I always had a fascination with fractal geometry, and with the Mandelbrot set and Julia sets in particular.
Below is a small interactive demonstration of the Mandelbrot set and its relation to Julia sets for different values of c.
For a primer on the math, please visit Wikipedia: Mandelbrot set and Julia set.
The top graph shows the Mandelbrot set. Use shift+scroll for zooming in and out. Clicking a spot causes the lower graph to display the Julia set corresponding to that value of c. Clicking the lower graph draws the orbit starting at the clicked point.
I find that this gives a nice intuition about how Julia sets work. Note in particular how some Julia sets give rise to orbits which converge towards a fixed cycle instead of just a single point. The order of the cycle is associated with the bulb on the Mandelbrot set from which c was picked.