of scientists and mathematicians conflict with African American

cultural orientation. Similar conflicts between African American identity and

mathematics education in terms of self-perception, course selection, and

career guidance have been noted (cf. Hall and Postman-Kammer 1987; Boyer

1983). But we should not assume that this constitutes a problem of "self-esteem."

The relation between cultural identity and learning is quite complex; it would

be naive to suggest that today's African American students have the same rela-

tion to ideas about their ancestry as did students in previous decades, and in

no case has there ever been a simple "mimicry" of African culture. Rather,

ethnographic research (Hebdige, 1987; Mercer 1988; Rose 1994) shows that

African American youth actively construct identity using a wide variety of cul-

tural signifiers.

223

224

Implications

For this reason, applications of African fractals will have to stress design tools

and guided discovery, and avoid passive presentation. While "interactive" has

become a catchword in multimedia, many of these systeins merely use the computer

like a slide projector, with students pressing different buttons to see various images.

Multimedia in this form has a distinctly "canned" feel to it. The design approach,

in contrast, offers students tools for constructing patterns of their own creation.

Thanks to many participants—in particular, programmers TQ Berg and Jaron

Sampson, and minority math education specialist Gloria Gilmer-we have started

development of an African fractals software math lab. The lab begins with simu-

lations of traditional African patterns and shows students how the mathematical

structure behind these designs offers them tools to create their own.

Again, it is important to stress that African American students are not

expected to be interested in the material out of a simple identity reflection,

anymore than they would necessarily be interested in wearing Dashiki shirts and

Afros. Rather, it is the opportunity to create new configurations and syntheses

that combine tradition and innovation that are significant. At the june 1996 meet-

ing of the Columbus Urban Youth Conference, we explored these connections

with a class of eighteen 12-year-old African American students. The first class

meeting introducing traditional architecture was a near disaster; despite multi-

media and manipulatives, it appeared that the primitivist associations with

"mud huts" were a strong deterrent. The following session, using the Ghanaian

log spiral-cellular automata--owari relations, was quite successful, probably

because the combination of traditional religious knowledge and mathematical

graphics sent a more clear antiprimitivist message. But in a design exercise

where the students began with computer graphics simulations of the Ghanaian

logarithmic spiral patterns, they showed little interest in producing further

imitations of the African designs. Rather, the students quickly caught on to

visual correlates of the equation parameters and began a free-for-all competition

to see who could make the most bizarre patterns. Their interest appeared to be

sparked by the African connections, but quickly went beyond them.

Perhaps more important than mitigating a direct conflict between ethnic

identity and mathematics, using African fractals in the classroom might help guard

against an overemphasis on biological determinism, which has been found

adversely to affect mathematics learning. Geary (1994) reviews cross-cultural stud-

ies that indicate that while children, teachers and parents in China and Japan

tend to view difficulty with mathematics as a problem of time and effort, their

American counterparts attribute differences in mathematics performance to

innate ability (which can then become a self-fulfilling prophecy). For African

Americans, biological determinism has been closely associated with mythic

Futures for African fractals

stereotypes about "primitive people" (e.g., the fable that Africans count "one,

two, three, many"). By showing the presence of complex mathematical concepts

in African culture, we can mend some of that damage. Since reductive myths of

biological determinism are detrimental to mathematics learning for students of

all ethnic backgrounds, all students could potentially benefit from this material.

Finally, we should note that the increasing use of multicultural curriculum

materials in the arts and humanities have not been matched in the sciences. This

could send a message to minority students that their heritage is only pertinent

to the arts and humanities, and that the sciences are really for people from

other ethnic groups. In addition, some texts such as Multicultural Mathematics (Nel-

son 1993) have emphasized only Chinese, Hindu, and Muslim examples, so

that even in cases where multiculturalism is used, African math may be left out

(see Katz 1992 for a similar critique). And of the few texts that do use African

math, almost all examples are restricted to primary school level. Again, this restric-

tion might unintentionally imply primitivism (i.e., that mathematical concepts

from African culture are only childlike). For this reason, our lab's inclusion of

advanced topics such as fractal geometry, cellular automata, and complexity

are worth the extra effort to tie into a secondary school curriculum (without over-

looking the use of standard topics such as logarithmic scaling, geometric con-

struction, and trigonometry).

While the multimedia lab's most significant potential for improving education

is in mathematics, we should not ignore African Studies. African art, for example,

is increasingly used in secondary schools across the nation, and use of our lab could

greatly enhance such courses. First, as noted above, it provides an alternative to

detrimental misrepresentations of Africans as "primitive" people. In art history

lessons, for instance, students often learn about the geometric basis for Greek...

architecture or Renaissance painting, while commentary on African works is

often restricted to discussion of "naturalness" or "emotional expression." Second,

the lab aids in inregrative curricula development (see Roth 1994 on difficulties

in this area). It would allow math teachers who would like to include ethno-

mathematics components in their teaching to refer to examples in which students

are already engaged, and would provide art teachers with new tools for design and

analysis. Similar advantages could be obtained in other African Studies areas.

225

Information technologies and sustainable development

The use of indigenous knowledge systems in development goes back to colo-

nial appropriations, but in the postcolonial context these systems have taken

on new meaning as a sign of either epistemological independence, or at least

226

Implications

a more egalitarian view of knowledge systems. In chapter 10, for example, we

saw the scaling spirals of Jola settlement architecture that arose from their

circular buildings; the French research organization ENDA has built an

impluvium created by the combination of modern materials and this tradi-

tional Jola design. Another of ENDA's rural development projects that incor-

porate both traditional fractal architecture and modern techniques is shown

in figure 14.6.

In chapter 6 we saw how the scaling patterns of kente cloth were created

to match the scaling of saccadic eye movements as they scan from the face to the

body. The Ghanaian Broadcasting Corporation, Ghana's national television

channel, has continued this practice in the context of modern information tech-

nologies, utilizing the scaling pattern of kente cloth in their test pattern (Ag. 14•7).

Whereas the traditional scaling was applied to the human visual scan, this tech-

nologized version makes use of the same pattern for testing the video scan. A simple

application, but it shows that African fractals are not just restricted to low-tech

adaptations; they can also provide some useful bridges between traditional and

high-tech worlds.

In chapter 1o we saw that there were ties between the traditional knowl-

edge systems supported by African fractals and the productive maintenance of

these societies in what Per Bak would call a state of self-organized criticality. This

suggests that most of the indigenous African societies were neither utterly anar-

chic, nor frozen in static order; rather, they utilized an adaptive flexibility that

could be applied to modern development. But decades of research have shown

FIGURE 14.6

Modernized

fractal village

This ENDA project in Burkina

Faso combined the traditional

fractal structure with modern

construction techniques.

