Sophisticated Mathematics Behind African Village Designs / Fractal patterns use repetition on large, small scale

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 in Long Island, N.Y. "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 vs. a panoramic symmetry instead of the common right vs. 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 American Indian 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.

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, N.Y., Eglash has written about the revelation in the 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.

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 that combines his African fractal materials with African American hair styles and other design elements such as quilts.