Wallpaper Patterns

Abstract

There are seventeen different wallpaper groups. To see why, we shall examine each of the five possible types of lattice in turn. Given a lattice L we first work out which orthogonal transformations preserve L. Such transformations form a group and, by (25.2), the point group of any wallpaper group which has L as its lattice must be a subgroup of this group. This limitation on the point group is then sufficient to allow us to enumerate the different wallpaper groups with lattice L. An exhaustive analysis of every case would take up too much space. So we concentrate our attention on a small number of examples, and defer the remaining calculations to the exercises. That all the groups we find are genuinely different, in other words that no two are isomorphic, will be shown at the end of the chapter.