For centuries, mathematicians and artists alike have been fascinated by tessellations, also called tilings. Tessellations of a plane can be found in the regular patterns of tiles in a bathroom floor, or flagstones in a patio. They are also widely used in the design of fabric or wallpaper. Tessellations of three-dimensional space play an important role in chemistry, where they govern the shapes of crystals.

A tessellation of a plane (or of space) is any subdivision of the plane (or space) into regions or "cells" that border each other exactly, with no gaps in between. The cells are usually assumed to come in a limited number of shapes; in many tessellations, all the cells are identical. Tessellations allow an artist to translate a small motif into a pattern that covers an arbitrarily large area. For mathematicians, tessellations provide one of the simplest examples of symmetry.

In everyday language...