Fractals

So talking to my little brother the other day led me to relearning some math and even a bit of chaos theory. Thanks, Dave!

The two of us were admiring our favorite shirts at Shirt.Woot (a fantastic site- if you've never been there, go) and we happened upon the "Fractal Tree" design. Now I was confused, because I didn't think this was a fractal. David disagreed, although I have a sneaking suspicion this was just for the sake of disagreement, not because he actually knew. However, it turns out he was right either way-- the tree is a fractal, because each of the branches is a smaller iteration of the tree as a whole. A fractal, by definition, is an infinitely detailed geometric shape with small sections similar to large ones.

What I was thinking of was a specific type of fractal, known as a Koch curve or a Koch snowflake. I was going to just copy and paste the description of one from Wikipedia, but it seemed a little more complicated than I'm going for, so I'll give it a go on my own. A Koch snowflake starts as an equilateral triangle. Then, imagine adding another triangle, 1/3 the size of the original, to each of the triangle's sides. This would give you a six-pointed star. Now add smaller triangles to each of the sides of this new shape. Repeat infinitely. I'll illustrate this description with the gif image Wikipedia had, since that makes it a little more clear.

You can see where this is a fractal, since each addition is merely a smaller version of the original shape. Koch snowflakes, and fractals in general, are geometrically interesting because they have infinite perimeter (since you can just keep adding triangles ad infinitum) but finite area. Now I know what you're thinking-- just what my little brother did. "But Amanda, if I keep adding little triangles, the area will keep getting larger too, so how can it be finite?" This is a very good point, but it actually only makes the area hard to calculate exactly, not infinite. Regardless of how many triangles you add, the area of the whole shape will always be smaller than a circle of the same diameter. Therefore, it is finite! Cool, huh?

So what does this have to do with chaos theory? Well, clearly I am not a chaos theorist, but I'll try to sum it up. As far as I can tell (and feel free to correct me on this if you know better), one of the major principles of chaos theory is that complex systems and reactions come from very simple origins. That's the basis behind the "butterfly effect", or the metaphor that the air movement caused by a butterfly flapping its wings is the initial impetus for hurricane strong enough to destroy cities. Fractals are, in a way, geometric demonstrations of this principle, because they start very simply but display infinite complexity.

If my analysis here doesn't suffice, you can also check out this website or James Gleick's book Chaos for some pretty readable overviews of chaos theory.

Pull this out next time you watch "Jurassic Park" and you'll be the life of the party.