Wallpaper Groups

Symmetry group 1 (p1)

This is the simplest symmetry group. It consists only of translations. There are neither reflections, glide-reflections, nor rotations. The two translation axes may be inclined at any angle to each other. Its lattice is parallelogrammatic, so a fundamental region for the symmetry group is the same as that for the translation group, namely, a parallelogram.

For many actual wallpaper patterns, translations are the only trasformations that leave the pattern invariant. One of the translation directions is vertical, up and down the wallpaper strip. Usually, horizontal transations are not invariant on commercial wallpaper. Instead, the pattern is raised or lowered on adjacent wallpaper strips.

Up to the 17 wallpaper groups
On to the next wallpaper group

David E. Joyce
Department of Mathematics and Computer Science
Clark University
Worcester, MA 01610

These files are located at http://aleph0.clarku.edu/~djoyce/wallpaper/