Ron Eglash has an odd job title – he’s an ethnomathematician whose has built his career studying fractals in African architecture. He starts by explaining fractals, walking us through George Cantor’s subdivision of lines to infinity by eliminating the middle third of each line. The result, on an infinite line divided into a set of points larger than infinity, literally drove Cantor mad. But other mathematicians built additive fractals like the Koch curve. These shapes were all self-similar – they look similar in large and small scales. And if you try to measure their surfaces, you’ll discover that they’re infinite in length.
These curves were unpopular with the mathematical establishment – they were termed “pathological curves”, and largely ignored for a hundred years, until Benoit Mandlebrot observed that fractals were extremely common in nature.
Egland’s work began by analyzing photos of African villages and seeing fractal patterns. With a Fulbright grant, he began travelling around Africa and asking people why people build the way they do. He found palaces made from fractile organizations of rectangles, villages of self-similar circular compounds, Nigerian villages apparently built from self-similar circles. The only straight lines in this village were associated with the altar used for annual sacrifice. These patterns have religious significance in many cases – a recursive stack of calabashes is topped by a tiny calabash, which contains a woman’s soul and is smashed when she dies.
Egland is clearly used to skepticism about his work. He asks and answers some of the questions he most often faces:
– Aren’t these patterns just indigenous to all architecture?
He’s studied indigenous architecture in native American and South Pacific culture as well, and while native American architecture has circular and four-fold symmetry, it lacks self-similarity.
– Aren’t you ignoring the diversity of African cultures?
There’s self-similarity throughout African architecture, but there are different algorithms in different places.
– Does this represent real math knowledge, or is it just intuitive?
Many of these patterns are algorithmic. Egland walks us through the production of woven grass mats – they’re woven loosely on the bottom, where the wind is less fierce, and more tightly at the top. There’s a clear, logarithmic relationship between efficiency and effort – the African mathemeticians building these mats are doing a very good job.
Egland looks closely at Bamana sand divination, a complex system of making patterns of dashes. The system, he learned, works as a psuedo-random number generator using modula 2 addition. He argues that this system migrated into Spain from the Islamic world, becoming “geomancy”… which influenced Leibinitz, and can be traced to the birth of the digital computer. Responding to Brian Eno’s complaint that there’s not enough Africa in computers, he quipps, “I don’t think there’s enough African history in Brian Eno”