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January 2002

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From:
Sam Anderson <[log in to unmask]>
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Science for the People Discussion List <[log in to unmask]>
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Wed, 9 Jan 2002 12:00:27 -0800
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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


RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRR
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
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.


[Patterns in African American Hairstyles

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

Gloria Gilmer http://www.math.buffalo.edu/mad/PEEPS/gilmer_gloria.html]



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 thatcombines
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."


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
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.



For more about the book see Dr. Eglash's webpage at

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


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."


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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