AFRICA KNOWS FRACTALS LIKE BRAIDS KNOW FINGERTIPS

http://www.ccd.rpi.edu/Eglash/csdt/african/CORNROW_CURVES/teaching/teaching.html

http://www.ccd.rpi.edu/Eglash/csdt/african/CORNROW_CURVES/teaching/crproinstr/cpi1.htm

http://www.ccd.rpi.edu/Eglash/csdt/african/CORNROW_CURVES/cornrow_software/cornrow_software.swf

http://www.math.buffalo.edu/mad/special/gilmer-gloria_HAIRSTYLES.html

–

CULTURALLY SITUATED DESIGN TOOLS

http://www.ccd.rpi.edu/Eglash/csdt/index.html

“Many cultural designs are based on mathematical principles. This

software will help students learn standards-based mathematics as they

simulate the original artifacts, and develop their own creations.”

–

SPECTREVISION RESENTS — RON EGLASH, ETHNO-MATHEMATICIAN

http://www.ted.com/index.php/talks/view/id/198

http://www.ted.com/speakers/view/id/142

Why you should listen to him:

“Ethno-mathematician” Ron Eglash is the author of African Fractals, a

book that examines the fractal patterns underpinning architecture, art

and design in many parts of Africa. By looking at aerial-view photos

— and then following up with detailed research on the ground —

Eglash discovered that many African villages are purposely laid out to

form perfect fractals, with self-similar shapes repeated in the rooms

of the house, and the house itself, and the clusters of houses in the

village, in mathematically predictable patterns.

As he puts it: “When Europeans first came to Africa, they considered

the architecture very disorganized and thus primitive. It never

occurred to them that the Africans might have been using a form of

mathematics that they hadn’t even discovered yet.”

His other areas of study are equally fascinating, including research

into African and Native American cybernetics, teaching kids math

through culturally specific design tools (such as the Virtual

Breakdancer applet, which explores rotation and sine functions), and

race and ethnicity issues in science and technology. Eglash teaches in

the Department of Science and Technology Studies at Rensselaer

Polytechnic Institute in New York, and he recently co-edited the book

Appropriating Technology, about how we reinvent consumer tech for our

own uses.

“Next time you bump into one of those idiots who starts asking you

questions like, ‘where is the African Mozart, or where is the African

Brunel?’ — implying that Africans do not think — send them a copy of

Ron Eglash’s study of fractals in African architecture and watch their

heads explode.” – http://mentalacrobatics.com

–

CONTACT

http://www.rpi.edu/~eglash/eglash.htm

e-mail : eglash [at] rpi [dot] edu

BOOK

African Fractals: Modern Computing and Indigenous Design

http://books.google.com/books?id=1t7KaHjLBA8C

PAPERS ETC

http://www.rpi.edu/~eglash/eglash.dir/afractal/develop.htm

http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm

http://www.ccd.rpi.edu/Eglash/csdt/african.html

http://www.rpi.edu/~eglash/eglash.dir/nacyb.htm

INTERNATIONAL STUDY GROUP ON ETHNOMATHEMATICS

http://www.rpi.edu/%7Eeglash/isgem.dir/isgem.2.htm

–

DOUBLING THE DOUBLE — BINARY TOO?

http://www.sciencenewsforkids.org/articles/20061108/Feature1.asp

http://www.ccru.net/digithype/Afrobinary.htm

Ron Eglash – on Africa in the Origins of Binary Code

“The relationship between what we do as physicists today and the

future that our work engenders is dialogical in the sense that not

only do our present actions determine what our future will be, but we

must be ever mindful of the impact of our present actions on the

future… Through our moral responsibility and awareness of signals

and trends, we in effect, ‘listen’ to what the future has to tell

us.” (Donnell Walton, Chair, National Conference of Black Physics

Students 1995.)

