Wallpaper Patterns

A wallpaper pattern is based on a motif (a small picture) that is repeated at the points of a regular lattice (a grid). The twelve that are shown here are the ones for square lattices. The overall pattern has to be symmetric: some combination of translations, reflections, and reflections brings the pattern back onto itself identically. The symmetry can be as simple as a pair of translations (shifts). It is a remarkable fact that there are exactly 17 such wallpaper patterns. In contrast, in 3D and 4D there are 219 and 4783 space groups.