African fractals
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”
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June 18th, 2007 at 7:02 am
[...] What a speaker line up. You walked out of the room rubbing your head wondering how you are going to process all that knowledge. Next time you bump into one of those idiots who starts asking you questions like, “where is the African Mozart, or where is the African Brunel, implying that Africans do not think send them a copy of Ron Eglash’s study of fractals in African architecture and watch their heads explode as they try to understand just what the hell is going on, and that is just one of many many examples that were shared at this conference. Sending txt messages in Amharic, no problem, want to build a computer that thinks like a brain, easy peasy. [...]
September 23rd, 2007 at 11:45 am
[...] Ron Eglash, the “ethnomathematician” who sees fractals everywhere he goes in Africa. [...]
June 4th, 2008 at 9:04 pm
“Where is the African Mozart?”
A funny thing to bring up since blacks invented Jazz, Blues, Soul, R&B, Rock, Reggae, Rap and even Techno. As far as I’m concerned Motzart was a clown and the notion that blacks are not good musically is utterly laughable.
Whites gave us Country Twang (stolen from blues), and classical. Please… What a ridiculous thing to bring up.
June 10th, 2008 at 7:34 am
Bernie, a funny thing to bring up since whites invented music theory and most instruments. How would those genres exist without the theory or instruments?
July 23rd, 2008 at 3:56 am
Mark:
Do you really think whites invented music theory?? I would love to see reference to support that crazy assertion.
Last I checked blues, jazz and soul employed a completely different scaling system than European classical. A new form of theory evolved from African music, and this is based on pentatonic scaling. Indian music uses another scaling, for example. Whites did not invent either
In addition to this; if you were to take jazz theory in university you will find many important differences between it and classical theory. Modern music is heavily based on African music theory, and the employment of the “blue notes†which derive from this theory.
Your hypothesis is absurd, and I suspect that you know absolutely nothing about music.
- Bernie
August 1st, 2008 at 9:55 am
Music theory is based on musical modes of the ancient Greeks, which were Caucasoid people. The modes are Ionian (Major scale), Aeolian (minor scale), Locrian (diminished scale), and the rest are categorized into the Major type (Lydian and Mixolydian) and the minor type (Dorian and Phrygian) by the Tonic (1st), Mediant (3rd), and Dominant (5th) degrees of the modes.
The Blues scale is an added Diminished 5th or Augmented 4th (enharmonic) note (the blue note) to the Major and minor pentatonic scales, which take particular notes from the Major (Ionian) and minor (Aeolian) scales.
Classical music often used the Harmonic and Melodic minor scales. The former is an augmentation of the 7th (leading tone) degree by one half-step (semitone), and the latter is the same but with an augmented 6th (submediant) degree. Both scales are variations of the minor scale (Aeolian).
Jazz uses chords like 9ths (2nd degree), 11ths (4th degree), 13ths (6th degree), diminished, and half diminished. As far as I know black people formulated the Blues and Jazz genres, but these genres are based on established music theory. Blues and Jazz are tritone heavy (Diminished 5th interval), which throughout much musical history has been shunned for its dissonant and evil sound. The diminished chord is also the tonic of the Locrian (diminished) mode, which was rarely used back then because of its anomalous sound. During the 18th century the Dimished 5th interval was called diabolus in musica. Times change and people become more open to different things. You can hear good examples of this from George Crumb’s Black Angels and Heavy Metal music. Considering all of this, what was it that I didn’t know, Bernie?
October 11th, 2008 at 12:15 pm
I thought the ancient Greek’s were aknowledged to have taken their music theory from the Eygptian / Sumerians? Pythagorus studied in Eygpt, and there is no evidence that he even came up with the material credited to him (emerging as written records at least 600 years after his death, at much the same time as the Christian Bible). Music Theory was later edited by the Christians for their own purposes, and as a result, the origin of modes has been completely lost.
Also Pythagorus was also supposed to have had a thing for the number 4, and squares, which must have horrified him when they found the circle of fifths gave a diminished 7th in relation to this number/ shape (and doubtless led to some swift revision of the theory in order to maintain the integrity of his philosophy). Overall i have to confer with Bernie that Music Theory is most likely African or Middle Eastern in origin … although since the library at Alexandria was burnt down, and the dark ages sort to redefine written history as we know it, maybe we will never know for sure…over to Mark to refute this … (although i will say now that stating that the Greek came up with the theory independantly of the Sumerians, about 2000 years of the facts is unprovable, and highly unlikely, given we still use Sumerian Clocks)…maybe it’s best to aknowledge that it’s not currently possible to be sure of any of the “facts”? …
April 26th, 2009 at 11:11 am
Mark, you are wrong. Ancient Greeks were not Whites. In fact, there is no scientific record every published in history which claims Ancient Greeks were Whites. Ancient Greece an Ancient Kemetian (Ancient Egyptian) colony and was inhabited by the same race of people (Black Africans) as those in Ancient Kemet. Furthermore, the first time White-Europe ever heard of the Ancient Greeks was from the Arabs (who know they were Black African). Most of the history you were taught was created in the mid-1700’s and 1800’s.