ABSTRACT
The simplest kinds of repeating patterns are purely one-dimensional. This chapter classifies patterns by their groups of symmetries. A frieze pattern is constructed by placing a repeating pattern of figures on a one-dimensional lattice in the plane. The symmetry group of a frieze pattern will contain a nontrivial translation. Each frieze group determines a plane group by simply repeating the frieze pattern to extend the lattice to a rectangular one. A wallpaper pattern is a two-dimensional lattice with some structure on the lattice points. A symmetry of a wallpaper pattern is a symmetry of the lattice that respects the structure of the lattice points. Two space groups are considered the same—or to have the same affine space-group type—if they are conjugate as subgroups of the affine group.
