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 by Gloria Gilmer]
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: firstname.lastname@example.org
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