http://www.gibney.de/does_anybody_know_this_fractal
Experimenting with the "divisibility" of complex numbers, I stumbled across this thingy. It looks like a fractal. Although the formula does not do any iterations.
Plot the Gaussian rationals (a + bi)/c, where the size of each dot depends inversely on the value of c (cf Ford circles). This creates a square grid, resembling graph paper with infinitely thin divisions. Gibney's 'fractal' is the inversion of this 'graph paper fractal' in the unit circle. That's why everything appears to lie on circles passing through the origin; they are the inversions of straight lines with rational gradient, which naturally contain many rational points. Other interesting things can be done by repeated circle inversions, such as this: http://cp4space.wordpress.com/2012/09/02/imitating-escher/ That reminds me, I need to make some Escher-style Christmas cards using this technique. Sincerely, Adam P. Goucher