Futures for African fractals

227

FIGURE 14-7

Kente cloth in the

Ghanaian Broadcasting

Corporation test pattern

Kente cloth pattern is used in

the upper right-hand quadrant

of the large circle.

that a top-down approach to development, even that making use of indigenous

knowledge, is often less effective than a bottom-up, "grass roots" approach.

Adopting information technology to rural areas could provide the opportunity

for putting African fractals to work in sustainable development.

In addition to the need for bottom-up authority, researchers have demon-

strated the critical role of women in African development (e.g., Boserup

1970; Nelson 1981; Adepoju and Oppong 1994); particularly in terms of the

gendered division of labor in rural societies (Beneria 1982). While much of

this analysis has focused on the vulnerability of women in bearing the brunt

of economic change, it has also started a new appreciation for the extensive

knowledge systems that existed in precolonial women's activities. Since many

of these practices continue today (albeit in modified form), women's indige-

nous knowledge systems have become an important resource in new approaches

to development.

Some obvious challenges include environmental damage (increasing salin-

ization, deforestation, and desertification), external economic pressures (the move

to cash-cropping, tourism, and migration to cities; abuse of power by private

corporations), increased disease (AIDS and other viruses), political unrest

(ethnic conflict, uncontrolled military force, abuse of authority), and damage

to the sociocultural system (disruptions of women's traditional authority, loss

of traditional knowledge systems). While all of these are far too large to be

addressed by any one approach, none of them can be viewed in isolation from

the others. In Nigeria, for example, the Shell Petroleum Development Company

228

Implications

began operations in Ogoniland that eventually led to widespread environ-

mental damage; attempts to protest through the press and other communica-

tion eventually led to the execution of Ogoni writer Ken Saro-Wiwa (Soynika

1994). Freedom of the press is not a separate issue from, protection of the

environment.

It is right to decry abuse of authority, but replacing one authority with another

is not necessarily going to provide a long-term solution. African fractals suggest

two alternative approaches. First, what is needed is not E. F. Schumaker's call

for "small is beautiful," but rather a self-organized approach to changes in the

relations between scale and the socioenvironmental systems—not just appro-

priate technology, but appropriate scaling. Second, more critical attention

needs to be paid to the artificial/natural dichotomy, which tends to be trapped

in either the organicists' desire for untouched nature (e.g., Hughes 1996), or the

techno-optimist's desire for resource extraction.

An alternative to these damaging extremes can be found in Calestous

Juma's 1980 classic, The Gene Hunters. Rather than a preservationist perspective,

in which indigenous society would be portrayed as natural elements of an

unchanging ecosystem, or a technocratic profiteering perspective, in which agri-

cultural development is merely a question of maximizing yields with imported

strains, Juma provides evidence for indigenous agricultural activity as sustainable

biotechnology. His studies show a long-standing African tradition of new seed

variety development that combined ecological sustainablilty with innovation

and experimentation. These practices have been threatened by corporate mono-

cropping, which can cause soil depletion, over-dependance on insecticides, loss

of genetic variation, and other social and ecological crises, as well as the appro-

priation of these genetic resources by a biotechnology industry with little inter-

est in indigenous legal rights. Juma notes that the challenge now facing African

agriculturalists is not just preservation of biodiversity, but also access to the legal,

technical, and financial apparatus that would allow them to reap the profit that

could sustain such ecologically sound efforts.

From the viewpoint of complexity theory, Juma's critique suggests that we

are trapped between the periodic stasis of the preservationists' limit cycle, and

the white noise of the profiteering positive feedback loop. As we saw in these

mathematical models, both are lacking in flexible interactions with memory;

the limit cycle being too tied to it, and the white noise being too free from it.

Information technologies have the potential to provide this memory, documenting

indigenous knowledge from seed varieties and soil types to gene sequences to

ecotopes. By providing informed rural access to information technologies,

African agriculturalists can take a step toward protecting their genetic resources

Futures for African fractals

from appropriation and move toward Juma's approach, which we might call

"biotech-diversity" (cf. Haraway 1997; Shiva 1997).

To view indigenous knowledge as a self-organizing system is one thing, but

creating the same bottom-up approach for a synthesis of ecological sustain-

ablility and rechnological development is a much greater challenge. For example,

Russel Barsh notes: "There is an implicit assumption in the research methodo!-

ogy used to elicit traditional pharmacological knowledge that this information

is recorded and transmitted digitally (numbers and/or words) ... (rather than]

internalizing an analog model" (1997. 33-34).

Native Seeds, a botanical organization dedicated to the continuation of

indigenous plant stock, has been creating a "cultural memory bank" that will record

both analog and digital information on Native American agriculture. The con-

cept, originating from Philippine ethnobotanist Virginia Nazarea-Sandoval

(2996), documents the combination of cultural and biological information about

the crops, seeds, farming, and utilization methods. The information, including

video interviews, is stored on CD-ROM, with access controlled entirely by the

indigenous farmers. In the U.S. context, which is overloaded with electronic tech-

nology and ethnocide, this approach makes sense, but the African context,

with its enormous indigenous population and sparse electronic technology, will

call for techniques that can have a wider impact, one that includes development

of a technological infrastructure as well.

If there is to be social transformation through grass-roots technological inno-

vation, it will require much more participation than agricultural systems alone.

Other kinds of information technology development could include flexible eco-

nomic networks, which allow small-scale business collaborate in the manu-

facture of products and services trey could not produce independently. These

Networks have created strong revitalization in certain rural areas of Europe

(Sabel and Piore 1990), and have shown promise in pilot stucies in the rural United

States as well (e.g., ACEnet in southern Ohio). The use of computers to orga-

nize production and vending and provide dynamic searches for the appropriate

market niche-one which would be environmentally and socially sustainable as

well as profitable—could spread the benefits of new information technologies to

the microbusiness level without having to put a laptop in every pushcart, and micro-

fancing programs have already proved successful in many Third World countries

(Serageldin 1997).

African traditions of decentralized decision making could also be com-

bined with new information technologies, creating new forms that combine

democratic rule with collective information sharing. The idea of "electronic

democracy" has slowly been developing over the Internet; but the efforts have

229

230

Implications

been hampered by the tendency to assume that virtual voting must he the same

as ordinary voting. Perhaps the neural net style of African decision making could

be utilized in the West as well, with voters indicating proportional strengths

for various options. Conversely, perhaps there are ways to apply computer

media to enhance African decision making. One approach would be the develop-

ment of community networks through public-access terminals (Schuler 1995).

And the enormous development in electronic security measures, creating sys-

tems that stymie even the most sophisticated hackers (encryption codes, finger-

print scanners, etc.), might find uses in preventing voter fraud that is so

common in unstable political regimes.

