Penrose Tiles with Matching
These are one version of Penrose tiles, which make a famous quasi-periodic tessellation. Quasiperiodic means that they fill the plane without ever becoming a repeating pattern, but any given chunk of them will repeat an infinite number of times. They were a mathematical curiosity until scientists discovered quasicrystals, which had these patterns in nature.
Some places to learn about these:
My favorite: Dave Austin's two articles, http://www.ams.org/samplings/feature-column/fcarc-penrose and http://www.ams.org/samplings/feature-column/fcarc-ribbons
Also: basics at http://intendo.com/penrose
A GeoGebra sketch to look at motions in the tiling http://www.geogebratube.org/material/show/id/10163
Here's the plain kites and darts with no matching help: http://www.geogebratube.org/material/show/id/35261