the klog

So I was reading this entry on the plog, where Paul talks about how Pollock's paintings aren't as easy as they look. I then found an article (forget where) about how mathematicians demonstrated that Pollock's paintings actually have fractal structure, the kind of structure you find all over the place in nature. Not only that, but his later paintings were more complex in fractal structure than his earlier work. Here is one of his most famous paintings: Blue Poles: Number 11, 1952. Here is a Discover Magazine article on the findings. And here, thanks to Paul, is a more detailed article on the topic. Good stuff.

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Fractal dimension

And now for a really big tangent. I also had Paul explain to me the concept of "fractal dimension", which I found ridiculously cool. Well, Paul first showed me this very mathy page about it, and I was just thoroughly lost and confused. He eventually managed to make me understand the concept, though, if not that page. :) You might just want to read this rather non-mathy explanation of fractal dimension, complete with pictures, but below is my attempt at an explanation of the math...

So take a normal 2-dimensional object, like a square or a triangle. If you double its size along each axis, how much more area does it have? It's 2² = 4 times as big. Now, if you take a cube, a sphere, or any other typical 3-dimensional object, and you double its size along each axis, it has 2³ = 8 times as much volume. (If you double the length of a line segment, it will have 2¹ = 2 as much length.) In a sense, this is what we really mean when we talk about dimension.

Now take a Sierpinksi Triangle. (As you may recall, it is formed by taking an equilateral triangle, "cutting out" the upside-down triangle formed by connecting its midpoints, and then cutting out the middles of each of the 3 remaining sub-triangles, iterating an infinite number of times.) Here's the weird thing. If you double its size along each dimension, the area doesn't quite quadruple. Notice how the new object is basically 3 of the original ones placed together!* So for the Sierpinksi Triangle, the new area is 3 times the old one, which means that its dimension can be found with the equation 2^d = 3 => d log 2 = log 3 => d = log 3 / log 2 = about 1.58. And there you go. Fractal dimensions. I hope that was reasonably understandable, though I'm not really confident that it was....

(*Aside: If the original were not a fractal, and were, say, a sierpinski triangle only up to a finite number of iterations, then each of the 3 sub-triangles in the new object would have one less iteration of little triangles removed than the original object, and so they would actually have a little more area than the original, and the 3 times bit doesn't apply. You can then use normal geometry to show that the new object indeed has 4 times as much area as the old one, as usual. It's the infinite iteration that makes the fractal exhibit different properties. As for real-world objects, including sea-shells, Mandelbrot Set images on computer screens, and Pollock paintings, they're obviously not actually infinitely complex, but if they exhibit the right pattern down to the point where the materials get in the way, we just consider them as good enough approximations and give them credit.)

Finally, that Discover magazine article links to this page, which has some nifty pictures, especially toward the end of using fractals to generate simulated trees.