While the temporal traditions of African societies were frequently

cited by colonialists as evidence for primitive superstition, they

appear today in quite the opposite context: mathematical analyses of

traditional African designs, techniques and knowledge systems indicate

a wide array of sophisticated indigenous inventions. Like Paul

Gilroy’s fractal Atlantic, Donnel Walton’s invocation of African

divination traditions- of listening to the future- is not only useful

in its call for greater ethical responsibility, but also as a reminder

fo the surprising links between traditional knowledge and modern

science.

The modern binary code, essential to every digital circuit from alarm

locks to super computers, was first introduced by Leibniz around 1670.

Leibniz has been inspired by the binary-based ‘logic machine’ of

Raymond Lull, which was in turn inspired by the alchemists’ divination

practice fo geomancy (Skinner 1980). But geomancy is Clearly not of

European origin. It was first introduced there by Hugo of Santalla in

twelfth century Spain and Islamic scholars had been using it in North

Africa since at least the 9th century, where it was first documented

in written records by the Jewish writer Aran ben Joseph.

Geomancy is traditionally practiced by drawing patterns in the sand.

It begins with four sets of random dashed lines. These are paired off

(i.e. summed by addition module two), and the even/odd results

recorded with two strokes or one stroke. Four of these binary digits

represents one of 16 possible divination archetypes (ruler, travel,

desire, etc.) Although the first four are generated by this random

process, the following 12 are created by recursively applying the same

pairing operation on the binary digits making up the four symbols (a

process which can be seen in many other areas of African knowledge

systems, c.f. Eglash 1995).

The nearly identical system of divination in West Africa associated

with Fa and Ifa was first noted by Trautman (1939), but he assumed

that geomancy originated in Arabic society, where it is known as ilm

alraml (“the science of sand”). The mathematical basis of geomancy is

however, strikingly out of place in non-African systems. Like other

linguistic codes, number bases tend to have an extremely long

historical persistence. 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 notation.

In ancient Egypt, on the other hand, base 2 calculation was

ubiquitous, even for multiplication and division, and Zaslavsky (1973)

notes archeological evidence linking it to the use of doubling in the

counting systems of sub-Suharan Africa. Kautzsch (1912) notes that

both Diodorus Siculus and Oblian reported that the ancient Egyptian

priests “employed an image of truth cut in halves.”

Doubling is a frequent theme in African divination and many other

African knowledge systems, connecting the sacredness of twins, spirit

doubles, and double vision with material objects, like the

blacksmith’s twin bellows and the double iron hoe given in

bridewealth. In a recent interview in Wired, 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

adaptive rhythm and flexibility 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. Take, for example, Henry Louis Gates’ (1988, pp. 30) use of a

recursive doubling description when discussing the relations between

divination and sexuality in West African traditions:

The Fon and Yoruba escape the Western cersion of discursive sexism

throught he 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.

The binary coding of traditional African knowledge systems, like that

of their antecedents in modern computing networks, is neither rigid

nor arhythmic; its beat is a heritage heard by those who listen to the

future.

References

Eglash, R. ‘African Influences in Cybernetics.” In C.H. Gray (ed) The

Cyborg Handbook.

Gates, H.L. The Signifying Monkey.

Gilroy, P. The Black Atlantic.

Kautzsch, T. “Urim”, Encylopedia of Religious Knowledge.

Skinner, S. Terrestrial Astrology.

Trautmann, R. “La divination a la Cote des Esclaves et a Madagascar,

Le Vodou, le Fa, le Sikidy.” Memories de l’institut Francais d’Afrique

Noire.

Zaslavsky, Cladia. Africa Counts.

