I thought in addition to the graph theory (which I’m enjoying writing, but doesn’t seem to be all that popular), I’d also try doing some writing about fractals. I know pretty much nothing about fractals, but I’ve wanted to learn about them for a while, and one of the advantages of having this blog is that it gives me an excuse to learn about things that that interest me so that I can write about them.
Fractals are amazing things. They can be beautiful: everyone has seen beautiful fractal images – like the ones posted by my fellow SBer Karmen. And they’re also useful: there are a lot of phenomena in nature that seem to involve fractal structures.
But what is a fractal?
The word is a contraction of fractional dimension. The idea of that is that there are several different ways of measuring the dimensionality of a structure using topology. The structures that we call fractals are things that have a kind of fine structure that gives them a strange kind of dimensionality; their conventional topological dimension is smaller than their Hausdorff dimension. (You can look up details of what topological dimension and Hausdorff dimension mean in one of my topology articles.) The details aren’t all that important here: the key thing to understand is that there’s a fractal is a structure that breaks the usual concept of dimension: it’s shape has aspects that suggest higher dimensions. The Sierpinski carpet, for example, is topologically one-dimensional. But if you look at it, you have a clear sense of a two-dimensional figure.
That’s all frightfully abstract. Let’s take a look at one of the simplest fractals. This is called Sierpinski’s carpet. There’s a picture of a finite approximation of it over to the right. The way that you generate this fractal is to take a square. Divide the square into 9 sub-squares, and remove the center one. Then take each of the 8 squares around the edges, and do the same thing to them: break them into 9, remove the center, then repeat on the even smaller squares. Do that an infinite number of times.
When you look at the carpet, you probably think it looks two dimensional. But topologically, it is a one-dimensional space. The “edges” of the resulting figure are infinitely narrow – they have no width that needs a second dimension to describe. The whole thing is an infinitely complicated structure of lines: the total area covered by the carpet is 0! Since it’s just lines, topologically, it’s one-dimensional.
In fact, it is more than just a one dimensional shape; what it is is a kind of canonical one dimensional shape: any one-dimensional space is topologically equivalent (homeomorphic) to a subset of the carpet.
But when we look at it, we can see it has a clear structure in two dimensions. In fact, it’s a structure which really can’t be described as one-dimensional – we defined by cutting finite sized pieces from a square, which is a 2-dimensional figure. It isn’t really two dimensional; it isn’t really one dimensional. The best way of describing it is by its Hausdorff dimension, which is 1.89. So it’s almost, but not quite, two dimensional.
Sierpinski’s carpet is a very typical fractal; it’s got the traits that we use to identify fractals, which are the following:
- Self-similarity: a fractal has a structure that repeats itself on ever smaller scales. In the case of the carpet, you can take any non-blank square, and it’s exactly the same as a smaller version of the entire carpet.
- Fine structure: a fractal has a fine structure at arbitrarily small scales. In the case of the carpet, no matter how small you get, it’s always got even smaller subdivisions.
- Fractional dimension: its Hausdorff dimension is not an integer. Its Hausdorff dimension is also usually larger than its topological dimension. Again looking at the carpet, it’s topological dimension is 1; it’s Hausdorff dimension is 1.89.