COLUMBUS, Ohio -- In everything from braided hairstyles to the design of housing settlements, the geometric structures known as fractals permeate African culture.
In a new book, an Ohio State University scholar examines the unlikely pairing of this mathematical concept and the culture and art of Africa.
"While fractal geometry is often used in high-tech science, its patterns are surprisingly common in traditional African designs," said Ron Eglash, senior lecturer in comparative studies in the humanities. Eglash is author of African Fractals: Modern Computing and Indigenous Design (Rutgers University Press, 1999).
Eglash said his work suggests that African mathematics is more complex than previously thought. He also says using African fractals in U.S. classrooms may boost interest in math among students, particularly African Americans. He has developed a Web page to help teachers use fractal geometry in the classroom. (http://www.cohums.ohio-state.edu/comp/eglash.dir/afractal.htm)
Fractals are geometric patterns that repeat on ever-shrinking scales. Many natural objects, like ferns, tree branches, and lung bronchial systems are shaped like fractals. Fractals can also 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.
In African Fractals, Eglash discusses fractal patterns that appear in widespread components of indigenous African culture, from braided hairstyles and kente cloth to counting systems and the design of homes and settlements.
Other researchers have studied bits and pieces of African mathematics in areas such as art, architecture, and religious practices, but Eglash said this is the first attempt to describe the common theme of fractal geometry among several different African cultures.
"There is no singular 'reason' why Africans use fractals, any more than a singular reason why Americans like rock music," Eglash noted in his book. "Such enormous cultural practices just cover too much social terrain."
He began this research in the 1980s when he noticed the striking fractal patterns in aerial photos of African settlements: circles of circular houses, rectangles inside rectangles, and streets branching like trees. Eglash confirmed his visual intuition by calculating the geometry of the arrangements in the photos -- they were indeed fractal.
At first he thought that only unconscious social dynamics were responsible. Later, however, he received a Fulbright grant for field work in west and central Africa, and found during his travels that fractals were a deliberate part of many African cultures' artistic expressions and counting systems, too.
In one chapter, Eglash described an ivory hatpin from the Democratic Republic of the Congo that is decorated with carvings of faces. The faces alternate direction and are arranged in rows that shrink progressively toward the end of the pin. Eglash determined that the design matches a fractal-like sequence of squares where the length of the line that bisects one square determines the length of the side of the following square.
In another chapter, he illustrated how divination priests of the Bamana people in Dakar, Senegal, calculate fortunes using a recursively generated binary code. Eglash explained that diviners use base-two arithmetic, just like the ones and zeros in digital circuits, and bring each output of the arithmetic procedure back in as the next input. This produces a string of symbols that the priests then interpret as the client's fortune. This technique is similar to a kind of random number generation in computing, Eglash said, and the Bamana's technique can produce over 65,000 numbers before the sequence repeats.
While fractals can be found in cultures on other continents -- Celtic knots are one example -- fractals are particularly prevalent in Africa.
Eglash pointed out that this does not mean African mathematics is more complex than Western mathematics, or that African cultures are "closer to nature" because fractals are present in nature -- these sweeping conclusions are just plain incorrect, he said.
"Creating a body of mathematics is about intellectual labor, not some kind of transcendental revelation. There are plenty of important components of European fractal geometry that are missing from the African version," Eglash said.
On the other hand, Eglash maintained, his work does show that African mathematics is much more complex than previously thought.
Knowing fractal geometry enables scientists to model complex processes in biology, chemistry, and geography on computer. It also helps generate realistic computer images of natural features such as rugged terrain or tangled tree branches. Still, most schools teach classical geometry -- the study of simple shapes like circles or squares -- not fractal geometry, Eglash said.
In studies of African-American students' poor math performance, researchers have suggested that computer-based teaching methods or the presentation of real-world math applications might encourage students to learn more. According to Eglash, the use of African fractals in math classes could combine both solutions.
Eglash's Web page contains links for obtaining both commercial products related to African fractals as well as free materials. For example, he has just written a program that allows students who visit the page to interact with a computer simulation of the patterns in cornrow hairstyles.
Even without computers, Eglash said, students can still learn about Fractals using common school supplies. In his book he explained how to fold a piece of paper to demonstrate the geometry of a traditional African tie-dye method, for example.
The Web page also has some materials that teachers can print out and use with their students. One lesson shows how students can derive fractal equations from their own photos of cornrow braid patterns using a protractor and some simple calculations.
Eglash cautioned that African-American students won't automatically be interested in fractals simply because they appear in African designs. He suggests that the most powerful potential of African fractal geometry comes from its opposition to biological determinism -- the assumption that math ability is genetically determined.
"Just think how often students are told by parents, 'Oh, don't worry about your bad scores, I was no good at math either.'" Such myths have their most devastating impact on minority children, Eglash said, but he makes a distinction between this kind of argument and more simplistic models of identity or self-esteem.
For instance, when Eglash introduced fractal geometry to a class of 12-year-old African-American students at a 1996 urban youth conference, the students used traditional African fractals only as inspiration for creating new designs of their own.
"The best thing we can do is give students the tools for constructing their own identities -- powerful new tools like African fractals -- and then just get out of the way," Eglash said.
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