Fractal Graphs, anyone? (another AI question)

Well for my AI utility thing, I've been toying with the idea of using fractals as a basis for some algorithms. I got the idea when I realized that for what I'm wanting to do, the perceptrons would not be storing normal geometric functions and things like that, but would each work independently to work with a seemingly random set of numbers to find some level of organization. When I pictured this, it reminded me of a fractal graph I had seen in math. I thought fractals, because of the iterative nature would be perfect for predicting things. Unfortunately, we didn't really do much with fractals in math, and I have no idea how to generate a fractal, based on the regular (x,y) (with a z in there somewhere) graph. I got on google and had a look around, but I can't find a tutorial that explains any of this well at all. Anyone?

If you look at our contest web site, there's a fractal example Prelude submitted, with pictures and source code.

One of the most famous fractal graphs is the graph of the Mandelbrot-set.

http://www.olympus.net/personal/dewey/mandelbrot.html
http://www.math.utah.edu/~alfeld/mat...andelbrot.html

The Mandelbrot set is based on the recursive relation z -> z^2 + c, where c is a complex number. I assume you know the mathematics of complex numbers? If c = a + b * i, then (a, b) is a point in the complex plane. Start with z1 = 0 and then calculate z2, z3, z4 etc. The value will go to infinity or be finite. Make those pixels in the plane at points c for which z is finite and you have a picture of the Mandelbrot set. In computer graphics they use often a lot of colors. This can be done by checking how many steps a value c, not part of the Mandelbrot set, can be calculated before it gets to infinity. For this check you need to define a value N which indicates finite, after N it is infinity.