http://scienceblogs.com/goodmath/2007/07/an_introduction_to_fractals_1.php
An Introduction to Fractals
Category: goodmath > topology > Fractals
Posted on: July 9, 2007 7:40 PM, by Mark C. Chu-Carroll
gasket.jpg 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 Seirpinski carpet, for example, is topologically
one-dimensional. But if you look at it, you have a clear sense of a
two-dimensional figure.
carpet.jpg 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.
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