An extremely short course on fractals

May 24, 2013

[This was an email to a mailing list I'm on to provide background for another discussion.]

Say you have a curve that you're looking at under a microscope. You do your best to measure the length of the line given that your scope doesn't have perfect resolution, so any tiny details get washed out and replaced by you going "well, it looks like a line".

For a straight line, ----, as you increase the resolution of your scope, the length you measure doesn't change. The same is true of a half circle, or most other such smooth things. Now, that when you double your resolution, what looked like a little bit of line turns out to be a squiggle, rather longer than the line it was blurred to, though with the same starting and ending points. And then for each part of the squiggle, when you double the resolution, each bit of line on it turns out to be a squiggle itself. And so on and so on.

If the length you measure grows in a regular fashion as you increase your resolution, so the length at one resolution is equal to the length at half that resolution to some power α, then we say that α is the Hausdorff (or fractal) dimension. For a straight line, the length is always the same, and α=1. It's a one dimensional object. There are ones like the Hilbert curve where α=2 (that is, if you take the infinitely fine version, it fills the whole of two dimensional space), and things like the Koch curve which is about α=1.262. Wikipedia has a nice list, ordered by Hausdorff dimension.

Yet all of these things are lines. You could grab them and straighten them out into a straight line with Hausdorff dimension 1. The notion of dimension we're used to is defined by what you can straight something out into, not by the resolution game above. If I can straighten it out into a line, it's one dimensional. If I can flatten it into a surface, it's two dimensional. This notion of dimension we call topological dimension.

Any time the dimension from how length grows with resolution exceeds the dimension of what you can straighten it out into, you have something weird and spiky with infinitely small noise.

That's all there is. They're cute. But why does anyone care? Because they showed up as a the shape of a bunch of weird things in dynamical systems that people didn't realize existed until well into the 20th century.

So, a brief digression on dynamical systems: Say I have a recipe that takes each point in a space and maps it to another point. If I repeat the recipe again and again, the points start moving along paths, point -> recipe applied to point -> recipe applied to recipe applied to point.

One question to ask about the behavior is where the points go. Do all the points in some region stay in that region, that is, are there basins of attraction? Up until the mid 20th century, everyone thought that the only basins that weren't really fragile mathematical artifacts were simple things with smooth boundaries with equal Hausdorff and topological dimension. Anything that wasn't would just go away if you distorted the recipe slightly, so wasn't important for modeling anything real.

Turns out that that's not true. There's a whole class of basins which behave in all kinds of new and strange ways that basically required a full rewrite of dynamical systems theory. Some of these systems and basins have boundaries which are fractals.

So the fractals aren't the interesting part. The interesting part is the behavior and classification of dynamical systems. The fractals are just an easy part to see that everyone's latched onto.

Did you enjoy that? Try one of my books:
Nonfiction Fiction
Into the Sciences Monologue: A Comedy of Telepathy