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Fractals: How to Decode the Geometry of Chaos І The Great Courses

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Let's begin by clarifying what's distinctive about fractals by comparing them with familiar, non-fractal shapes and processes. The key point is that fractals behave oddly when you magnify them. Here is a schematic of a fractal shape. It's some kind of wiggly thing. I've drawn a little box around a piece of it to suggest that this is a region that I want to magnify and blow up into the next figure. What appeared here to be sort of a rounded, curvy feature, under magnification, you start to see it has wiggles on it that weren't visible at the earlier resolution. If we blow up one of those rounded features at that resolution, then we see that that rounded feature itself has wiggles on it. So that would be typical of a fractal, to see some kind of theme—in this case, wiggles—repeating at smaller and smaller scales. The structure is inexhaustible. Finer structure is revealed no matter how far down you look.

In contrast, Euclidean shapes, by which I mean things that you would have studied in high school geometry: circles, straight lines, squares, spheres, things like that. Imagine them in their most perfect form. I'm not talking about a real circle, say, with a loop of string. I mean an abstract, Platonic circle, a perfectly thin line formed into a circle. So think of the perfect circle or the perfect sphere. A shape like that, if you look at that closely enough, if you imagine zooming in with a microscope on a shape like that, it starts to look smoother and smoother, and more and more featureless. That's what I'm trying to indicate here in this lower panel, a typical non-fractal shape—I'm using the circle as illustration. Again, if I blow up the curved region, it now starts to look straighter but also smoother. Then, if I blow up that region, pretty soon it really starts to look featureless. You will just eventually see a line with nothing more to see. The show's over. That's the first property: that fractals have structure that goes all the way down.

The second defining feature of fractals, then, is that the small features resemble the larger ones. In that sense, the structure can be said to be "self-similar." For an edible example--it's a head of Romanesco broccoli. It's this unbelievable fractal vegetable. Look at this thing. You can see it has all kinds of florets coming off of it. It has subcones, smaller cones, and those cones have cones. In fact, they're not really cones. The way to think of it is that this whole shape is kind of a fractal version of a cone. It's a very bumpy cone, and each of the florets is a mini-Romanesco. If we were to zoom in on this mini-structure, you would see it's pretty much a faithful copy of the original larger structure. What is even more stunning is that that mini-Romanesco has little florets on it. If you were to zoom in on one of those, again, [you would see] the same structure.

It's incredible, and as I say, it illustrates the scheme of self-similarity. There are tiny, tiny florets near the top, bigger ones at the bottom. The structure is architecturally identical all the way around and also at different levels.

Strictly speaking, any real fractal, like this one, as opposed to this idealized Platonic—I shouldn't say Platonic because Plato never thought about fractals, but you know what I mean—idealized, perfect mathematical fractals. Those are truly inexhaustible and self-similar, whereas real things like this are not. They couldn't possibly be for all kinds of reasons. Real objects, like the Romanesco, are only approximately fractal in various ways.

An interesting way to think about self-similarity, as I mentioned briefly in the last lecture, is as a kind of symmetry. I didn't really say what a symmetry means. I think you probably have an instinct for what a symmetry means, but the way a mathematician would put it is that an object has a symmetry if it stays the same despite a change. It's sort of a cool way to think of it. Something that stays the same despite a change has a symmetry. For example, a sphere clearly has some kind of symmetry. It has rotational symmetry because if I rotate a sphere, it stays the same. Our bodies have approximate mirror symmetry, at least on the outside, in that our left side looks sort of like our right side, approximately. Not on the inside. Likewise, a fractal has a symmetry, except it's not a symmetry through a mirror, and it's not a symmetry of rotation. It's a symmetry under magnification. Small things look like the big thing. That is sometimes called the scale symmetry, a symmetry under a change of scale. It stays the same if you magnify it.

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