Classification of wallpaper groups and frieze groups

Abstract

The aim of this paper is to classify wallpaper groups and frieze groups. To do so, we will first define symmetry groups and some of their invariants. Then, we will restrict ourselves to plane symmetry groups (the wallpaper groups) and define when two of these groups are equivalent. Then, we will show that there are exactly 17 equivalence classes. We will also see a few examples in order to show how to find the corresponding equivalence class of the wallpaper group for a given periodic design. Moreover, we can also restrict the definition of symmetry groups to frieze groups. Since the defined equivalence relationship will remain valid, we will repeat the same process to see that there are exactly 7 frieze groups, of which we will also study some concrete examples. Finally, we will relate wallpaper groups and frieze groups in the last chapter of this bachelor thesis.