Tiling Theory

The interplay between local constraints and global structure of mathematical and physical systems is both subtle and important. A rich history of work on tilings of the Euclidean plane (and higher dimensional or non-Euclidean spaces) has brought to light numerous examples of finite sets of tiles with rules governing local configurations that lead to surprising global structures.
Joan Taylor and I have discovered a single tile that can fill space, but only in a non-periodic pattern. Here are some pictures of it and some links to pictures other people have made. In the 3D case, the colored bars are really just guides to the eye. The rules for how tiles fit together is completely determined by the tile shape. For details, see our preprint.
A 3D rendering by Edmund Harriss.
An artistic 2D rendering by Yoshiak Araki.

Last modified: 23-Oct-07