The Mandelbrot set is named after the pioneer in fractal theory Benoit Mandelbrot. The set is a two dimensional fractal obtained from studying the convergence of the sequence:
Zn+1 = Zn^2 + C (1)
and is defined as all complex values of C that makes (1) bounded for all n.
The Mandelbrot wallpapers are simply obtained by iterating (1) pixel by pixel, starting with z0 = 0. If the sequence stays bounded after a large number of iterations it is assumed that the sequence is bounded and that the present pixel value is in the Mandelbrot set and is then colored black. If the sequence diverge then the pixel is colored according to how fast the divergence is. Below are different color maps used for the visualization.