The most well-known of the fractals is the infamous Mandelbrot set. It’s one of the first things that was really studied as a fractal. It was discovered by Benoit Mandelbrot during his early study of fractals in the context of the complex dynamics of quadratic polynomials the 1980s, and studied in greater detail by Douady and Hubbard in the early to mid-80s.

It’s a beautiful example of what makes fractals so attractive to us: it’s got an extremely simple definition; an incredibly complex structure; and it’s a rich source of amazing, beautiful images. It’s also been glommed onto by an amazing number of woo-meisters, who babble on about how it represents “fractal energies” – “fractal” has become a woo-term almost as prevalent as “quantum”, and every woo-site that babbles about fractals invariably uses an image of the Mandelbrot set. It’s also become a magnet for artists – the beauty of its structure, coming from a simple bit of math captures the interest of quite a lot of folks. Two musical examples are Jonathon Coulton and the post-rock band “Mandelbrot Set”. (If you like post-rock, I definitely recommend checking out MS; and a player for brilliant Mandelbrot set song is embedded below.)

So what is the Mandelbrot set?

Take the set of functions
where for each , is a complex constant. That gives an infinite set of simple functions over the complex numbers. For each possible complex number , you look at the recurrence relation generated by repeatedly applying , starting with :

If doesn’t diverge (escape) towards infinity as gets larger, then the complex number is a member of the Mandelbrot set. That’s it – that simple definition – repeatedly apply for complex numbers – produces the astonishing complexity of the Mandelbrot set.

If we use that definition of the Mandelbrot set, and draw the members of the set in black, we get an image like the one above. That’s nice, but it’s probably not what you expected. We’re all used to the beautiful colored bands and auras around that basic pointy black blob. Those colored regions are not really part of the set.

The way we get the colored bands is by considering *how long* it takes for the points to start to diverge. Each color band is an escape interval – that is, some measure of how many iterations it takes for the repeated application of to diverge. Images like the ones to the right and below are generated using various variants of escape-interval colorings.

My personal favorite rendering of the Mandelbrot set is an image called the Buddhabrot. In the Buddhabrot, what you do is look at values of which *aren’t* in the mandebrot set. For each point before it escapes, plot a point. That gives you the escape path for the value . If you take a large number of escape paths for randomly selected values of , and you plot them so that the brightness of a pixel is determined by the number of escape paths that cross that pixel, you get the Budddhabrot. It’s fascinating because it reveals the structure in a particularly amazing way. If you look at a simple unzoomed image of the madelbrot set, what you see is a spiky black blob; the actually complexity of the structure isn’t obvious until you spend some time looking at it. The Buddhabrot is more obvious – you can see the astonishing complexity much more easily.

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:

1. 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.
2. 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.
3. 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.