When you zoom in on a piece of the Mandelbrot set, you realize that that piece contains, and consists of, another Mandelbrot set. Zoom in again, and you see that that piece also contains and consists of another Mandelbrot set. Zoom in again. Same thing. In fact, you can zoom in forever and you will always see more Mandelbrot sets!
The Mandelbrot set doesn't iterate over these simple numbers. Instead it iterates over complex numbers.
Complex numbers come in two parts: a real part and an imaginary part. The real part is easy to grasp. They are regular numbers that you know and love: 1, 0, -5, 4.534343, 232423432.4787865, -0.0000000000002, etc. The imaginary part of a complex number is a real number (like above) multiplied by a unique little number called i.
For the Mandelbrot set, we start Z at zero...but our choice for C will be a complex number. Our exact choice for C is what determines the Mandelbrot set. The Mandelbrot set consists of all the choices for C we can find (where Z starts at zero and C is a complex number) so that the iterations never grow beyond the number 2. That is the mathematical definition of the Mandelbrot set.
By color coding the numbers output by the equation, software is able to create the intense visualizations we've seen on so many T-shirts.
Mandelbrot himself said, ""Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
To make more complex fractals, mathematicians and tinkerers alike have altered the recursive equation. An easy example is raising Z to the third or fifth power, which adds more spirals to the visual output.
However, little scientific rigor has been applied to why humans find fractals so cool. But we do. There are websites to make your own fractals, and fractal software that does the same task at a more complex level.
There are even fractal shows, held in the planetarium of the New Mexico Museum of Natural History and Science.
Check out a gallery of some more cool fractals. If you're interested in making your own fractal, we recommend the free software Xaos. It takes a few minutes to figure out how each variation of Mandelbrot's equation changes the look of the fractal, but the controls are simple and the results are awesome.