–

OWARI SIMULATION

http://www.ccd.rpi.edu/Eglash/csdt/african/owari.htm

SELF-ORGANIZING SYSTEMS

http://www.ccru.net/digithype/recursivenum.htm

Ron Eglash on Recursive Numeric Sequences in Africa

1 ) Nonlinear additive series in African cultures. 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

numbers from a recursive loop; that is, a function in which the output

at one stage becomes the input for the next: X n + 1 – X n + 1 . But

this is a strictly linear series, increasing by the same amount each

time — the numeric equivalent of a staircase. Addition can, however,

produce 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 pp. 11 1 ). Figure 1 shows how these numbers are

derived from the shape of triangles of increasing size, and how the

numeric series can be created by a recursive loop. As in the case of

certain formal age-grade initiation practices, the simple versions are

used by smaller children, and the higher iterations picked up with

increasing age. While there is no indication of a formal relationship

in this instance, there is still an underlying parallel between the

iterative concept of aging common to many Africa cultures — each

individual passing through multiple turns of the “life-cycle” — and

the iterative nature of the triangular number series.

Another nonlinear additive series in 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, I found a visually distinct

example of this series in the successive chambers of the temple of

Karnak, as shown in figure 2a. Figure 2b shows how these numbers can

be generated using a recursive loop. This formal scaling plan may have

been derived from the non-numeric versions of scaling architecture we

see throughout Africa (cf. Eglash 1995, Eglash et al 1994).

An ancient set of balance weights, apparently used in Egypt, Syria and

Palestine circa 1200 B.C.E., also appear to employ the Fibonacci

sequence (Petruso 1985). This is a particularly interesting use, since

one of the striking mathematical properties of the sequence is that

one can create any positive integer through addition of selected

members — a property that makes it ideal for application to balance

measurements (Hoggatt 1969 pp 7f). There is no evidence that ancient

Greek mathematicians knew of the Fibonacci sequence. There was use of

the Fibonacci sequence in Minoan design, but Preziosi (1968) cites

evidence indicating that this could have been brought from Egypt by

Minoan architectural workers employed at Kahun.

2) Discrete self organization in Owari

Figure 3a shows a board game that is played throughout Africa in many

different versions variously termed “ayo,” “bao,” “giuthi,” “lela,”

“mancala,” “omweso,” “owari,” “tei,” and “songo” (among many other

names). Boards cut into stones, some of extreme antiquity, have been

found from Zimbabwe to Ethopia (see Zaslavsky 1973 figure 11-6). The

game is played by scooping pebble or seed counters from one cup, and

sequentially placing one each in the cups that follow. The goal is to

have the last counter land in a cup with only one or two counters

already in it, which allows the player to capture those 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 decrease 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 left uninterrupted it can propagate in

this way as far as needed (figure 3b). 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 outlook. Heims’ (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 mathematical order

in 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 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

computing machines could be built such that they could repair

themselves if “damaged in air raids,” and 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 AEC in 1954-5 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

suggestion of his colleague Stanislaw Ulam, von Neumann settled for a

graphic abstraction; “cellular automata” as they came to be called. In

this model (figure 4a) each square in a grid is said to be either

alive or dead (that is, in one of two possible states). The iterative

rules for changing the state of any one square are based on the eight

nearest neighbors (e.g. if 3 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 table cloths

with poker chips and dozens of human helpers (Mayer-Kress, personal

communication), 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” column was an instant hit, and computer

screens all over the world began to pulsate with a bizarre array of

patterns (figure 4b). As these activities drew increasing professional

attention, a wide range of mathematically-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 so elusive in studies

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 more

simple scaling structure, the logarithmic spiral (figure 5), has

garnered much of the attention. Even back in the 1950s 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 (1991) 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 text for CALAB, the first

comprehensive software for experimenting with cellular automata,

mathematician Rudy Rucker ( 1989, pp. 168) refers to systems which

produce paired log spirals as “Zhabotinsky CAs,” after the chemist who

first observed such self organizing patterns in artificial media:

