Penrose tiling generator
Craig Kaplan has an online interactive Penrose tiling generator. It is easy enough to tile the plane with identical tiles of one shape. The tilings tend to be periodic. If you use hexagonal tiles, for example, the tiling looks the same no matter which hexagon you start from. A Penrose tiling uses two different shapes, and is not periodic. The pattern repeats more or less, but never exactly.
Well. Either you find this totally fascinating, or you’re bored silly. If you find it fascinating, try out the online generator. You can tweak one tile and see how the other tile has to change in order to maintain the tiling.
Breathes there a man with soul so dead,
Who never to himself hath said,
“A non-periodic tiling of the plane?
Damn! I’d like to see that!”