Ethan Zuckerman’s online home, since 2003

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”

12 Responses to “African fractals”

  1. Bernie says:

    “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.

  2. Mark says:

    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?

  3. Bernie says:

    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

  4. Mark says:

    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?

  5. Chris says:

    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”? …

  6. TruthTeller says:

    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.

  7. Tj says:

    Well said Truthteller, we should all do our own research and stop relying on what we’ve been told.

  8. yogourt says:

    TruthTeller is right Mark. Ancient greeks were black. (greek statues in museums with caucasoid facial features are fake). As a matter of fact , blacks invented everything. music is just a detail really. Blacks invented western civilisation too. and all the most important scientific discoveries in human history were discovered by black slaves (white slave owners took all the credit for these inventions)…Most ironic of all , the very language you’re speaking (english) was also invented by blacks.

  9. michael carson says:

    i think thats so very true,also when i look at tv history chanel i always see them try to pass off roman mumification as ancient when they know egyptians with the right hand over the left are the real egytian mummies.

  10. reni(g)2 says:

    Damn. This is the reason all people are primitive. “We did this first” arguments stunts growth. Far as I’m concerned people didn’t discover shit. They just sampled the order of life. Calculus is based on this logic. Anyone comfortable with themselves does not need to make enemies unnecessarily. To clarify my point here’s a quote from Eglash.

    “Creating a body of mathematics is about intellectual labor, not some kind of transcendental revelation. There are plenty of important components of European fractal geometry that are missing from the African version,” Eglash said.

    This obviously was said not offend his fellow European mathematicians and scholars.

    On the other hand, Eglash maintained, his work does show that African mathematics is much more complex than previously thought.

    This obviously was said not to offend [any deeper] the African’s who’s work he was so astonished by.

    Sorta like throwing them an extra ishango bone.
    “That’s a pretty decent idea, while not a complete concept we see you though.”

    Africans cheer with joy, “we somebody! :)”

    Modern civilization is fucking sick. Epic.

    Also since transcendence has no point in logical science I guess inspiration is irrelevant too.

  11. reni(g)2 says:

    So are African’s the stalk or the tip in the fractal of life?

  12. milo gardbner says:

    My interest in the topic dates to three time periods:

    1. Had African fractal villages been common before 5,000years ago, the myth of Egyptian base 10 being created from the merger of Egyptian base 2 and base 5 cultures (Upper and Lower Egypt), about the time of the birth of hieroglyphic writing takes on a special meaning. Egyptian scribes would have understood infinite math aspects of its Eye of Horus 1 = 1/2 + 1/4 + 1/16 + 1.32 + 1/64 + …

    This possible historical event saw Pharaoh adopting the binary side of African fractal villages as a metaphor for eternity … further encoded in the Book of the Dead.

    2. Had African fractal villages been first created after contact with Egyptian Old Kingdom lovers of the Eye of Horus myth, another type of historical event would have taken place. Book of the Dead connections to eternity would have been modified by African village to infinite binary family line bisections (from 3,000 BCE to 2050 BCE)

    3. Had African fractal villages been first been created after contact with Egyptian Middle Kingdom, New Kingdom, Demotic and more recent Egyptian cultures (from 2050 BCE to 800 AD) the infinite series aspect would have been modified by considering Egyptian finite arithmetic. Egyptian, Greek and medieval scribes considered infinite binary series family statements … as Archimedes wrote down the first calculus ,,, stated the problem of an area of a parabola by a 1/4 geometric series, one-phase of the Eye of Horus:

    4A/3 = A + A/4 + A/16 + A/64 + …

    and looked for a finite series solution, reported by Archimedes

    4A/3 = A + A/4 + A/12

    with the later not being reported in African villages. Africans seemed not to go beyond family. Abstract mathematical aspects arrive in other ways.

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  2. afrotechie » Blog Archive » With the cheetahs at TEDGlobal 2007 - [...] Ron Eglash, the “ethnomathematician” who sees fractals everywhere he goes in Africa. [...]
  3. African fractals | She Dreams In Digital - [...] http://www.ethanzuckerman.com/blog?p=1469 [...]

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