“When you look at Zhabotinsky 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 stretching all the way

down to the river bottom…. In three dimensions, a Zhabotinsky

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 6 shows how the owari marching group system can be used as a

one-dimensional cellular automaton to demonstrate many of the dynamic

phenomena produced on two-dimensional systems. The Akan and other

Ghanaian societies had a remarkable pre-colonial use of logarithmic

spirals in iconic representations for self organizing systems (figure

7a). The Ghanaian spirals and the four-armed computer graphic in

figure 5b are quite distant in terms of the machine technologies that

produced produced them, but there may well be mathematical connections

between the two. Since cellular automata model the emergence of such

patterns in modern scientific studies of living systems, and certain

Ghanaian log spiral icons were also intended as generalized models for

organic growth, it is not unreasonable to consider the possibility

that the self organizing dynamics observable 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

chapter on owari, but unfortunately it only covers the rules and

strategies of the game. Recently Kofi Agudoawu (1991) of Ghana has

written a booklet on owari “dedicated to Africans who are engaged in

the formidable task of reclaiming their heritage,” and he does note

its association with reproduction: “wari” in the Ghanaian language Twi

means “he/she marries.” Herskovits ( 1930), noting that the “awari”

game played by the descendants of African slaves in the new world had

retained some of the pre-colonial 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 (figure 7b) can be commonly 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 enjoying 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.

References

Agudoawu, Kofi. Rules for Playing Oware. Kumasi: KofiTall 1991.

Badaway, A. Ancient Egyptian architectural design: a study of the

harmonic system. Berkeley: University of California Press, 1965.

Eglash, R., Diatta, C., Badiane, N. “Fractal structure in jola

material culture.” Ekistics pp. 367 371, vol 61 no. 368/3fi9, sept-dec

1994. Eglash, R. “Fractal geometry in African material culture.”

Symmetry: Culture and Science. Vol 6-1, pp 174-177, 1995.

Fagg, W. “The Study of African Art.” Bulletin of the Allen Memorial

Art Museum, Winter 1955 56, 12, 44-61.

Gies, F., Gies, J. Leonard of Pisa and the New Mathematics of the

Middle Ages. NY: Thomas Orowell 1969.

Heims, S.J John von Neumann and Norbert Wiener The MIT Press,

Cambridge, 1980.

Herskovits, Melville. “Wari in the new world.” paper read at the

Americanist Congress, Hamburg 1930.

Hoggatt, V.E. Fibonacci and Lucas Numbers. NY: Houghton Mifflin 1969.

Markus, Mario. “Autonomous organization of a chaotic medium into

spirals.” pp. 165-186 In Istvan Hargittai and Clifford Pickover (eds)

Spiral Syrmmetry, London: World Scientific 1991.

Petruso, K.M. “Additive Progression in Prehistoric Mathematics: A

Conjecture.” Historia Mathematica 12, 101-106, 1985.

Preziosi, D. Minoan Architectural Design. Mouton 1968.

Rucker, Rudy. CALAB. San Jose: Autodesk 1989.

Von Neumann, John. Collected works. General editor, A. H. Taub. New

York, Pergamon Press, 1951.

Zaslavsky, Claudia. Africa Counts. Boston: Prindle, Weber & Schmidt

inc. 1973.

–

RANDOM NUMBERS GENERATOR

http://www.math.buffalo.edu/mad/special/eglash.african.fractals.html

African Fractals: Modern Computing and Indigenous Design

IN 1988, Englash was studying aerial photographs of a traditional

Tanzanian village when a strangely familiar pattern caught his eye.

The thatched-roof huts were organized in a geometric pattern of

circular clusters within circular clusters, an arrangement Eglash

recognized from his former days as a Silicon Valley computer engineer.

Stunned, Eglash digitized the images and fed the information into a

computer. The computer’s calculations agreed with his intuition: He

was seeing fractals.

Since then, Eglash has documented the use of fractal geometry-the

geometry of similar shapes repeated on ever-shrinking scales-in

everything from hairstyles and architecture to artwork and religious

practices in African culture. The complicated designs and surprisingly

complex mathematical processes involved in their creation may force

researchers and historians to rethink their assumptions about

traditional African mathematics. The discovery may also provide a new

tool for teaching African-Americans about their mathematical heritage.

