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African Fractals: A Book Review


Sam Anderson <[log in to unmask]>


Science for the People Discussion List <[log in to unmask]>


Wed, 9 Jan 2002 12:00:27 -0800





text/plain (838 lines)

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.

In Struggle,

S. E. Anderson

African Fractals: Modern Computing and Indigenous Design

by Dr. Ron Eglash

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
of circular

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
a new

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
a small

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
up it,

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
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

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

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
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

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

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

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
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

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,
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

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

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
him to

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

he said.

"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
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

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

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
biting wind

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.
"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

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

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

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
a formal

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
that this

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
dashes and

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,
the two

"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,
which is

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
and fit

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.
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

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
much more

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
that the

Africans might have been using a form of mathematics that they

even discovered yet." Eglash said educators also need to rethink
the way

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
in his

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

African-American culture.

[Patterns in African American Hairstyles

Gloria Gilmer]

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
talk drew

so many people the conference organizers worried about fire code


"What Ron is helping us understand is how mathematics pervades
all that

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
type of

mathematical thinking and creativity that has often been ignored

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 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,

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,

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

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
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."

Book now available from Rutgers University Press: Order by phone

800-446-9323. Order book from

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
into the

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
study of

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
case, the

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
in the

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:

Assistant Professor

Department of Science and Technology Studies

Rensselaer Polytechnic Institute (RPI)

Troy, NY 12180-3590

email: [log in to unmask]


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created and maintained by

Dr. Scott W. Williams

Professor of Mathematics

CONTACT Dr. Williams

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