Since I can't stand to just post a video without any explanation:
A fractal is a figure with a self-similar pattern. What that means is that there is some way of looking at it where a piece of it looks almost the same as the whole thing. In this video, what they've done is set up three screens, in a triangular pattern, and set them to display the input from a camera. When you point the camera at the screens, what you get is whatever the camera is seeing repeated three times in a triangular pattern - and since what's on the screens is what's being seen by the camera; and what's seen by the camera is, after a bit of delay, what's on the screens, you're getting a self-similar system. If you watch, they're able to manipulate it to get Julia fractals, Sierpinski triangles, and several other really famous fractals.
It's very cool - partly because it looks neat, but also partly because it shows you something important about fractals. We tend to think of fractals in computational terms, because in general we generate fractal images using digital computers. But you don't need to. Fractals are actually fascinatingly ubiquitous, and you can produce them in lots of different ways - not just digitally.
So I'm trying to ease back into the chaos theory posts. I thought that one good
way of doing that was to take a look at one of the class chaos examples, which
demonstrates just how simple a chaotic system can be. It really doesn't take much
at all to push a system from being nice and smoothly predictable to being completely
This example comes from mathematical biology, and it generates a
graph commonly known as the logistical map. The question behind
the graph is, how can I predict what the stable population of a particular species will be over time?
Sorry for the slowness of the blog; I fell behind in writing my book, which is on a rather strict schedule, and until I got close to catching up, I didn't have time to do the research
necessary to write the next chaos article. (And no one has sent me any particularly
interesting bad math, so I haven't had anything to use for a quick rip.)
Anyway... Where we left off last was talking about attractors. The natural question
is, why do we really care about attractors when we're talking about chaos? That's a question
which has two different answers.
As pointed out by a commenter, there are some really surprising places where fractal patterns can
appear. For example, there was a recent post on the Wolfram mathematica blog by the engineer who writes
the unlimited precision integer arithmetic code.
In the course of the series of posts I've been writing on fractals, several people have either emailed or commented, saying something along the lines of "Yeah, that fractal stuff is cool - but what is it good for? Does it do anything other than make pretty pictures?"
That's a very good question. So today, I'm going to show you an example of a real fractal that
has meaningful applications as a model of real phenomena. It's called the logistic map.
When you mention fractals, one of the things that immediately comes to mind for most people
is fractal landscapes. We've all seen amazing images of mountain ranges, planets, lakes, and things
of that sort that were generated by fractals.
Seeing a fractal image of a mountain, like the one in this image (which I found here via a google image search for "fractal mountain"), I expected to find that
it was based on an extremely complicated fractal. But the amazing thing about fractals is how
complexity emerges from simplicity. The basic process for generating a fractal mountain - and many other elements of fractal landscapes - is astonishingly simple.
Aside from the Mandelbrot set, the most famous fractals are the Julia sets. You've almost definitely seen images of the Julias (like the ones scattered through this post), but what you might not have realized is just how closely related the Julia sets are to the Mandelbrot set.
One of the most fundamental properties of fractals that we've mostly avoided so far is the idea of dimension. I mentioned that one of the basic properties of fractals is that their Hausdorff dimension is
larger than their simple topological dimension. But so far, I haven't explained how to figure out the
Hausdorff dimension of a fractal.
When we're talking about fractals, notion of dimension is tricky. There are a variety of different
ways of defining the dimension of a fractal: there's the Hausdorff dimension; the box-counting dimension; the correlation dimension; and a variety of others. I'm going to talk about the fractal dimension, which is
a simplification of the Hausdorff dimension. If you want to see the full technical definition of
the Hausdorff dimension, I wrote about it in one of my topology posts.
So, in my last post, I promised to explain how the chaos game is is an attractor for the Sierpinski triangle. It's actually pretty simple. First, though, we'll introduce the idea of an affine transformation. Affine transformations aren't strictly necessary for understanding the Chaos game, but by understanding the Chaos game in terms of affines, it makes it easier to understand
Most of the fractals that I've written about so far - including all of the L-system fractals - are
examples of something called iterated function systems. Speaking informally, an iterated function
system is one where you have a transformation function which you apply repeatedly. Most iterated
function systems work in a contracting mode, where the function is repeatedly applied to smaller
and smaller pieces of the original set.
There's another very interesting way of describing these fractals, and I find it very surprising
that it's equivalent. It's the idea of an attractor. An attractor is a dynamical system which, no matter what its starting point, will always evolve towards a particular shape. Even if you
perturb the dynamical system, up to some point, the pertubation will fade away over time, and the system
will continue to evolve toward the same target.