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.”