z_{n+1} = z_{n}^{2} + c

remains ** bounded ** in the complex plane, where z_{0}=0.

As usual, points near the Mandelbrot set are colored to indicate how quickly their iterates go to infinity. In this example, the limit on the number of iterations can be specified. The resolution can also be controlled: unless your machine is very fast, it seems best to work at resolution 64 and about 150 iterations until you find an interesting spot, and only then draw the picture at resolution 256 and perhaps 500 or 1000 iterations.