OMG, fractals are the coolest thing! It's not that they are pretty or art, but that they can be so incredibly complex but so incredibly simple. The Mandelbrot set is probably the grandaddy of them all, and can be represented with just ONE equation.

zn = (zn - 1)^2 + c

(It's hard to represent that in here, but basically reads z (sub n) = (z(sub n-1))^2 + c)

What this means, is take a number, square it, then add the original number to the result.
Then take that result, square it, then add the original number to the result.
Then take that result, square it, then add... you get the point.

You can see that if you started with the number 2, it would approach infinity very fast. The progression would be:

2, 6, 38, 1446, 2090918

But if you started with .25, it would eventually get to a point of stasis, somewhere around .36 or so. This number never "escapes" to infinity, but stays around a certain result.

Now apply the same calculation on imaginary numbers like 1 + 1i. Make the real part of the number the x-axis of a graph, and the imaginary part of the number the y axis of a graph. Now calculate points at regular intervals, and display them as pixels on a screen. If the number never escapes to infinity, then paint that pixel black. If the number does escape to infinity, then paint it a color depending on how fast that number escapes to infinity (how many iterations did it take to get beyond a certain limit, like 1000?). You will get the Mandelbrot set!



Explore the Mandelbrot set on this webpage (where the above image comes from)

Here is a paper I did on this topic in college for a writing class. Fractals may become more useful as we begin to understand what they mean.