Crystallographic group

Summary

A crystallographic group is a discrete group of isometries of Euclidean space $\R^n$ whose quotient space is compact. These groups describe the symmetries of periodic patterns, such as crystals in dimension $3$ and wallpapers in dimension $2$. By the Bieberbach theorems, each one contains a lattice of translations of finite index. In each dimension there are only finitely many of them up to isomorphism. In dimension $3$ they are the space groups of crystallography, and in dimension $2$ the wallpaper groups.