Wednesday, July 13, 2016

Fractal Geometry in African Villages: Lessons from an Outsider

In the 1960s, the Italian architecture firm Superstudio proposed the eradication of all architectural difference under a ubiquitous grid that they called a "Model for Total Urbanization." This is one of their posters.

"The exception proves the rule." How many times have we heard that without really understanding it? It was Cicero in ancient Rome who first said it, and what he implied was that exceptions presuppose the existence of a norm (to which they are the exception). We always look to outsiders, the exceptions to the rule, to help define ourselves as normal. Outsiders have always defined insiders. An old military adage says that you need a great enemy to create a great army. For ancient Greece, it was the Spartans and Macedonians who provided the "savage" exception that proved the superiority of the civilized Athenian polis, or city-state. For the West, a monstrous "other" occupying the rest of the world provided evidence of the dubious superiority of its own worldview.

The Western gridded "ideal city," as imagined by Fra Carnevale

In popular imagination, the exceptions to Western civilization still represent everything uncivilized, chaotic and savage in the world. When Edward Said and Claude Levi-Strauss came along in the '60s and pointed out that, hey, just because other cultures are different doesn't mean that they are devoid of their own internal logic and structure, we began to open our eyes to the fact that maybe, in the end, we had something to learn from "them." We'd gotten so blind to the veil we've fabricated as a frontier between city/country, order/chaos, insider/outsider, West/Rest, that we'd lost sight of the fact that ours is just one of many ways of being in the world, and that the veil is an illusion.

Somewhere in America

Case in point: the gridded and bordered modern city that sprang up in Europe towards the end of the Roman Empire - with its straight lines and corners retained by a circumscribing wall - quickly became the template for Western life. They were initially walled defensive positions (called Oppida), but the Romans soon became aware of their greater (in the long run) symbolic significance: The walled city is a locus of power. Those outside the walls were the medieval "wildmen," or savages beneath consideration. It established a clear - if fictional - boundary between the order of Man and the chaos of Nature.

The Universe is Euclidean, its rigid geometry tells us. The very linearity of modern cities came to represent the mythic Western advancement from barbarian to citizen, from chaos to order. The more squared-off the space, the more civilized its occupant. The barbarians lived outside the polis amid the chaos of nature, and "Nature," as Katherine Hepburn reminded Bogart in African Queen, "is what we are put in this world to rise above." Consequently, linear gridded space became the Western standard.

Hardly a coincidence that Star Trek automatons, "The Borg," occupy a cube.

By the 15th Century, Linear Perspective gained favor as a way of reifying what had already become a pervasively geometric worldview, literally set in stone by the Romans.  Perspective appealed to rich patrons because it backed up their notions of the hierarchy of social power (think majestic cathedrals with impossibly soaring trompe l'oeil ceilings inspiring awe among the plebs). Artists of Europe clamored to learn the rules of Linear Perspective as a way of codifying a "civilized" worldview, which flattered their clients by portraying them as higher-order citizens. Even if anyone had been aware of another way of seeing the world, they wouldn't have cared for it. West is best, and all that.

Every day I walk the streets of New York City, I'm aware that to get across town means zig-zagging at right angles across an artificially imposed grid that by its very inorganicness was designed to position the works of man as superior to those of nature.

What would an alternative even look like? We're so used to what we've got that it's hard to picture it, but we don't have to: When the fractal geometry of African villages was "discovered" by the West, it provided the exception that exposed the tenuousness of the norm we've come to accept. Certain villages in Africa (such as Tiébelé on the Ghanian border) have been organizing themselves for centuries according to mathematical principles that were only discovered in the West in the 1900s.

The classic Mandelbrot fractal set

Whereas the Western gridded polis denies Nature by proposing space as a system of stackable finite blocks, fractal architecture suggests an unfolding of space according to principles of organic growth. Each unit of fractal geometry relates intrinsically to its neighbor, regardless of scale. There is no frontier, no hierarchy of space as in the West.

Fractal sets are everywhere in Nature

Fractal architecture could never produce a Versailles, for example, that so self-consciously set itself apart from the populace that surrounded it. The "self-similarity" of fractal geometric modules would preclude it. Western architecture relies on the grid as an exclusionary device - you are either inside or outside the square - but fractal architecture seems to suggest that we all have the same potentiality.

I always worry that I'm writing too much in blog posts, that they're too long (because who the hell reads blog posts?), so I won't be getting into the weeds about what fractals are, except to say that they began as an outlier set of mathematical rules once considered to be useless oddities - Euclid's outsiders.

The fractal architecture of Ba-ila village, in Southern Zambia

The realization that whole communities live according to a spatial map that is entirely different to ours was an eye-opener. Watch this TED talk (and buy the book) by this fascinating mathematician who traveled Africa on a Fulbright scholarship, standing on rooftops and recording the fractal geometry he saw all around him. And next time you're sitting at a traffic light, imagine a world that has no right angles.

Ron Eglash mapped the fractal set at the heart of community life for Ba-ila villagers

Tiébelé, Ghana


  1. Fascinating Alan! ( as I type this in my rectangular box provided for text) Westerners definitely equate the grid with higher civilization.

