"Clouds are not spheres, mountains are not cones, and lightning does not travel in a straight line. The complexity of nature's shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry."

This quote is taken from The Fractal Geometry of Nature by Benoit B. Mandelbrot, creator of the Mandelbrot set and fractals themselves. There are other types of fractals than the Mandelbrot set (perhaps you've heard of the Jurasic Park fractal), but this entry is concerned only with the former.

The Mandelbrot set is a fractal created using complex numbers that are graphed on the complex plane according to the equation Z²+C->Z where Z and C are both complex numbers. This equation is iterated many times to discover whether or not point C is located within the graph of the Mandelbrot set.

What's Iterating?

The term "iteration" refers to repeating the process over again. The equation Z²+C is evaluated, the resulting number is stored as Z, and the equation is evaluated again. This is refered to as one iteration. 100 iterations would involve evaluating Z²+C 100 times.

What Determines points within the Mandelbrot set?

Well, in the Mandelbrot set, Z begins as (0,0) and C is the point to be tested. Iterating it will determine whether or not point C is contained in the Mandelbrot set. When repeatedly iterated, Z will either escape to infinity, or stay within a set of bounds. If it does not excape to infinity, then point C is contained within the Mandelbrot set. Any number Z with a magnitude of 2 or greater (magnitude=sqrt(a²+b²) or distance from the origin) will escape. On a graph, any point contained within the set will be colored black. Here is a picture of the Mandelbrot set from this site about the Mandelbrot set.

Another interesting way to add a color scheme to the Mandelbrot Set is to color points that do escape, based on how long it takes for them to reach a magnitude greater than 2. For example, consider coloring every point that escapes after 200 iterations red, after 300 iterations orange, 400-yellow, etc. and and that have not escaped after 10,00 iterations black. This creates interesting designs on the graph (see above link).

Whats So Interesting About The Mandelbrot Set?

Well, one of the reasons mathmatitions (or Math-a-Magicians) are so interested in the Mandelbrot Set is because it is infinitly complex. This means that one can zoom in on the graph hundreds of billions of times without losing any pixel quality. In this way, it opens up many doors for enhancing digital photos an similar uses.

Another reason is because it is self-similar. This means that one can zoom in indefinatly on certain parts to get images that are completely identical to the original. For more information on this, see referenced site.