Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature.
To understand the Mandelbrot-Set it is necessary to have some mathematical background, especially about complex numbers. Fractals stand in relationship with the Chaostheory.

With the aid of computer graphics, Mandelbrot who then worked at IBM's Watson Research Center, was able to show how Julia's work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics.

The Mandelbrot set is a connected set of points in the complex plane. Pick a point Z0 in the complex plane.

Z1 = Z02 + Z0
Z2 = Z12 + Z0
Z3 = Z22 + Z0
. . .
If the sequence Z0, Z1, Z2, Z3, ... remains within a distance of 2 of the origin forever, then the point Z0 is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set. How fast the sequence diverges can be translated into a color.

Black Points are in the Mandelbrot Set