    1. Thanks Theresa! Now I know at least one person read it and I wasn't up until 1:30AM writing for nothing! ;)

  2. Excellent! Fractals are fascinating and daunting but I appreciate them on an aesthetic level.

  3. I read these posts, everyone, and love them. But I have a bone to pick ... It was the environment that presupposed the use of the linear grid in Europe as elsewhere. This article sneakily suggests that a more organic organizational style was not tried, stress tested and found wanting. The people who laid the groundwork for western civilization did not come to rely on hierarchal, granular organization because they lacked the ability to muse, dream or attend TED talks and contemplate African villages, but because they wanted to survive. Whatever its faults, the grid afforded our ancestors this privilege - the organic style could not withstand the stresses they encountered.
    Great post as always- awesome pics.

    1. Firstly, thank you so much for reading all my rantings. And secondly, for taking the time to give me your thoughts.

      I should point out that I've never been one to let the facts get in the way of my point. Given, also, that I'm a layman rattling a hornet's nest and compressing the entire story of urbanization into a few paragraphs, then it's not surprising that there are some gaping holes.

      I totally agree with you that medieval Europe formed walled enclaves from a survival standpoint. Even before Julius Caesar attacked and wiped them out, they were threatened on all fronts by swarthy invaders like the Huns, moors etc. So they built walls to survive, with everything inside those walls a hastily cobbled mishmash (and not initially a grid at all).

      I guess I bent the truth to make the point that I believe there were other forces at work too. Perhaps not primarily, but they were there nonetheless. For example, the gates of Oppida were bigger and more imposing than required for pure defense, implying (to my amateur brain at least) that those who did the planning were aware that walled cities served a symbolic purpose as well as a practical one.

      Their primary purpose, as you say, was survival, but their secondary one was to gather citizens under a collective identity. Just as the encircling colonnades of St. Peter's basilica were deliberately designed to give the impression that the assembled throng were held within the protective and symbolic embrace of a unified ideal (Christianity), the imposing fortifications of early cities gave their citizens a sense of belonging and identity. Cities are no longer defensive in nature of course, but they still provide this symbolic role in addition to their practical one. [I'm generalizing enormously again, and I welcome your thoughts].

      I've always felt that the physical manifestation of any human expression (whether it's Art or City Planning) reveals something of our deepest wiring. I can't help feeling that ever since we huddled for survival around fires on the Savannah, divisions of Insider versus Outsider have been orchestrating our every action at a fundamental level.

      I should add that I don't believe that African communities living within "fractally designed" living spaces are any less likely to see divisions between Us and Them than the rest of us. As human beings I believe that figuring out who's on your side and who's not, and where the line lies between the two, is our most deeply ingrained survival tactic.

  4. I should probably also add that I'm not advocating a kind of Social Darwinism; that there's an inevitability to excluding certain Others from our worldview. For my part, I think that while it's all but impossible to truly get inside someone else's head, it should not stop us trying.

  5. Hey Alan, I discovered your blog a couple of weeks ago and have been working my way through the archives. The one on reflections in water was the one that got me hooked. It's a treatise. There's no way a treatise can be three paragraphs long. I mean: keep the long posts coming.

    As for the schools of thought and imposing one world view on another as better - oh, well - we are only human and that is what we do, it is one of our major flaws or qualities. Maybe this is a subtext I read that isn't even there, but anyway. What comes to my mind on this subject is the book "Guns, Germs and Steel" by Jared Diamond. Whether you are talking about "cargo" or about the supremacy of grids over fractals I think the best explanation I heard is in there. Spoiler: chance, reinforced by feedback loops, some more chance.

    I live in Sao Paulo, by the way, an immense place that has a loose relationship with the grid. It was insane to navigate before GPS and maps on your phone. Organic is fine, but concrete organic with no skyline and curving streets that follow no rhyme nor reason, not so much. :)

  6. Hi Leticia,

    Thank you for your comments.

    I think #Etype (in the blog comments) was making a similar point, that it was essentially trial-and-error that led to the adoption of our current model. I do agree. To a point. But I think that a defensive (or survival) strategy was not the only thing at play in the development and look of the urban landscape. If that had been the case, then cities would all have looked like the star-shaped fortress of Palmanova in Northern Italy.

    True, Palmanova has a central gathering point and wide avenues in common with many post-medieval towns, but that is where the similarity ends - at least with respect to the grid. Its avenues are radially designed for easy access to the defensive boundary. If survivalist trial-and-error thinking was at the root of city planning, then surely all cities would be star-shaped to some degree. [And I'm aware that one could argue that Paris, the first truly modern city under Hausmann, is star-shaped around the Etoile - which of course means star].

    What I'm trying to imply in the post is that yes, defensive reasoning was a major factor, but that it dovetailed with something deeper: the need for mankind to create artificial frontiers. These are mental constructs, that in the West are tied more to Imperial thinking inherited from the Greeks than they are to a purely practical impulse to protect city-dwellers from attack from without.

    I'm thinking for example of the American "Frontier Myth," a purely abstract concept of an Us versus Them boundary between civilization within and savagery without. This Frontier mythology has been widely cited as being the quintessential foundational story at the heart of modern American identity. This wasn't a defensive posture but it's polar opposite; an Imperial one that positioned Insiders as superior to Outsiders. And it was adopted wholesale from the Greeks.

    The poet Shelley said that "we are all Greeks," and we're it not for their worldview we would all be "savages and idolaters." Even the Romantics, though they sided with the "wildman" outsiders of the world, acceded to the same polar division of the world.

    You're correct that the subtext of my post is that it's a very human trait. I should re-state that it's not one I stand behind, as it's caused untold suffering (including what is perhaps the greatest atrocity in modern history, the genocide of the Native Americans).

    I just find it fascinating that this thinking is at the heart of everything we do.

  7. Article on African Fractals: