Remember those plastic spirographs you could draw patterns with as a kid? I spent hours lying on the floor with sheets of paper covered in wonky, colored circles, trying to give up control of the pen and let the template guide the arcs so they would come out right. Oh, the fascinating sensation when you allowed the drawing to make itself!
Planets, on the other hand, are locked in to their orbits by the physics of the universe. They don't wobble and jerk around, or drink too much Fanta Grape.
This program by Howard Arrington makes their orbits visible in relation to each other, and I find their secret symmetries incredibly beautiful. Not just the patterns, but the fact that they exist - direct evidence of order in the overwhelming complexity of the universe.
Via a blog post at The Long Now Foundation
Visit Ensign Software's webpage to see more orbit pairings