One of the most fun and curiousity-inspiring things in mathematics are Penrose Tilings. They are tessellations, filling the plane with a finite number of tiles, but in a non-regularly-repeating way. The student sketch has links for further reading, and there's a pretty big section on them at Wikipedia, too.
There's motions to think about, especially with the edge modifications, and geometric measurements of angle and proportion (strong connections to regular pentagons). If these weren't fun enough, it turns out that the golden ratio is all over the place.
Here's the plain kites and darts with no matching: [url]http://www.geogebratube.org/material/show/id/35261[/url]