In contrast to the relatively ordered world of Euclidean geometry

taught in most classrooms, fractal geometry yields less obvious

patterns. These patterns appear everywhere in nature, yet

mathematicians began deciphering them only about 30 years ago.

Fractal shapes have the property of self-similarity, in which a small

part of an object resembles the whole object. “If I look at a mountain

from afar, it looks jagged and irregular, and if I start hiking up it,

it still looks jagged and irregular,” said Harold Hastings, a

professor of mathematics at Hofstra University. “So it’s a fractal

object-its appearance is maintained across some scales.” Nearly 20

years ago, Hastings documented fractal growth patterns among cypress

trees in Georgia’s Okefenokee Swamp. Others have observed fractal

patterns in the irregular features of rocky coastlines, the ever-

diminishing scaling of ferns, and even the human respiratory and

circulatory systems with their myriad divisions into smaller and

smaller branches. What all of these patterns share is a close-up

versus a panoramic symmetry instead of the common right versus left

symmetry seen in mirror images.

The principles of fractal geometry are offering scientists powerful

new tools for biomedical, geological and graphic applications. A few

years ago, Hastings and a team of medical researchers found that the

clustering of pancreatic cells in the human body follows the same

fractal rules that meteorologists have used to describe cloud

formation and the shapes of snowflakes.

But Eglash envisioned a different potential for the beautiful fractal

patterns he saw in the photos from Tanzania: a window into the world

of native cultures.

Eglash had been leafing through an edited collection of research

articles on women and Third World development when he came across an

article about a group of Tanzanian women and their loss of autonomy in

village organization. The author blamed the women’s plight on a shift

from traditional architectural designs to a more rigid modernization

program. In the past, the women had decided where their houses would

go. But the modernization plan ordered the village structures like a

grid-based Roman army camp, similar to tract housing.

Eglash was just beginning a doctoral program in the history of

consciousness at the University of California at Santa Cruz. Searching

for a topic that would connect cultural issues like race, class and

gender with technology, Eglash was intrigued by what he read and asked

the researcher to send him pictures of the village.

After detecting the surprising fractal patterns, Eglash began going to

museums and libraries to study aerial photographs from other cultures

around the world.

“My assumption was that all indigenous architecture would be more

fractal,” he said. “My reasoning was that all indigenous architecture

tends to be organized from the bottom up.” This bottom-up, or self-

organized, plan contrasts with a top-down, or hierarchical, plan in

which only a few people decide where all the houses will go.

“As it turns out, though, my reasoning was wrong,” he said. “For

example, if you look at Native American architecture, you do not see

fractals. In fact, they’re quite rare.” Instead, Native American

architecture is based on a combination of circular and square

symmetry, he said.

Pueblo Bonito, an ancient ruin in northwestern New Mexico built by the

Anasazi people, consists of a big circular shape made of connected

squares. This architectural design theme is repeated in Native

American pottery, weaving and even folklore, said Eglash.

When Eglash looked elsewhere in the world, he saw different geometric

design themes being used by native cultures. But he found widespread

use of fractal geometry only in Africa and southern India, leading him

to conclude that fractals weren’t a universal design theme.

Focusing on Africa, he sought to answer what property of fractals made

them so widespread in the culture. “If they used circular houses, they

would use circles within circles,” he said.

“If they used rectangles you would see rectangles within rectangles. I

would see these huge plazas. Those would narrow down to broad avenues,

those would narrow down to smaller streets, and those would keep

branching down to tiny footpaths. From a European point of view, that

may look like chaos, but from a mathematical view it’s the chaos of

chaos theory-it’s fractal geometry.” Eglash expanded on his work in

Africa after he won a Fulbright Grant in 1993.

He toured central and western Africa, going as far north as the Sahel,

the area just south of the Sahara Desert, and as far south as the

equator. He visited seven countries in all.

“Basically I just toured around looking for fractals, and when I found

something that had a scaling geometry, I would ask the folks what was

going on-why they had made it that way,” he said.

