Ambigrams, Explosions, and Fractals

Today I started reading Martin Gardner’s collection of essays, The Night is Large, and the first chapter is about symmetry. He points out the pervasive symmetry of our universe, found in fish and crystals and human beings, and wonders why this is so. Mathematics also obeys all kinds of symmetries. But why should the universe be like this? His answer is that mathematics arose from observing the universe. That’s true, but I don’t think it really answers the question. We start out with some simple counting and measuring, but then the shocking thing is that it all works so well, so consistently. We never get to the bottom of it. As he says later, “God is a geometer.” But isn’t it interesting that we are, too? That is the insight expressed by Jacques Maritain, following Thomas Aquinas, when he says that our intelligence is made capable of knowledge; its nature is to know. It possesses an uncanny ability to understand.

Gardner also discusses breaking symmetry. This is when a system of greater symmetry changes to a system of lesser symmetry–for instance, a drop of milk falling into a full bowl. The drop is symmetrical, and the bowl is circular, so they both possess infinite symmetry: they can be reflected by any great circle or diameter, respectively. Contrast this to, say, a five-point star, which can be reflected in only five ways. A circle has more symmetry than any other two-dimensional figure, and a sphere more than any three-dimensional one. But when the drop of milk hits the surface, it creates a crown-like shape with twenty-four points. The system goes from infinite symmetry to twenty-four-fold symmetry.

Scientists imagine a similar loss of symmetry at the Big Bang. Somehow the universe went from a tiny, compact pinpoint to what we see today. The strange thing is how chunky everything is. Matter is organized into galaxies, with vast empty space between them. Within the galaxies are solar systems, again with empty space intervening. Instead of a regular consistency, we see chunks, but also extreme order. Perhaps symmetry breaking can explain how this came about.

I’ve heard similar things before, but it’s always seemed like hand-waving to me. If the universe started symmetrical, how did expansion introduce any asymmetries? This only seems possible if you admit initial impuries or asymmetries. But then you haven’t explained anything, because why should things have started out asymmetrical? That seems inelegant. But upon reading Gardner’s essay, I thought that perhaps quantum variation could be the answer. Given the erratic movement of atomic particles–or to be more precise, their probabilistic location–you would get asymmetries as soon as time starts moving. And those tiny asymmetries could blossom into the chunkiness we see today.

Gardner remarks that this is one instance of chaos theory. People love to throw around that word, and it was especially popular in the 90s (when this essay was published), but sometimes it’s hard to tell what it means. As far as I’ve been able to gather, chaos theory studies systems in which tiny details can produce unexpected large-scale results. (Sometimes you hear the opposite: that chaos theory studies how order arises from chaos. But I think these people are just making it up.) The classic example is a butterfly flapping its wings and causing a hurricane. The details are significant and interact in irreducible ways, so you cannot simplify the picture with formulae or rules. All you can do is work out the details algorithmically and see what you get. In the case of the Big Bang, the idea is that you get large-scale effects (galaxies) from small-scale details (quantum fluctations).

Gardner also connects chaos theory to fractals. Now I hear this all the time, so that it seems I never get one without the other, but I’ve never understood their relation. A fractal is just some pattern that is the same at any level of magnification. It is a pattern that contains itself repeated again and again in smaller and smaller copies. A Sierpinski triangle is a simple example; a Mandelbrot set a more complex one. But what does this have to do with chaos theory? Are all fractals chaotic? You could say that a Mandelbrot set is kind of chaotic, but not a Sierpinski triangle. Do chaotic systems act like fractals? The opposite actually seems true: if in chaotic systems the details afford no way to deduce the big picture, then they are precisely not fractals, because there the big picture is always a perfect mirror of the details, and vice versa. So I don’t get it. Why do people connect fractals with chaos theory? What is their relation?

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