Geometry out of Latvia and Africa

Both of my parents were born and grew up in the tiny 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 geometric pattern to cloth, creating a wall hanging, a pillow cover, or some other decorative article.

Example of Latvian needlework featuring a geometric design.

Geometric patterns with a high degree of symmetry are characteristic of much of traditional Latvian folk art (see also Latvian Folk Art Museum and Latvian Needlework).

Latvian weaving (above) and vase (below) with highly symmetric designs.

See "The Investigation of the geometry and composition of Latvian folk art designs" by Silvija Kukle and colleagues.

Examples of traditional Latvian silver (and amber) jewelry, including Namejs rings and bracelets.

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 (Platonic solids) and the 17 wallpaper symmetries are good examples.

My introduction to geometry in African folk art came with the beautifully illustrated book Geometry from Africa: Mathematical and Educational Explorations by mathematician and educator Paulus Gerdes. He provided 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 noted in the first chapter. He described 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.

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.

Example of a Tellem textile pattern.

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

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

Example of a sona sand drawing.

The sand drawing above illustrates a fable: Sambálu, the rabbit (positioned at point B, lower right), discovers a salt mine (point A, center). Immediately, the lion (point C, upper left), the jaguar (point D, upper right), and the hyena (point E, lower right) 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. Note that only from B can you go to point A, without going beyond the line that represents the fence.

To many people, African geometry remains an unknown, rarely glimpsed realm. This book helped dispel some of the mystery, revealing a rich tapestry of geometric designs and concepts.

Originally posted November 29, 1999