In some cases Eglash found that fractal designs were based purely on

aesthetics-they simply looked good to the people who used them. In

many cases, however, Eglash found that step-by-step mathematical

procedures were producing these designs, many of them surprisingly

sophisticated.

While visiting the Mangbetu society in central Africa, he studied the

tradition of using multiples of 45-degree angles in the native

artwork. The concept is similar to the shapes that American geometry

students produce using only a compass and a straight edge, he said. In

the Mangbetu society, the uniform rules allowed the artisans to

compete for the best design.

Eglash found a more complex example of fractal geometry in the

windscreens widely used in the Sahel region. Strong Sahara winds

regularly sweep the dry, dusty land. For protection from the biting

wind and swirling sand, local residents have fashioned screens woven

with millet, a common crop in the area.

The windscreens consist of about 10 diagonal rows of millet stalk

bundles, each row shorter than the one below it.

“The geometry of the screen is quite extraordinary,” said Eglash. “I

had never seen anything like it.” In Mali, Eglash interviewed an

artisan who had constructed one of the screens, asking him why he had

settled on the fractal design.

The man told Eglash the long, loosely bound rows forming the bottom of

the screen are very cheap to construct but do little to keep out wind

and dust. The smaller, tighter rows at the top require more time and

straw to make but also offer much more protection. The artisans had

learned from experience that the wind blows more strongly higher off

the ground, so they had made only what was needed.

“What they had done is what an engineer would call a cost-benefit

analysis,” said Eglash. He measured the length of each row of the non-

linear windscreen and plotted the data on a graph.

“I could figure out what the lengths should be based on wind

engineering values and compared those values to the actual lengths and

discovered that they were quite close,” he said. “Not only are they

using a formal geometrical system to produce these scaling shapes, but

they also have a nice practical value.” Eglash realized that many of

the fractal designs he was seeing were consciously created. “I began

to understand that this is a knowledge system, perhaps not as formal

as western fractal geometry but just as much a conscious use of those

same geometric concepts,” he said. “As we say in California, it blew

my mind.” In Senegal, Eglash learned about a fortune-telling system

that relies on a mathematical operation reminiscent of error checks on

contemporary computer systems.

In traditional Bamana fortune-telling, a divination priest begins by

rapidly drawing four dashed lines in the sand. The priest then

connects the dashes into pairs. For lines containing an odd number of

dashes and a single leftover, he draws one stroke in the sand. For

lines with even-paired dashes, he draws two strokes. Then he repeats

the entire process.

The mathematical operation is called addition modulo 2, which simply

gives the remainder after division by two. But in this case, the two

“words” produced by the priest, each consisting of four odd or even

strokes, become the input for a new round of addition modulo 2. In

other words, it’s a pseudo random-number generator, the same thing

computers do when they produce random numbers. It’s also a numerical

feedback loop, just as fractals are generated by a geometric feedback

loop.

“Here is this absolutely astonishing numerical feedback loop, which is

indigenous,” said Eglash. “So you can see the concepts of fractal

geometry resonate throughout many facets of African culture.” Lawrence

Shirley, chairman of the mathematics department at Towson (Md.)

University, lived in Nigeria for 15 years and taught at Ahmadu Bello

University in Zaria, Nigeria. He said he’s impressed with Eglash’s

observations of fractal geometry in Africa.

“It’s amazing how he was able to pull things out of the culture and

fit them into mathematics developed in the West,” Shirley said. “He

really did see a lot of interesting new mathematics that others had

missed.” Eglash said the fractal design themes reveal that traditional

African mathematics may be much more complicated than previously

thought. Now an assistant professor of science and technology studies

at Rensselaer Polytechnic Institute in Troy, Eglash has written about

the revelation in a new book, “African Fractals: Modern Computing and

Indigenous Design.” “We used to think of mathematics as a kind of

ladder that you climb,” Eglash said. “And we would think of counting

systems-one plus one equals two-as the first step and simple shapes as

the second step.” Recent mathematical developments like fractal

geometry represented the top of the ladder in most western thinking,

he said. “But it’s much more useful to think about the development of

mathematics as a kind of branching structure and that what blossomed

very late on European branches might have bloomed much earlier on the

limbs of others.

