Simple exploration of the Mandelbrot Set (and the orbits of the iteration with different 'c' values).[br][br]This is the iteration: [math]z_{n+1} = z_{n}^{2}+c[/math][br][br]Try moving 'c' around to see the orbits change.[br][br]Stable orbits are coloured black. The colours 'outside' the set are determined by how quickly the iteration diverges...

For examining the fundamental calculation of the Mandelbrot and Julia Sets.[br][br]The Mandelbrot set is the set of values C for which the function [br]f(z)=z^2+C converges when iterated starting at z=0.[br]a_0=0, a_1=f(0)=C, a_2=f(C)=C^2+C... a_n+1=f(a_n)[br][br]The Julia set, which can be defined for a broad class of complex functions,[br]has a related idea: keep C constant, a make a set based on the behavior of[br]iterations of a_0=z for different z in the complex plane.[br][br]This sketch shows you 22 iterations of a_0 = A, and f(z)=z^2 + C, where we[br]think of points (a,b) as representing the complex number a+bi.