What’s Fractal and What Isn’t

Some comments on Ron Eglash’s very interesting TED presentation on the concept of fractals and and the history of their investigation in math and science, with special emphasis on Ron’s investigations into African village design. I hope these notes don’t read as any kind of dismissal of this fascinating work — in fact, I think this kind of dialog is what TED is all about. A fractal-inspired friend asked for my thoughts on this video, so here they are. (The following won’t make a lot of sense unless one views the TED talk before reading further.)

I should start by saying that overall, in his speech “Self-similarity” is the key concept; “fractal geometry” is the mathematical idealization of that concept, and “fractal design” is pretty much not defined other than by implication.

“Symmetry” is another key concept, since it’s one of several fundamental examples of pattern recognition (and pattern recognition may be the most fundamental behavioral capability in systems both alive and arguably not alive). So I find it interesting to consider fractal self-similarity to be a kind of symmetry. This leads to many interesting areas in math and physics and biology that aren’t addressed in the presentation (and which I am notably unqualified to discuss).

In particular, symmetry manifests in many different ways, some of which aren’t immediately obvious. Fractal symmetry could be defined as the arrangement of elements according to patterns of self-similarity, in which case I would agree that he has indeed found examples of fractal symmetry in, for example, the African village layouts. But this is not a formal manifestation of fractal geometry per se. It’s “fractal design,” or what one might call “fractal-like.” This doesn’t diminish its interest, or his insight in noticing this in the aerial photos, or his delight in connecting it to other universal principles from mathematics.

“Self-organizing patterns” is a broader concept than self-similar “fractal” geometry, but he uses it as evidence that fractal geometry is used as a technique. This notion I found a bit loose. Self-Similar and Self-Organizing aren’t necessarily related. Self-Similar is one side-effect of a system of Self-Organizing components, but you can postulate any number of self-organizing systems that don’t involve much (if any) self-similarity, especially since all living things from viruses to human civilizations are self-organizing.

The fact that rural agrarian communities have discovered binary counting is not (to me) surprising. Using it for esoteric spiritual purposes is also quite natural, since virtually anything has been used that way. But I see no connection between that fact and certain African cultures brilliantly applying “self similarity” in their designs and architecture. Calling their scheme for generating random numbers by a fancy mathematical name isn’t required to make it interesting — it’s already a testament to the deep innovative intelligence of these cultures. His conclusion that “computers started in Africa” is, to my mind, cute but not insightful. Computation doth not a computer make, and in European culture “computer” originally just meant “someone who does the computing, with pen and paper.” 

Ron’s argument — that there is no connection between African fractal-inspired creations and the fact that these people often live very close to highly fractal ecologies — seems unnecessary and fallacious. Just because other cultures are different, and also live close to nature, doesn’t mean that closeness to nature therefore has no role in the African sensibilities. It’s faulty logic.

He’s surprised that Africans use systematic approaches to building wind fences, based on experience and experimentation. I’m hoping that his surprise is not based on mere chauvinism. Why would Africans not be as innovative, experimental, systematic, insightful, and creative in their engineering (even with things made of grass) as, say, Europeans? Why, in fact, is it even mildly surprising?

That said, it is certainly a good thing that he’s trying to improve math education in Africa, or anywhere else, and for that he deserves credit and acclaim.

“Self-organization is in the brain.” This is so astonishingly self-evident, that I’m amazed at his determination to abstract this principle as if it were something not utterly and inescapably intrinsic to life itself, and all of biology. Perhaps it’s because he’s coming to sociology and biology from math, and not from the roots of the “life sciences.”

I don’t really find fault with his positions or his presentation, but I had to conclude that he’s getting a lot of mileage out of drawing connections between disciplines while those connections don’t do full justice to the respective disciplines. I think his access to approval and grants owes a lot to this approach, and I certainly don’t blame him for it. But I also don’t see a compelling thesis in what he’s saying. His observations about African art and architecture are very interesting, and I would have gotten a lot more from his presentation if he hadn’t tried to make something more “TED-worthy” than his main points seem to justify.

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