Nigerian American computer engineer Egondu Onyejekwe has started

efforts to apply information technology networking in African developmental

projects using complexity theory as a guiding principle. One area she cites is the

problem of land ownership (for example, see Charnkey 1996). She notes that the

continual division of land promoted by the colonial legacy often results in

unproductive economies of scale, but that government ownership tends to make

conditions worse by adding more hierarchy. "Resolving the land problem requires

a non-hierarchical method of organization, a system in which cooperative behav-

• ior is rewarded at the same time that individual innovation can flourish; a com-

bination of cooperation and competition like we see in cellular automata and other

computational models of self-organizing systems. What better way to encourage

this than through computing and information networks?"4

Neither the African fractals framework nor dissemination of information

technologies offers panaceas. My point is, rather, that the shift in perspective often

called for in development need not be either conservative return to the past, nor.....

the epistemological equivalent of an alien invasion. African fractals offer a

framework that is both rooted in indigenous cultures and cross-pollinates with

new hybrids.

APPENDIX

Measuring®

-the fractal

-dimension

-of African-

-settlement-

-architecture

.. There are several different ways to estimate the fractal dimension of a spatial

pattern. In the case of Mokoulek (fig. 2.4 of chapter 2) we have a black-and-

white architectural diagram, which allows us to do a two-dimensional version

of the ruler size versus length plots we saw in chapter 1. By placing the archi-

tectural diagram of Mokoulek under grids of increasing resolution, and count-

ing the number of grid cells that contain some part of the diagram, we can plot

the increase of aren with decrensing cell size (just as we obtained a plot of the

increasing length with decreasing ruler size). Figure a. shows the results, indi-

Cating a fractal dimension of 1.67-not too far from the 2.53 fractal dimension

that is obtained analytically from the computer simulation.

For the aerial photo of Labbazanga (fig. 2.5 of chapter 2) we have an

image in shades of gray, and the simple grid-counting method cannot be applied

It is possible to reduce the gray scale to black and white, but an alternative

method allows us to make a more direct measure of the scaling properties. Fig-

ure A.za shows the method for finding the scaling slope of 1/F noise in a one-

dimensional time series by applying a Fourier transform. In figure A.2b we see

how this can be applied to a two-dimensional spatial distribution by sweep-

ing the same spectral density measure around in polar coordinates. Rather than

the line of one-dimensional 1/F noise, a rwo-dimensional distribution is

23I

232

Appendix

characterized by a cone. It is difficult to show the entire cone, but we can take

horizontal slices (fig. A.2b), which show similar characteristics for both Lab-

bazanga and its fractal simulation (fig. A.3).

1000.

500.

log (number of cells containing image)

slope ₫ -1.67

200.

100.

50.

0.02

0.05

0.1

0.2

log (cell size)

FIGURE A.J

Measuring the fractal dimension of Mokoulek

0.5

power

time

One-dimensional time series for 1/F noise.

frequency

1/F noise spectral density

from 1-D Fourier transform.

low frequencies at high power

power

cut slices from the cone

2-D Fourier transform, with frequency in polar

coordinates: wider circle = higher frequency.

The line of 1/F noise is rotated to become a cone.

high frequencies a: low power

FIGURE A.2

Using a 2-D Fourier transform to detect fractal spatial distributions

high frequencies at low power

low frequencies at high power

high frequencies at leny power

low frequencies at high power

FIGURE 1.3

Results of a 2-D Fourier transform applied to aerial photo of Labbazanga

(a) Spectra for aerial photo of Labhazanga (kg. z.ga from chapter 2). (b) Spectra for fractal image

(fg. 2.gh 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 Labhazanga.

r:al

Notes-

CHAPTER 1 Introduction to fractal geometry

1. 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-yun) and the binary yin/yang.

3. Michael Polanyi (1966) referred to this as "tacit knowledge."

CHAPTER 2 Fractals in Africon seutlement architecture

1. On triangular churches, see Norberg-Schulz (z965, i7z); for the Pantheon, see

ibid., 124-

2. Another passage, "path of the serpent," is used only by royalty. Ir alcernates left and

righe as it approaches the center of the palace, and chus creates a scaling zigzag pattern.

The implication seems to be that even royalty must negotiate the fractal ranking, bur

they can traverse it in a more direer route.

3. American readers are probably most familiar with nuclear families, but in Africa the

family structure typically exrends to much larger networks. The English term "cousins,"

for example, emphasizes the nuclear family by lumping all these relatives together, while

many African kinship sysrems have distinct terms for paternal parallel cousins, mater-

nal parallel cousins, paternal cross cousins, etc.

4. The starus 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 (1920) vol. 1, 304).

5. This is another meaning for the rerm "participant simulacion." In the first meaning, briefly

mentioned in the introduction, I defined it as an effort in cooperative modeling and

analysis, a rechnologized version of recent attempts in collaborative ethnography by

some anthropologists and their informants. In that sense it supports the humanist goals

235

236

Notes

of self-governing autonomy. But in the Mokoulek case I am also using it in the post-

modernist sense, a participant in a virtual workd. The contrasting meanings and their

consequences are discussed in detail in chapter to, where the two are brought together.

6. The results were published in Eglash and Broadwell (198g), and are reproduced in

the appendix.

CHAPTER 3 Fractals in cross-cultural comparison

1. In general, anchropologists 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 pofitical hierarchy. The term

"tribe" is controversial, however, since colonialists often «sed it to deny the existence

of indigenous state societies, so it is important to separate the technical designation

from its colloquial use.

2. 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 occurs in the neocolonial context to valorize the "van-

ishing native" while appropriating their cultural resources. See Minh-ha (1986),

Anzaldúa (1987), Clifford (1988), and Bhabba (1990) for discussion of some of these

3. Crowe and Nagy (1992), for example, have done extensive analysis of Fiji decoration,

and found 12 of 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 on the Celtic interlace patterns, and some Italian foor 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 buift

up its particular repertoire of designs--from whatever sources—as part of a dynamic

yet culturally sp. cific practice.

GHAPTER 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. The 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 system.s

1. It is unfortunate that an otherwise excelfent paper comparing African and Australian

ethnomathematics (Warson-Verran and Turnbull 1994) fails to make this distinction

berween the iterative generation of linear and nonlinear number series.

2. Readers who recall the definition of nonlinear functions as involving, at minimum, some-

thing like x' 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 addirion.

3. After giving a lecure 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."

Notes

237

4. The series was first introduced as an example of a recursively comporable periodic string

by Axel Thue (1863-1922), using the replacement rules 0 → 01, 1 → 10, with an ini-

tial 0. Morse discovered its application to dererminisric chaos, in which it models the

fractal time series produced by certain nonlinear equations. See Schroeder (199s,

264-268) on these aspects of the sequence.

5. One-dimensional versions can show all'che dynamics of two dimensions, and can

even be used as a kind of parallel computer. Consider, for example, a rule that in each

iteration the number of counters in a cup is replaced by the sum of itself and its left

neighbor. Starting with one: 0100000 → 0110000 → 0121000 → 0133100 → 0146410

This fourth iteration gives us che binomial coefficients for expansion of (a + b)*,

which equals at + 4a36 + 6a262 + 4ab3 + 64.

