Below is a comprehensive review essay of a pioneering work on
the mathematics that's embedded within African cultures. The
book helps reinforce new innovative pedagogical approaches to
the teaching of math AND a further verification of the importance
of inclyding math and the sciences within a Black/Africana Studies
program or department.
The dilemma here is that there are so few folk who can handle
(i.e. teach) the subject.
S. E. Anderson
African Fractals: Modern Computing and Indigenous Design
by Dr. Ron Eglash http://www.rpi.edu/~eglash/eglash.htm
IN 1988, RON EGLASH was studying aerial photographs of a traditional
Tanzanian village when a strangely familiar pattern caught his
The thatched-roof huts were organized in a geometric pattern
clusters within circular clusters, an arrangement Eglash recognized
his former days as a Silicon Valley computer engineer. Stunned,
digitized the images and fed the information into a computer.
computer's calculations agreed with his intuition: He was seeing
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
researchers and historians to rethink their assumptions about
traditional African mathematics. The discovery may also provide
tool for teaching African-Americans about their mathematical
In contrast to the relatively ordered world of Euclidean geometry
in most classrooms, fractal geometry yields less obvious patterns.
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
part of an object resembles the whole object. "If I look at a
from afar, it looks jagged and irregular, and if I start hiking
it still looks jagged and irregular," said Harold Hastings, a
of mathematics at Hofstra University. "So it's a fractal object-its
appearance is maintained across some scales." Nearly 20 years
Hastings documented fractal growth patterns among cypress trees
Georgia's Okefenokee Swamp. Others have observed fractal patterns
irregular features of rocky coastlines, the ever-diminishing
ferns, and even the human respiratory and circulatory systems
myriad divisions into smaller and smaller branches. What all
patterns share is a close-up versus a panoramic symmetry instead
common right versus left symmetry seen in mirror images.
The principles of fractal geometry are offering scientists powerful
tools for biomedical, geological and graphic applications. A
ago, Hastings and a team of medical researchers found that the
clustering of pancreatic cells in the human body follows the
fractal rules that meteorologists have used to describe cloud
and the shapes of snowflakes.
But Eglash envisioned a different potential for the beautiful
patterns he saw in the photos from Tanzania: a window into the
Eglash had been leafing through an edited collection of research
articles on women and Third World development when he came across
article about a group of Tanzanian women and their loss of autonomy
village organization. The author blamed the women's plight on
from traditional architectural designs to a more rigid modernization
program. In the past, the women had decided where their houses
But the modernization plan ordered the village structures like
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.
for a topic that would connect cultural issues like race, class
gender with technology, Eglash was intrigued by what he read
the researcher to send him pictures of the village.
After detecting the surprising fractal patterns, Eglash began
museums and libraries to study aerial photographs from other
around the world.
"My assumption was that all indigenous architecture would be
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,
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
fractals. In fact, they're quite rare." Instead, Native American
architecture is based on a combination of circular and square
Pueblo Bonito, an ancient ruin in northwestern New Mexico built
Anasazi people, consists of a big circular shape made of connected
squares. This architectural design theme is repeated in Native
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
of fractal geometry only in Africa and southern India, leading
conclude that fractals weren't a universal design theme.
Focusing on Africa, he sought to answer what property of fractals
them so widespread in the culture.
"If they used circular houses, they would use circles within
"If they used rectangles you would see rectangles within rectangles.
would see these huge plazas. Those would narrow down to broad
those would narrow down to smaller streets, and those would keep
branching down to tiny footpaths. From a European point of view,
may look like chaos, but from a mathematical view it's the chaos
chaos theory-it's fractal geometry." Eglash expanded on his work
Africa after he won a Fulbright Grant in 1993.
He toured central and western Africa, going as far north as the
the area just south of the Sahara Desert, and as far south as
equator. He visited seven countries in all.
"Basically I just toured around looking for fractals, and when
something that had a scaling geometry, I would ask the folks
going on-why they had made it that way," he said.
In some cases Eglash found that fractal designs were based purely
aesthetics-they simply looked good to the people who used them.
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
tradition of using multiples of 45-degree angles in the native
The concept is similar to the shapes that American geometry students
produce using only a compass and a straight edge, he said. In
Mangbetu society, the uniform rules allowed the artisans to compete
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
and swirling sand, local residents have fashioned screens woven
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.
never seen anything like it." In Mali, Eglash interviewed an
had constructed one of the screens, asking him why he had settled
The man told Eglash the long, loosely bound rows forming the
the screen are very cheap to construct but do little to keep
and dust. The smaller, tighter rows at the top require more time
straw to make but also offer much more protection. The artisans
learned from experience that the wind blows more strongly higher
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
plotted the data on a graph.
