You’ve seen the Mandelbrot Set:
But what is it? Here is a video that explains it pretty clearly:
What, honestly… did you have anything better to be doing just now?
Aside: I often wonder about the guys who discover things like this. Mandelbrot, Kepler, Galileo, Einstein and the like… doing their thing, puzzling over the math, finding something unexpected and beautiful that seems to delve into the very heart of reality. I’m not religious, but I gotta think that the discovery of something like this must seem like a glimpse into the mind of God to someone who is. I can still remember the day I found a book on orbital mechanics that explained the math succinctly and clearly, and I sat down and for the first time scratched out some basic math about the orbital periods of planets. My tiny little young mind was well and truly blown by the simple ability to predict where a planet would be an arbitrary time in the future. That seems as nothing compared to Mandelbrot’s discovery.