CHAPTER 8 Recursion

1. The standard cerminology is somewhat ambiguous, since "recursion" is sometimes

used to refer specifically to what we will call "self-reference," and at other times it is

used in the more general sense applied here. "Iteration" is used in its normal definition,

and for the least powerful we will use the term "cascade." Technically, these three types

of recursion roughly correspond to Turing machines, push-down automata, and finite-

state automata, but these models are a little too abstract to be directly useful in help-

ing readers develop a sense of the distinctions that are of interest here.

2. Sagay (1983) explicitly mentions starting with the small shape in the center, whereas

the Ipako Elede rows look like they might be better described as a preestablished

linear sequence (although Sagay does not give details here).

3. Actually, it is not wax that is used in much of Africa, but rather a latex created by boil-

ing the sap of che Euphorbia plant. Williams notes that it can produce long, delicate

chreads that are impossible for wax.

4. Pelton (1980, 230) contrasts the singular random events of the Native American

trickster myths with "the less episodic, more narrative myths of Legba and Ogo-Yuruga

(in Africa]." The reason for the difference is parcly mathematical. The Native Ameri-

can concept of unpredictability is based more on chance (see Ascher 1991, 87-94),

while the African concept tencis to be closer to deterministic chaos, as we saw in Bamana

sand divination.

5. Curtin (1971) shows that the slave trade from what is now northern Senegal dimin-

ished after 1700, and that the Nigerian area did not begin major activity until after 1730.

This still leaves the possibility chat Fuller came from the area of present-day Benin and

Chunk, which would be too for south to have directly shared influences with the Bis-

sari, but Holloway (1990, 10) notes that Virginians showed some preference for

Africans from the Senegambian region.

6. I qualified this as "standard" because there has been a growing concern that anthro-

pologists may have overemphasized the importance of age-grade and kinship by pro-

jecting their own desires as well as the interests of their informants. Shaw (1995), for

example, shows how Louis Leaky's description of the extreme obedience of the Kikuyu

to their age-grade system was colored both by Leaky's desire for the order of a "small

English village" that he never experienced (having grown up with missionary parents)

and the Kikuyu elders' own interests in receiving the initiation payments that were over-

due ro them.

7. In addition to the association of the vertical with the spiritual, Fernandez suggests that

the spatial distinction derives from the Fang's periodic clan fission/relocation. The frag-

mentation of a social group comes with horizontal movement and is seen as the result

of stagnation or strife, while the establishment of the group in a new location is seen

as positive regeneration, building from the ground up.

8. Maurer and Roberts (1985, 25) describe the Tabwa belt, a leather strip with bands of

beads or wire as representations of a single descent line. Since the Tabwa use the mpande

238

Notes

disk to represent the expansion of all kinship groups from a singular origin, it is not

unreasonable to think of the mukaba belt as a lower-dimensional projection of the

mpande disk. If one is willing to speculate so wildly that even I 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'öne-dimensional

projection of the mpande spiral. A similar practice, the "Poincaré slice," is used in non-

linear dynamics (see Abraham and Shaw 1982).

9. It is important to understand that the problem is not one of "authenticity." I 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 ethnomathe-

matics 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 the math, and it is here that the interpretive

dexibility offered by narratives presents problems.

10. 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-reference. 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 smarty-pants programmer could again make a meta-metaloop, and

so on. 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 Furing proved for computing, and as Kurt Gödel 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).

I1. The most specific connection made by Taylor is the possibility that the material attrib-

uted to Hermes-Thoth was derived from some of the Egyptian priesthood writings men-

tioned by Clement of Alexandria.

12. Stéphanidès (1922, 192) suggests a more direct sub-Saharan origin of alchemy, enter-

ing Egypt around 718 B.c.e., following the invasions of Ethiopia.

13. That's not to say that the Legba drum beats were random; but the drumming did

indeed have an unexpected change of pace.

CHAPTER 10 Complexity

1. 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 notes,

for example, are excellent examples of analog communication, but they are entirely dis-

crete. See Eglash (1993) for details.

2. Bium et al. show that an analog Turing machine would be susceptible to the halting

problem. See Eglash (1002, 1008c) for more details on this recent history of chernetics.

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 neinory is pathologi-

cally simplified: the neurotic slavishly follows memory, while the 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, but Band C voters say, "If only

I had known A was going to win, I would have been willing to vote the other way."

Notes

Tank and Hopfield (1987, 106) contrast this one-shot majority rule voting with the

collective-decision-making process in neural ners: "In a collective-decision commit-

tee the members vore cogether and can express a range of opinions; the members

know all about the other vores and can change their opinions. The committee gener-

ares... what 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 be a fraction, which reduces Py--in fact, 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

stops missteering, even while the 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 the

Lorenz attractor consisted of only positive feedback, but this tuins out to be incorrect.

In terms of dynamical systems theory (Abraham and Shaw 1982; Devaney 1986),

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 dehnition into a control theory formulation.

7. I've oversimplified the relations here. For example, a fner 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 the 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 berween one and two dimensions we have objects like the Koch

curve. This is slightly different when we think 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 precolonial catastrophe 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 the colonial encounter

makes it clear thar these are entirely different orders of chacs."

1). There is also a good illustration of collective fractal generation in the arts: the Mbuti

bark-cloth design shown in chapter 3 is acrually the produer of multiple artists.

CHAPTER 11 Theoretical frameworks in cultural studies of knowledge

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 artributes (scaling, recursion, infinity, and dimension) were rested in a more or less

falsifable 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 Europcan fractals before making that judgment.

2. This should not necessarily be assumed to mean "closer to nature," since it could also

refer to an indigenous knowledge system that promotes good ecological practices; but

the ambiguity is problematic.

3. In fact I'm not--my master's degree is in systems engineering, and although I took a

few graduare seminars in mathematics for my interlisciplinary Ph.D. (thanks to the flex-

ibility of the History of Consciousness board at the University of California at Santa

Cruz), I wouldn't dare call myself a mathematician in professional company. I have always

239

240

Notes

tried to introduce myself as an ethnomathematician during held 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 near 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. On the role of neologisms in the work of Cesaire, see Clifford (1988). On the construction

of negritude as a set of binary oppositions, see Mudimbe (3988).

6. For example, the octopus arose millions of years before vertebrates but has a nervous

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.

CHAPTER 12 The politics of African fractals

i. 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) Lesbian Body takes a topic that was often treated as the unassailahle 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

2. Angela Davis has pointed out Ellison's denaturalizing tropes in lectures at UCSC; her

recent work contimues 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 1 1 of Audre Lourd's Zami. Like Witrig (1973), she describes the sclf-assembly of a

lesbian body, but her techniques for this artifcial reconstruction come through the ana-

log media of scent, vibration, and form. See Eglash (1g9s) for other examples.

4. Consider, for example, the mojo hand/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 be built un by prayer' (by 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

are, of course, many centralist critiques of decentralization, but Joreen's text took 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 be true, in most cases there are hints that it also

serves to question the humanist concrol enacted in a political democracy that can sup-

port such deep economic subservience (see Queen Latifa's "Queen of Royal Badness"

in Smith 1900).

Notes

24I

6. In fact, chis was how I 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

hoinestead: ot ka ot, or "house within the house."

8. The 1993 Supreme Court ruling in Shaw v. Reno used the terms "bizarre" and "snake-

like," the larter echoing historian John Fiske's 18r2 characterization of a "dragonlike"

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 chink

that individuals or groups can indeed create such associations, just as they can create

the opposite (e.g., chat 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 1086) 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 1986; Anzaldúa 1987;

Bhabba 1990; Sandoval 1095; Haraway 1006).

12. Digital and analog are also confusing ters 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. The resulting acoustic waveform can

be analog, digital, or—especially in the case of rap music-somewhere in berween. See

Eglast (8993) for details.

CHAPTER 13

Fractals in the European history of mathematics

1. According to ancient accounts, the discovery of irrationals was in the middle of the

fifth century в.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 Theets, 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 that 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 its contemporary diversity, which would matter for the intuition and practice of

modern machematicians.

"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 fifch ... and it is possible to do this order for an

infinite number of months" (crans. Maxey Brooke).

242

Notes

4. Similar analysis was proviced 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

acultural profanity but rather modern variations of a long-standing black tradition of

public sexual commentary.

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 Hartouni

1997). Again, the African version is in strong contrast; recall from chapter 8 chat 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 197os from the stable

postmodernism of the 198os (Eglash 1998c).

7. The qualification is not inaccurate; the problem is that sometimes the authors of this

text (The Science of Fractal 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 for African fractals

1. For more on cyborgs, see Haraway (1996) and Gray (1995).

2. In fact, if I had used a large enough size difference, self-intersection could have been

avoided altogether, but I 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 part because it implied a Platonic view,

it made sense to the students chat religious symbolism would be mathematical, while

something as concrete as a 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. Onyejekwe's African Women Global Network is available from http://www.osu.edu/

org/awognet.

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-Index-

abbia, 120, 138-140, 145-146

Abraham, Ralph, 193, 238n8, 23906

abstraction, 17, 51, 53, 62, 78, 102, 109, 131,

133.202,212, 213,214,216

Achebe, Chinua, 1 73

adlition modulo. See mod 2

aclitive series, 186-180

mestherics. See estheties

aftine transformation, 75

Ahocentrises, 18o-181, 288, 222

age-grinde, 68, 87, 121, 124, 23706

agriculture, 24.31,125,227-224

Agudoawn, Koh, 107

Akan, 77-78,81, 104

alchemy, 99, 100, 101, 140, 141, 238n 12

algorithin, 38, 47, 6t, 68, 27, 97, 113, 118,

133,153-154, 170,174.206

analog, 151-154, 158-161, 164, 102-194, 200,

202, 214, 220, 238001, 2, 24003, 241712

Ananse, 137

Angola, 68, 186

animism, 104

anthropology: authority in, 183-184; function-

alist, 170; mathematical, 185-187, 101;

modernist, 131, 238; reflexive, OS; struc-

turalist, 181, 188. See also ethnography

apartheid, 184, 200

aperiodicity, 108, 172

Arabic culture, 98-99, 205

archaeology, 61, 87,80

architecture: African, 4-8, 19-40, 87-80,

110-111, 124, 126-128, 131, 135,

148-149, 162-164, 166, 174, 195-199,

205,210, 216-222, 224, 226; American,

3-5, 39, 49-50, 55. 197-190; Chinese, 4;

European, 3, 20, 39, 48-51, 55,89, 174,

195-196, 225; Indian, 47-48; Native

Alaris: 0, 39-42; South Pacifc, 47

Aristotle, 8, 51, 147-148, 205-206, 242

anthmerie, 86-108

arithmetic series. See additive series

art education, 225

artificial incelligence, 213

Ascher, Marcia, 45, 47, 186, 23704

Auhenticity, 74, 184, 193-194, 217,33602,

238n0, 240n1

authority, 31, 133, 183, 186, 203, 227-228

Babbage, Charles, 211-212

Badiane, Nfally, 162, 164

Ba-ila, 26-29, 55, 110, 23504

Bak, Per, 161, 170, 226

Baka, 183, 24014

Baker, Houston, 196

Bakubis, 172-173, 221

Baluba, 130, 166, 210

Bambara, Toni Cale, 194

Bamileke, Benimin, 55, 90, 182, 183

253

254

Bantu, 62

Banyo, 34-36

basket weaving, 45-46, 222

Bassari, 121-122, 23705

Batammaliba, 121, 126, 135

Batty, Michael, 49-50

beadwork, 113, 139, 166, 23705

Bell, Eric T., 207-208

Bembe, 123

Benin, 91, 124, 141-143, 166, 182, 216,

23705

Berg, TQ, 224

Bernoulli, Jacobo, 210

Bey, Hakim, 241n9

binary code, 95, 98, 101

binomial coefficients, 23705

biological determinism, 187, 191, 224-225

biology, 3, 34, 84, 102-105, 107-108, 124,

131, 133, 141, 159, 189, 191, 227-229,

240n6

biotechnology, 228-229

birth, 34, 90, 109, 127, 131, 133, 168, 170,

208, 210, 212, 242715

Blixen, Karen (Isak Dinesen), 197

Blyden, E. W., 200

body, 12, 63-65, 75-76, 131-133, 164, 226,

240001, 3

Boggs, James, 240n1

Bourdier, Jean-Paul, 32-33

braiding. See hairstyles

bridewealth, 89

Broadwell, Peter, 31

bronze sculpture, 138-139

Brown, James, 199-200

brown noise, 23907

Burkina Faso, 31-33, 182

Butler, Octavia, 194

Bwami, 52, 123

Bwiti, 129

Cairo, 37-38, 201-202

Cameroon, 21-25, 29-31, 34-36, 113,

119-120, 138-139, 145, 149-150, 182,

190, 216, 23907, 24014

Cantor, Georg, 8-10, 197, 206-208

Cantor, Moritz, 208

Cantor set, 12-13, 15, 17,93. 99, 147-148,

206-208

Caplan, Pat, 195, 241n6

Carby, Hazel, 194

Carver, George Washington, 194

carving, 7, 43-44. 45. 62-63, 68, 108, 113.

117, 120, 138, 143, 166, 187, 189

Casamance, 162, 164

cascade, 109-110, 111-114, 145

Cayley tree, 222

Cayuga, 186

cellular nutomata, 102-108, 143, 154-155.

