African Fractals: The Mathematics in Villages, Hairstyles and Textiles

In the 1980s, an American graduate student named Ron Eglash was looking at aerial photographs of African villages when he noticed something unexpected. In a photograph of a settlement of the Ba-ila people of southern Zambia, the village formed a large ring of homes. At the back of the ring was the chief's family enclosure, itself a smaller ring. Within that ring was the chief's own home, a ring within a ring within a ring. The same pattern appeared at different scales. It looked like a fractal.

What is a fractal?

A fractal is a shape in which a similar pattern repeats at different scales: if you zoom in on part of it, you see a smaller version of the whole. Fractals are found throughout nature, in ferns, trees, coastlines, river networks, snowflakes and the branching of our blood vessels and lungs.

The mathematics of fractals was developed in the 20th century, especially by the mathematician Benoît Mandelbrot, who coined the word "fractal" in 1975. Computers made it possible to create fractals by repeating simple rules many times, a process called iteration or recursion.

Fractals in African design

After his discovery in Zambia, Eglash spent time in several African countries, measuring settlement layouts, studying textiles, sculptures and hairstyles, and interviewing builders and artists. In his book African Fractals (1999), he argued that fractal patterns appear widely in African design, often intentionally. Among his examples:

• Settlement layouts. Besides the Ba-ila village, Eglash studied the palace of the chief of Logone-Birni in Cameroon, where enclosures are built within enclosures, each leading to a more sacred or important space, and the circular compounds of the Mokoulek and others.
• Architecture. Some buildings repeat shapes at several scales: walls with decorations echoing their own outline, or roofs built in layers.
• Hairstyles. Cornrow and braiding patterns that repeat and shrink in spirals or rows.
• Textiles and carvings. Kente and other woven patterns, Ethiopian crosses, and wooden sculptures with recursive designs.
• Divination. In some sand divination systems, such as those of the Bamana, symbols are generated by repeating a set of rules, similar to recursive computation. Eglash noted links between these systems and Ifá divination of the Yoruba, described in our story on Ifá, which also uses a binary code of marks.

Eglash suggested that these designs reflect deep ideas in African cultures, such as the belief that things at a small scale reflect things at a large scale, or the importance of repetition and renewal.

Not an accident?

One of Eglash's key claims is that, in many cases, the fractal patterns were intentional. He reported that builders and artists sometimes explained their designs in terms that showed awareness of scaling: for example, describing how a pattern grows or repeats. He was careful to say that African builders did not have the modern mathematical theory of fractals, but that they used fractal ideas in their own ways.

Debate and criticism

Eglash's work has been widely praised, but it has also faced questions:

• Some critics argue that it is easy to see patterns where we look for them, and that not all the examples are truly fractal in the mathematical sense.
• Others note that fractal-like designs also appear in other parts of the world, from Hindu temples to Gothic cathedrals, so they are not uniquely African.
• Some anthropologists worry about generalising across a whole continent.

Eglash himself has emphasised that he is not claiming Africans are "naturally" mathematical, or that fractals are found everywhere in Africa. His point is that African design traditions include sophisticated mathematical thinking that has often been overlooked.

Mathematics education

Eglash's work has inspired educational projects. He and his colleagues developed "Culturally Situated Design Tools", online software that lets students explore mathematics through designs such as cornrows, beadwork and African textiles. The idea is that students may engage more with mathematics when they see it in their own cultures.

He builds on a longer tradition. Mathematics educator Claudia Zaslavsky's 1973 book Africa Counts explored counting systems, games and patterns in African cultures, and the Mozambican mathematician Paulus Gerdes studied the geometry of basket weaving and the sona sand drawings of the Chokwe people of Angola.

Why it matters

The idea of African fractals challenges an old prejudice: that mathematics was invented only in Europe and Asia, and that African cultures had no mathematical thinking. Whatever one thinks of every example, the work of Eglash, Zaslavsky, Gerdes and others shows that mathematical ideas, of pattern, symmetry, scale and recursion, are part of the designs of African villages, hairstyles, textiles and divination. It invites us to look again at familiar things and see the mathematics hidden in plain sight.

Further reading

• Ron Eglash, African Fractals: Modern Computing and Indigenous Design (Rutgers University Press, 1999)
• Ron Eglash, "The fractals at the heart of African designs" (TED Global, 2007)
• Claudia Zaslavsky, Africa Counts: Number and Pattern in African Cultures (3rd ed., Lawrence Hill Books, 1999)
• Paulus Gerdes, Geometry from Africa: Mathematical and Educational Explorations (Mathematical Association of America, 1999)