"I could figure out what the lengths should be based on wind
values and compared those values to the actual lengths and discovered
that they were quite close," he said. "Not only are they using
geometrical system to produce these scaling shapes, but they
also have a
nice practical value." Eglash realized that many of the fractal
he was seeing were consciously created. "I began to understand
is a knowledge system, perhaps not as formal as western fractal
but just as much a conscious use of those same geometric concepts,"
said. "As we say in California, it blew my mind." In Senegal,
learned about a fortune-telling system that relies on a mathematical
operation reminiscent of error checks on contemporary computer
In traditional Bamana fortune-telling, a divination priest begins
rapidly drawing four dashed lines in the sand. The priest then
the dashes into pairs. For lines containing an odd number of
a single leftover, he draws one stroke in the sand. For lines
even-paired dashes, he draws two strokes. Then he repeats the
The mathematical operation is called addition modulo 2, which
gives the remainder after division by two. But in this case,
"words" produced by the priest, each consisting of four odd or
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
do when they produce random numbers. It's also a numerical feedback
loop, just as fractals are generated by a geometric feedback
"Here is this absolutely astonishing numerical feedback loop,
indigenous," said Eglash. "So you can see the concepts of fractal
geometry resonate throughout many facets of African culture."
Shirley, chairman of the mathematics department at Towson (Md.)
University, lived in Nigeria for 15 years and taught at Ahmadu
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
them into mathematics developed in the West," Shirley said. "He
did see a lot of interesting new mathematics that others had
Eglash said the fractal design themes reveal that traditional
mathematics may be much more complicated than previously thought.
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
climb," Eglash said. "And we would think of counting systems-one
one equals two-as the first step and simple shapes as the second
Recent mathematical developments like fractal geometry represented
top of the ladder in most western thinking, he said. "But it's
useful to think about the development of mathematics as a kind
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
Africans might have been using a form of mathematics that they
even discovered yet." Eglash said educators also need to rethink
in which disciplines like African studies have tended to skip
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
minorities more interested in the subject. Eglash has consulted
Gloria Gilmer, a well-respected African-American mathematics
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
book and agreed they presented a good way to connect with contemporary
[Patterns in African American Hairstyles
Gloria Gilmer http://www.math.buffalo.edu/mad/PEEPS/gilmer_gloria.html]
Jim Barta, an assistant professor of education at Utah State
in Logan, remembers a recent conference in which Eglash gave
a talk on
integrating hair braiding techniques into math education. The
so many people the conference organizers worried about fire code
"What Ron is helping us understand is how mathematics pervades
we do," said Barta. "Mathematics in and of itself just is, but
different cultures of human beings use it, they impart their
identities on it-they make it theirs." Joanna Masingila, president
the North American chapter of the International Study Group on
Ethnomathematics, said Eglash's research has shed light on a
mathematical thinking and creativity that has often been ignored
western concepts of mathematics. "It's challenging stereotypes
people think of as advanced versus primitive approaches to solving
problems," she said. "Sometimes we're limited by our own ideas
counts as mathematics." Eglash has now written a program for
site that allows students to interactively explore scaling models
photograph of a corn-row hair style.
Eventually, he'd like to create a CD ROM-based math lab thatcombines
African fractal materials with African-American hair styles and
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,
Biological determinism is the theory that our thinking is limited
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,
myth of biological determinism is a very prevalent myth," he
repeat it even when we don't realize it." Eglash said using the
fractals research to combat the biological determinism myth benefits
students. "On the other hand, there is a lot of interest in how
might fit in with African-American cultural identity," he
said."Traditionally, black kids have been told, 'Your heritage
the land of song and dance.' It might make a difference for them
that their heritage is also from the land of mathematics."
Book now available from Rutgers University Press: Order by phone
800-446-9323. Order book from Amazon.com
Description from the back cover:
Fractal geometry has emerged as one of the most exciting frontiers
the fusion between mathematics and information technology. Fractals
be seen in many of the swirling patterns produced by computer
and have become an important new tool for modeling in biology,
and other natural sciences. While fractal geometry can take us
far reaches of high tech science, its patterns are surprisingly
in traditional African designs, and some of its basic concepts
fundamental to African knowledge systems.
African Fractals introduces readers to fractal geometry and explores
ways it is expressed in African cultures. Drawing on interviews
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
social implications of the existence of African fractal geometry.
clear and complex, this book makes a unique contribution to the
mathematics, African culture, anthropology, and aesthetic design.
For more about the book see Dr. Eglash's webpage at
On the cover is the iterative construction of a Fulani wedding
for instance, embeds spiritual energy, Eglash argues. In this
diamonds in the pattern get smaller as you move from either side
the blanket's center. "The weavers who created it report that
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
middle they would risk death. The engaged couple must bring the
food and kola nuts to keep him awake until it is finished."
Dr. Ron Eglash:
Department of Science and Technology Studies
Rensselaer Polytechnic Institute (RPI)
Troy, NY 12180-3590
email: [log in to unmask]
BACK TO SPECIAL ARTICLES:
BACK TO Mathematicians of the African Diaspora
The website MATHEMATICIANS OF THE AFRICAN DIASPORA is brought
to you by
The Mathematics Department of The State University of New York
created and maintained by
Dr. Scott W. Williams
Professor of Mathematics
CONTACT Dr. Williams http://www.math.buffalo.edu/mad/00madcontact.html