158, 162, 164, 168, 120

Celtic design, 7, 48

Césaire, Aimé, 188, 24005

Index

Chaitin, Gregory, 153

•chaos, 93,95, 103, 108, 143, 159, 162, 168,

174, 182, 190, 193, 197, 199, 214, 23744

chi wara, 124-125, 127, 134, 209

Chinese mathematics, 4, 47-48, 185, 225,

23502

Chokwe, 61, 68, 69, 70, 84, 187

Chomsky, Noam: cognitive theory of, 211:

hierarchy of, 156-158

Christianity, 20, 48, 90, 127, 135-136, 149

cities. See architecture

class, 81

Clifford, James, 131, 183, 193. 236n2, 24005

coastlines, 15, 17

colonialism, 195-197

complexity, 5, 45, 68, 146, 151-176, 184, 189.

225, 228, 230

computer: analog, 151-155, 158-161, 164-166;

calculation by, 74, 89, 97, 151; in develop-

ment, 229-230; education, 223-225; hard-

ware, 95, 98, 101; programs, 110-912, 132,

135, 137-138, 188, 211; simulation, 3, 12,

21, 28, 31, 32, 34, 38, 61, 71, 77, 101-104.

147, 172; theory, 146, 156-158, 212-214

Congo. See Democratic Republic of Congo

Conway, John Horton, 103-104, 170

coordinate systems: Cartesian, 3-5, 42, 85,

196; polar, 231-234; spherical, 83

Coptic design, 236n4

cornrows. See hairstyles

cosmology, 43-44, 48, 131-135, 204, 210

counting: base six, 122; hase ten, 4, 99, 23502;

base two, 89-91, 100

Crowe, Donald, 47. 48

Crowley, Aleister, 99

Crutchfield, James, 1 59-160, 174

cybernetics, 236n2, 238n2

cyhorgs, 216, 242n1

Dan, 141-143, 166, 170, 175

Dangbe. See Dan

Danhen, J. W., 208

Davis, Angela, 240n2

de Sousa, Martine, 141

de Souza, Francisco, 141

death, 34, 164, 170, 204, 214

decentralization, 31, 39, 180, 197, 222, 229,

236n1

Delany, Samuel R., 194

Democratic Republic of Congo, 61, 127, 166

Derrida, Jacques, 192-193

Descartes, René, 195-196

descent, 8, 124-131, 149, 206, 23718

design themes, 3, 4, 6, 27, 39-40

Desta, Gebre Kristos, 216

deterministic chaos. See chans

development, 225-230

diaspora, 55, 180, 199

Diarta, Christian Sina, 7. 161-162, 164

Dínz, Rogelio, 43-44

differential equations, 236n2

Index

255

cliffusion limited aggregation, 49

digical, 101, 104, 151-152, 156-158, 166, 190,

102-194, 200, 211-213, 229, 238nn1, 2,

241012

dimension, 12, 15, 18-19, 32, 43, 81, 83-84,

93.104, 113, 115. 154, 170-172, 176, 209,

23808, 230007, 1

discase, 17,227

disequilibrium, 170

divination, 31, 93-108, 108, 122, 124, 133,

143, 151, 183, 190, 209, 23774

Dogon, 231-134, 138, 140, 146, 170, 175

doubling. See counting: base two

Du Bois, W.E.B., 200

dynamical systems theory, 239nб

East Africa, 86, 99, 216

economics, 180, 196, 211, 217,223, 227, 220,

24005

education. See art education; mathemarics.

education

Egypt, 37-38, 87-89, 99, 134-135, 137.

140-141, 188-180, 191, 204-208, 23604

Ellison, Ralph, 194

engineering, 5, 73-74, 85, 143, 230

Eno, Brian, 101

environment, 20, 39, 50-51, 133, 219,

227-22, 240n6

Epimenides of Crete, 111, 137

epistemology, 180, 180, 193, 225,230

Eshu, 174, 175

essentialism, 180-182

esthetics, 7, 38, 50, 52, 53-57, 62-63, 81, 113.

ethics, 192, 194-195, 210, 2404

ethnography, 28, 31, 45, 127, 131, 181-184,

200, 203, 223, 23505. See also anchropol-

ethnophilosophy, 149, 180-190

Ethiopia, 1o1, 135-136

Euclidean construction method, 65, 68-69:

113, 118

Eulerian parh, 48, 68, 70, 186

cvolution, 161, 187, 180-190, 24016

Foge, William, 7, 84, 139, 1090

falsiliability, 6, 179, 24001

Fang, 127, 129, 149, 210, 23707

forus. See birth

Fibonacci series, 87-89, 110-111, 156,

205-206

finite state automaton, 456-158, 23701

fluid How, 47-48, 77-78, 97, 104, 200, 213

Fon, 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; infinicy;

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

funeral rituals, 164

Gabon, 127

Gambia, 121, 182, 23705

game of life. See cellular automata

game theory, 101

Garcia, Linda, 93

Garvey, Marcus, 200

Gates, Henry Louis, go, 190, 219, 24204

Gauss, Carl Friedrich, 206

Geertz, Clifford, 181-182

gender, 190, 212-213, 227

generics, 124, 161, 180, 188, 228, 24006

geometry. See afhne transformation; computer:

simulation; coordinate systems; dimension;

Euclidean construction method; Eulerian

path; fractal geometry; graphing; helix;

hexagon; iterated function systems; nondif-

ferentiable curve; pentagon; Poincaré slice;

quincunx; scaling; self-similárity; Sierpin-

ski gasket; sinusoidal waves; spiral; tiling:

trigonomerry

geomancy, 98-101

Gerdes, Paulus, 68, 122, 186, 222

Getz, Chonat, 222

Ghana, 74, 77-80, 101, 104-108, 113, 115,

124, 182, 226-227, 23705

Gilmer, Gloria, 224

Gleick, James, 182

Gödel, Kurt, 199, 214, 238n10

graphing, 4, 12, 14, 47, 73-74, 79, 81, 83-85

graphics. See computer: simulation

Greek culture, 76, 89, 99, 147, 147-148,

203-206, 210,225

Griaule, Marcel, 138, 133

griot, 164

Guinea-Bissau, 44, 121

hairstyles, 7, 63. 81-84, 112-114

Hausdorff, Felix, 12

Hausclorff-Besicovirch measure. See dimension

Heaver, Hannan, 38, 200, 202

Heighway, John, 113

helix, 112, 114

Hermes Trismegistus, 99, 134, 141, 238n11

Herskovics, Melville, 107

hexagon, 4, S, 121-122, 214, 222, 2350)

hierarchy, 39, 120, 122, 156-158, 189, 197.

