Patterns are fascinating. I probably was addicted when I played with a Spirograph as a kid. Someone has taken a lovely photo of one here.
Spirographs explained by Mathematica. Nature's spirograph and embroideried one. Lisa Solomon's Spirographs. Camera spin creates wonderful patterns.
Draw yourself a SpiroGraphs here or here.Cool things for kids - Super Spiro.
Fractal art by univrsltransl8r, below.
Moiré Pattern explained by Wolfram Mathworld and Moiré Pattern Generator.
This will give you a headache, so don't watch it very long.
Geometric Tessellations and Labyrinthine Tessellation.
Pattern sampler with SymmetryWorks.