Geometry Out of Africa

Both of my parents were born and grew up in the little Baltic country of Latvia. I remember, as a young child in northern Ontario, intently watching my father painstakingly color in tiny squares of a grid to create a symmetric design. Using yarn and needle, my mother would then transfer that highly geometric pattern to cloth, creating a wall hanging, a pillow cover, or some other decorative article.

Geometric patterns with a high degree of symmetry are characteristic of much of traditional Latvian folk art. See http://www.webwm.com/w/h/frame0.htm for some striking examples of Latvian cross-stitch design.

I have long been intrigued by the geometric designs created by various cultures, both past and present, throughout the world. I'm impressed by the variety of such patterns. At the same time, there are wonderful similarities among designs in different parts of the world, even when there's no evidence of direct contact between the groups. That's a consequence of the underlying mathematics. Given a set of rules, there are many instances in which the number of possibilities is finite. The five regular polyhedra and the 17 wallpaper symmetries are good examples.

Two recent, beautifully illustrated books have introduced me to African geometry. To many people, that's an unknown, rarely glimpsed realm. The books help dispel some of the mystery, revealing a rich tapestry of geometric designs and concepts.

In Geometry from Africa, mathematician Paulus Gerdes of the Universidade Pedagógica in Maputo, Mozambique, provides a fascinating guided tour of geometric ideas encoded in carved patterns, woven designs, sand drawings, and other products created by people south of the Sahara.
"The development of geometrical thinking starts early in African history," Gerdes notes in the first chapter. He describes a variety of geometrically decorated artifacts, from rock paintings and engravings to decorated pots and textiles, some of which are more than 2,000 years old.

Example of a Tellem textile pattern. Symmetry is a prominent trait of many of these patterns. Textiles woven by the Tellem people in an area that is now in the Republic of Mali, for example, feature intricate combinations of white and indigo cotton threads to produce symmetric strip and planar patterns of various types.

Geometrical knowledge also played a crucial role in the shaping of articles of clothing, such as tunics, from the woven material, Gerdes says. A Tellem tailor would work with several different kinds of knots and stitches, for instance.

In succeeding chapters, Gerdes shows how one can discover or derive the Pythagorean theorem from African designs, explores the role of symmetry (with suggestions for home and classroom crafts projects), and focuses on the geometry of the sona sand drawings among the Chokwe in south-central Africa.

Note that only from B can one go to point A, without going beyond the line that represents the fence. The following sand drawing illustrates a fable: Sambálu, the rabbit (positioned at point B), discovers a salt mine (point A). Immediately, the lion (point C), the jaguar (point D), and the hyena (point E) demand possession, asserting the rights of the strong. The rabbit, affirming the inviolable rights of the weak, then quickly makes a fence to isolate the mine from all usurpers.

I have found that a handy, fun way to explore symmetric designs and reproduce examples from African art and crafts is to use the KaleidoMania! software package, developed by Kevin D. Lee of St. Paul, Minn. (See Plane Patterns, March 15, 1999.)

In African Fractals, Ron Eglash of the Rensselaer Polytechnic Institute in Troy, N.Y., investigates links between fractal geometry and traditional African designs, emphasizing their self-similar nature.

"Fractals can be seen in many of the swirling patterns produced by computer graphics, and they have become an important new tool for modeling in biology, geology, and other natural sciences," Eglash writes.

"While fractal geometry can indeed 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," he contends.

Eglash provides a gentle introduction to fractal geometry and tries to demonstrate how various aspects of fractals are expressed in African culture.

A Fulani wedding blanket from Mali.
Courtesy of Ron Eglash. 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."

Eglash sees similar recurring patterns on different scales in ivory sculptures, palace and village layouts, and braided hair designs. His argument for the ubiquity of fractals in African design and thought is provocative though not, I think, entirely convincing. Nonetheless, his book calls attention to intriguing facets of African culture.

Both African Fractals and Geometry from Africa vividly display the amazing variety of inventive designs and patterns that can emerge from simple geometric ideas and everyday activities.

"In any region or country, the diversity of activities in which geometrical considerations are involved may be extremely great," Gerdes says.

In Kenya, he remarks, the "Turkana and the el-Molo weave semi-spherical basket fish traps; Wata woodcarvers produce containers, fashioned from raw logs, decorating them with strip patterns; Kamba women decorate gourds; Luyia on the eastern shore of Lake Victoria build and decorate boats; Luyia basket makers weave conical quail baskets with a hexagonally woven bottom; Meru woodcarvers on the eastern slopes of Mount Kenya shape symmetrical spoons and Meru drummers use log-drums painted with symmetrical designs; the Pokomo [fashion] dugout canoes; Boran and Taita women weave mats out of strips; and Sakuye weave hats with a decorative motif with fourfold symmetry."

Geometric exploration goes hand in hand with artistic design.

References:

Eglash, R. 1999. African Fractals: Modern Computing and Indigenous Design. New Brunswick, N.J.: Rutgers University Press.

Gerdes, P. 1999. Geometry from Africa: Mathematical and Educational Explorations. Washington, D.C.: Mathematical Association of America.

Check out Ron Eglash's African fractal Web site at http://www.rpi.edu/~eglash/eglash.dir/afractal.htm.

Information about KaleidoMania! is available at http://www.keypress.com/x6173.xml.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.