210,230,23601

Hindu culture, 99, 185, 187, 225

Hofstadter, Douglas, 110, 213, 238n10

homosexuality, 213-214

homunculus, 127, 24205

Hughes, David, 218-222

humanism, 194-195, 209

Hurst, H. E., 12, 208-201

256

Hurston, Zorn Neale, 188

hybrids, 187, 200, 230, 241h11

1h0, 197

Ifa, 93-95

India, 7, 47-48

infinity, 8-9, 12, 13, 18, 34, 41-42, 70, 76-77,

91, 111, 135, 138-139, 146-150, 153,

157-159, 170, 190, 204-207, 210, 222,

239007, 1, 241009, 3

information technology. See computer

initiation, 68, 87, 100, 121-123, 133, 23706

intentionality, 5-6, 19, 49-57, 81, 113, 123,-

162, 165, 174, 184-187, 219-220, 225

intuition, 53,56-57,68,71,113, 154,24102

iron work, 61, 89-90, 141, 143

irrational numbers, 97, 204, 2410n1, 2

Ishango bone, 89, 91

Islam, 29, 31, 38, 93, 162, 202, 205

iteration, 15, 17, 18, 21, 22, 25, 26, 28, 29, 30,

31, 34.37,38, 45,48, 63, 67,68, 69, 76.

29,83, 86-88, 91,95.103-104, 110-130,

132-137, 145, 155,170, 172, 176, 210,

212, 222, 23701, 238010, 24207

iterated function systems, 76, 222

ivory sculpture, 62, 63, 65-68

Japan, 47-48

jewelry, 53-54

Jews, 99, 101, 200, 202, 207-208, 241010

Jola, 162-165

Juma, Calestous, 228-229

Kabbalah, 99

Kamil, abu, 205

Karnak, 88

Kauffman, Stuart, 270

kente cloth, 74-76, 226-227

Kenyetta, Jomo, 188

Kepler, Johannes, 206

Kikuyu, 209, 237n6

King, Martin Luther Jr., 199

kinship, 24, 113, 124, 127, 130-131, 145, 164.

186, 200, 235n3

Kirdi, 29

knot theory, 48

Koch, Helge von, 9-15, 17-18

Kolmogorov, A. N., 152-153, 155

kora, 217-218

Kotoko, 21, 24.32

Kronecker, Leopokl, 208

Kuba. See Bakuba

Kuti, Fela, 200

Kwele, 122-123, 127-128

Labbezanga, 31-32, 231-232, 234

labor, 24, 39, 113, 187, 189, 196, 227

Leaky, Louis, 23706

Legha, 143-144, 166, 175, 216, 27704,

238n13

Leibnitz, Gottfried, 100-101

Index

lightning, 91-03

limit cycle, 106, 143, 228

lineage, 24, 124, 127

linearity, 40-42, 71, 74, 76-77, 86, 121,4.

129-130, 196, 197, 211, 222, 237

linguistics, 193

Ingic, 4, 28, 20, 98, 111-112, 135, 204, 231, 213

Logone-Birni, 21-24

lotus, 135, 137

Lourde, Audre, 240n3

Lovelace, Adh, 211-212

Luba. Sec Baluba

.. Lucas, Edouard, 206

Lull, Raymond, 99-101

lungs, 15-17,34

Malagasy, 98

Malawi, 196

Mali, 8,31-32, 71-72, 133, 182

mancala, 101

Mandelbrot, Benoit, 12, 15, 17, 47, 51, 93,

176, 197, 208-209, 214

Mandiack, 44, 52

Mangberu, 61-68, 70

marriage, 119, 124

masks, 80-8t, 84, 121-123

mathematics education, 222, 223-225.

236nn2, 3

Mauritania, 113, 115, 218-219

May, Robert, 159, 168

Mayer-Kress, Gottfried, 239n6

Mbuti, 54, 23909

measurement, 4, 9, 12-18, 38, 72-74, 79, 89,

122, 151, 153-155, 159-160, 172,

174-175, 23901

medicine, 17, 127, 196, 24205

memory, 34, 97, 156-159, 161, 166, 174, 228,

220, 238n3

metalwork, 7, 112, 216. See also bronze sculp-. a

ture; iron work

Mezzrow, Mezz, 241010

migration, 121, 227

mimesis, 50-53,56

Mitsogho, 127-129, 149, 210

mud two, 95, 08-09

Mofoa, 29-31

morphogenesis. See biology

Morse, Marston, 97-98, 23704

Mozambique, 222

Mudimhe. V. Y., 149, 180, 189-190, 194.

24005

multiculturalism, 206, 225

music, 64-65, 143, 149, 154, 174, 193. 194.

200, 204, 200, 23804, 241П10, 242012

Mveng, Engelhert, 149-150, 190

Nankani, 32-34. 148-149, 210

natrative, 93, 95, 96, 133, 137, 146, 148, 149,

179, 186, 202, 206, 23704, 238n0

Native American culture, 40-46, 48, 116, 184,

186, 229, 23704

nature, 17, 18, 47,48, 50-53-56-57, 62, 141,

149, 180, 181, 190, 193. 228, 23602, 23902

Nazarea-Sandoval, Virginia, 229

"negritude," 188, 190, 191, 2400g

neural nets, 152, 154,165

neurobiology, 157, 156, 187, 199, 238, 24016,,

New Age mysticism, 187

Nigeria, 24, 94, 137, 173-175, 187, 200, 227,

230

Nile river, 99. 208-209

nomads, 115

nondifferentiable curve, 239n7

onlinearity, 40-43, 70, 71, 70-77, 80-82, 84

36-86, 97, 108, 113, 118, 122, 143, 162

182, 190, 200, 216, 222, 236n2, 23704,

238n8

numbers, 4, 5, 6, 8, 18, 31, 41, 42, 76, 86-108.

122, 153, 157, 159, 186, 190, 203-206,

212, 229, 235112

numerỏlogy, 4, 20, 95, 121-122, 134-135, 204,

23502

Nummo, 131, 133, 175

Nupe, 137

Nyangula, Alex, 220-222, 24202

Odum, Howarl, 214

Ogoni, 228

Ogotemmêli, 131

1/F noise, 159, 161, 166

Onyejekwe, Egondu, 23o

optimization, 73-74

orientalisin, 188

ORSTOM, 25, 29

owari, 101-108

Palestine, 89

paradox, 12, 111-112, 164, 203-205

participant simulation, 29, 182-184, 23505

pentagon, 204, 24101

periodicity, 103, 106, 141 - 143, 153, 156,.

158-160,172-173.228

Peter, Rozsit, 212-213

phase space, 239n6

philosoply, 149, 179, 181)-190, 203, 235112

physics, 7, 15,50,113,151-155, 158-176, 194

pi, 206

Plato, 203-205, 210, 241001, 2, 24203

plotting. See graphing

Poincaré slice, 238n8

point attractor, 106

polar coordinates, 231-233

politics, 31, 34, 101-102, 120, 124, 145, 174,

179, 180, 180-190, 192-202, 227-230,

2400g, 241hn8, 9

Popper, Karl, 6, 179, 23901

population, 5, 25, 49-50, 97, 159, 168, 196,

197, 205, 229, 236n6

Portland Baseline Essays, 188-189

positivism, 179

postmodernism, 193-194, 199, 216, 23605.