“When Europeans first came to Africa, they considered the architecture

very disorganized and thus primitive. It never occurred to them that

the Africans might have been using a form of mathematics that they

hadn’t even discovered yet.” Eglash said educators also need to

rethink the way in which disciplines like African studies have tended

to skip over mathematics and related areas.

To remedy that oversight, Eglash said he’s been working with African-

American math teachers in the United States on ways to get minorities

more interested in the subject. Eglash has consulted with Gloria

Gilmer, a well-respected African-American mathematics educator who now

runs her own company, Math-Tech, Inc., based in Milwaukee. Gilmer

suggested that Eglash focus on the geometry of black hairstyles.

Eglash had included some fractal models of corn-row hair styles in his

book and agreed they presented a good way to connect with contemporary

African-American culture.

Jim Barta, an assistant professor of education at Utah State

University in Logan, remembers a recent conference in which Eglash

gave a talk on integrating hair braiding techniques into math

education. The talk drew so many people the conference organizers

worried about fire code regulations.

“What Ron is helping us understand is how mathematics pervades all

that we do,” said Barta. “Mathematics in and of itself just is, but as

different cultures of human beings use it, they impart their cultural

identities on it-they make it theirs.” Joanna Masingila, president of

the North American chapter of the International Study Group on

Ethnomathematics, said Eglash’s research has shed light on a type of

mathematical thinking and creativity that has often been ignored by

western concepts of mathematics. “It’s challenging stereotypes on what

people think of as advanced versus primitive approaches to solving

problems,” she said. “Sometimes we’re limited by our own ideas of what

counts as mathematics.” Eglash has now written a program for his Web

site that allows students to interactively explore scaling models for

a photograph of a corn-row hair style.

Eventually, he’d like to create a CD ROM-based math lab that combines

his African fractal materials with African-American hair styles and

other design elements such as quilts.

One of the benefits of including familiar cultural icons in

mathematics education is that it helps combat the notion of biological

determinism, Eglash said.

Biological determinism is the theory that our thinking is limited by

our racial genetics. This theory gets reinforced every time a parent

dismisses a child’s poor math scores as nothing more than a

continuation of bad math skills in the family, said Eglash. “So for

Americans, this myth of biological determinism is a very prevalent

myth,” he said. “We repeat it even when we don’t realize it.” Eglash

said using the African fractals research to combat the biological

determinism myth benefits all students. “On the other hand, there is a

lot of interest in how this might fit in with African-American

cultural identity,” he said.”Traditionally, black kids have been told,

‘Your heritage is from the land of song and dance.’ It might make a

difference for them to see that their heritage is also from the land

of mathematics.”

Description from the back cover:

“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 have become an important new tool for modeling in

biology, geology, and other natural sciences. While fractal geometry

can take us into the far reaches of high tech science, its patterns

are surprisingly common in traditional African designs, and some of

its basic concepts are fundamental to African knowledge systems.”

African Fractals introduces readers to fractal geometry and explores

the ways it is expressed in African cultures. Drawing on interviews

with African designers, artists, and scientists, Ron Eglash

investigates fractals in African architecture, traditional

hairstyling, textiles, sculpture, painting, carving, metalwork,

religion, games, quantitative techniques, and symbolic systems. He

also examines the political and social implications of the existence

of African fractal geometry. Both clear and complex, this book makes a

unique contribution to the study of mathematics, African culture,

anthropology, and aesthetic design.

On the cover is the iterative construction of a Fulani wedding

blanket, for instance, embeds spiritual energy, Eglash argues. In this

case, the diamonds in the pattern get smaller as you move from either

side toward the blanket’s center. “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,” Eglash notes.

“Releasing this 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.”