The Mandelbrot Set
Fancy applet that draws the well-known Mandelbrot set (see a talk at Camden College for details about the mathematics). This is a fancy version of a relatively simple program, illustrating - aside from the beatiful mathematics - how to generate custom icons for Java lists
Make sure to also check out some really nice movies about Julia sets and the Mandelbrot set. I will write up a more detailed explanation of what Julia sets and the Mandelbrot set are shortly, and post the Java programs used to create these movies, but for now just enjoy them "unencumbered by the thought process". Or, for a head-start, check out the following PowerPoint presentation put together by Jonathan Arena and Joeseph O'Conner.