241009, 11, 242n6

Index

power law, 71-74, 89-93, 159-161

primitivism, 53, 89, 180, 188-189, 194,

196-197,224-225

probability, 94. See also chaos; randomness;

statistics; stochastic variation

programming. See computer: programs

pseudorandom number generation, 97-96

ush-down automaton, 157-151

Pythagoras, 203-20.

Queen Latifa, 240n

quincunx, 55, 18

racism, 180, 187, 188

randomness, 31, 93-99, 152-155, 158-161, 174,

186, 196, 197, 118, 23704, 238013, 23902

ratios, 204

rebirth. See birth

recursion, 8-12, 16-17, 34,43, 45, 47-48, 55,

77. 86,89, 93.95. 98, 99, 108, 109-147,

149, 151, 155-159, 161, 176, 187, 190, 192,

194-195, 199-200, 202, 205, 209-214, 217,

23701, 23803. 239011, 24107, 242006, 7.

See also cascade; iteration; self-reference

reflexive anthropology. See anthropology:

reflexive

religion, 7, 20, 28, 31, 47, 48, 53, 78, 90, 92,

93, 99. 124, 127, 129, 131-132, 135,

141-143, 164, 166, 170, 180, 189, 194,

202, 204, 205, 207, 208, 211, 24203

reproduction, 107-108, 124, 125, 134, 138,

140, 200-210, 212-214. See also birth

rite of passage, 34

ritual, 31, 68, 99, 121, 123, 126, 127, 162, 164,

165, 180, 186

romantic organicism, 194

Rosicrucianism, 95, 208

Rousseau, Jean Jacques, 192-193

Rucker, Rudy, 104, 162

Russell, Bertrand, 211

Sahara, 38, 71

Sahel, 71-74

Sampson, Jaron, 224

Saro-Wiwa, Ken, 228

scaling, 12, 17-19, 21, 26, 28-29, 31-35, 38,

41, 43, 43-48, 52, 54, 56, 61-63, 65, 68,

70, 71-85, 86, 89, 104, 110, 112-114,

116-118, 120-124, 126-128, 130-135,

137, 141, 148-149, 156, 166, 174, 175,

190, 196, 200, 202, 208, 216, 225, 226,

227, 228, 23502, 23901

Schumaker, E. F., 228

Schyler, George, 194

sculpture, 7, 52, 63, 66, 68, 79, 80, 81, 84.

112, 113, 127, 133, 134, 138-139, 216

secrets, 93, 97, 121-122, 200, 204

self-generation, 95, 97, 100, 135, 140, 206, 209

self-organization, 101, 104, 107-108, 161,

164-166, 168, 170, 176, 195-197, 218,

220, 226, 228-230, 242n6

257

258

Index

self-organized criticality, 16t, 170, 226

self-reference, 110-112, 135, 137-140, 146

self-similarity, 4, 18-19, 21, 24, 29, 31, 34, 38,

42, 43,93, 100, 124, 125, 140, 176, 195.

209, 218

Senegal, 8, ss, 81, 93, 140, 161-162, 174, 182,

183, 190, 117-118, 23705

Senghor, Léopold, 7, 190

sexuality, 209-214

Shammas, Anton, 200, 202

Shango, 90-93, 175

Shaw, Carolyn Martin, 200-210, 2370б

Sierpinski gasket (or triangle), 113, 115,

218-219

Sims, John, 222

sinusoidal waves, 141-142

slaves, 108, 122, 200, 23514, 23705

Solomonoff, Ray, 153

Songhai, 31-32, 195

Sotho, 200

soul, 33-34, 124, 126

South Africa, 15, 184, 200

South Pacific, 39, 47-48, 186

Sow, Fatou, 183

spectrum, 5-6, 49,51-92,56, 172-173, 176,

231-234

Spillers, Hortense, 194

spiral, 23-24, 29, 31, 45, 47-48, 76-79, 81,

86, 104-105, 107-108, 112, 129-130, 148,

162, 164, 210, 216, 224, 226, 238n8,

24203

spirit, 4, 28, 31, 89-90, 113, 119, 121, 124,

126, 127, 129, 831, 141, 148, 174, 175,

186, 188, 193, 194, 200, 204, 23707

Spivak, Gayatri, 184

square root, 205

state, 39-40, 51, 189, 236n1

statistics, 18, 241ng

status, 26-29, 55, 68, 2350g|

stochastic variation, 93, 241n9

Stoller, Paul, 3t, 195

stonework, 29, 101, 113, 135-137, 185, 196,

stools, 55-56

structuralism, 18t, 188

synlols, 6, 7,8, 20, 24, 34, 42, 43,55, 71,

77-78, 93-101, 108-109, 120, 126-128,

131, 139, 145, 147, 151-152, 156-158,

164. 170. 181-182, 186, 188, 192-194,

196, 208, 211, 24001, 24203

symmetry, 7,31, 42-43, 45-47, 79, 113, 118,

186-187, 190, 197, 222, 236n3

Syria, 80

Tabwa, 127, 130, 23708

tallies, 121-122

Tang, Chao, s61

Tanzania, 80, 195

tarumbeta, 86-87, 106, 10819.

tattoo patterns, 47

textiles, 7, 172-173

Thompson, D'Arcy, 190

Thue, Axel, 237114

tiling, 172

Togo, 124

tourism, 34, 217-218

triangular numbers, 86-87, 106, 108

tribe, 40, 189, 203

trickster, 99. 116, 137, 174.175, 182, 216

trigonometry, 68

Trinh, Min-ha, 32-33

Tswana, 200

Turing, Alan, 213-214

Turing machine, 157-159, 23812

twins, 89-90, 181-182

Ulam, Stanislaw, 102

Van Wyk, Gary, 200

viceo, 99, 226-227, 229, 24203

virtual construction, 21, 29, 183-184, 213,

230, 23505

vodun, 90-93,94-95, 141-143, 144, 166, 170,

174, 175, 183, 190, 194, 216, 238013,

24004

von Neumann, John, 101-102, 108

voodoo. See vodun

voting, 164-165, 229-230

Washburn, Dorothy, 48, 187

West, Cornel, 194

white noise, 154-155, 158-161, 173-174, 228,

23907

Wiener, Norbert, 214

Wolfram, Stephen, 106, 155, 158

Wolof, 162

womb, 34, 133, 212

woinen, 24, 32-34, 90, 124, 195, 200, 204,

212-213, 222, 227, 24005, 24204

Yoruba, 81, 82, 112, 113, 118, 174. 183, 190,

196, 24004, 24107

Zaire. See Democratic Republic of Congo

Zambia, 8, 26, 220-222

Zeno of Elen, 203-205

Zhahotinsky reaction, 104, 162

Zimbabie, 100, 196, 200

Zulu, 222

Untitled Note

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