(After its discoverer, Benoit Mandelbrot) The set of all complex numbers c such that

        | z[N] | < 2

for arbitrarily large values of N, where

        z[0] = 0
        z[n+1] = z[n]^2 + c

The Mandelbrot set is usually displayed as an Argand diagram, giving each point a colour which depends on the largest N for which | z[N] | < 2, up to some maximum N which is used for the points in the set (for which N is infinite). These points are traditionally coloured black.

The Mandelbrot set is the best known example of a fractal - it includes smaller versions of itself which can be explored to arbitrary levels of detail.

The Fractal Microscope.