>Date: Fri, 28 Dec 2001 08:55:23 -0800
>From: Davis Oldham <[log in to unmask]>
>To: MOSAIC <[log in to unmask]>
>Subject: African Fractals
>
>I found this article at the Mathematicians of the African Diaspora
>website (URLs below). Very interesting, especially for those interested
>in innovative ways to teach math to African American students, but also
>just as an antidote to our society's general stereotypes about African
>culture. My favorite passage from the article:
>
>"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."
>
>--
>Davis Oldham English Undergraduate Advising
>[log in to unmask] Box 354330
>Padelford A2B University of Washington
>206-543-2634 Seattle, WA 98195-4330
>FAX: 206-616-9318 http://depts.washington.edu/engl
>
>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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Department of Environmental Health
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