[chestervonwinch]

Functional iteration is actually a fun way to draw fractal images in the plane. It goes like this (for anyone interested):

1) Pick a function f: R^2 ==> R^2

2) Pick a region of R^2 (this could be the unit square for instance).

3) For each point in the region do the following:

  a) Plug the point into f. Then plug f(x) into f. Then plug f(f(x)) into f, etc....

  b) The norm of f(f(...f(x)...)) will either run off to infinite or stay bounded.

  c) Record for the original point, x, how many iterations it took the process to run off to infinite (or the maximum if the sequence stayed bounded).

4) Paint by number after assigning a unique color to each possible number of iterations.

Here's the result of this process for the function:

f(x,y) = ( exp(x) * cos(y), exp(x) * sin(y) )

http://i.imgur.com/LZKavio.png