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
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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 thatchedroof 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 geometrythe
geometry of similar shapes repeated on evershrinking scalesin
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 AfricanAmericans 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 selfsimilarity, 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 objectits
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 everdiminishing
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 closeup 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
gridbased 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 bottomup, or
selforganized, plan contrasts with a topdown, 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 theoryit'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 onwhy they had made it that way," he said.
In some cases Eglash found that fractal designs were based purely
on
aestheticsthey simply looked good to the people who used them.
In many
cases, however, Eglash found that stepbystep 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 45degree 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 costbenefit
analysis," said Eglash.
He measured the length of each row of the nonlinear 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 fortunetelling system that relies on a mathematical
operation reminiscent of error checks on contemporary computer
systems.
In traditional Bamana fortunetelling, 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
evenpaired 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 randomnumber 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 systemsone
plus
one equals twoas 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
AfricanAmerican math teachers in the United States on ways to
get
minorities more interested in the subject. Eglash has consulted
with
Gloria Gilmer, a wellrespected AfricanAmerican mathematics
educator
who now runs her own company, MathTech, Inc., based in Milwaukee.
Gilmer suggested that Eglash focus on the geometry of black hairstyles.
Eglash had included some fractal models of cornrow hair styles
in his
book and agreed they presented a good way to connect with contemporary
AfricanAmerican culture.
[Patterns in African American Hairstyles
http://www.math.buffalo.edu/mad/special/gilmergloria_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 itthey 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 cornrow hair style.
Eventually, he'd like to create a CD ROMbased math lab thatcombines
his
African fractal materials with AfricanAmerican 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 AfricanAmerican 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
8004469323. 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 121803590
email: [log in to unmask]
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