A rotation fixes one point in the plane and turns the rest of it some angle around that point. A reflection fixes one line in the plane, called the axis of reflection, and exchanges points on one side of the axis with points on the other side of the axis at the same distance from the axis. Finally, a glide reflection is a little more difficult to define. It's composed of a reflection across an axis and a translation along the axis. Gallian uses successive footprints in wet sand to illustrate a glide-reflection. By Theorem 27.1, Finite Symmetry Groups in the Plane, the only finite plane symmetry groups are Zn and Dn. Therefore every planar symmetry group is either cyclic or dihedral. .
The Frieze Groups.
Discrete frieze groups are the plane symmetry groups of patterns whose subgroup of translations is isomorphic to Z. In previous discussions of isomorphic groups, we were accustomed to viewing them as equal. However, some frieze groups are isomorphic but geometrically different. It is important to treat them differently. .
There are exactly seven different types of frieze patterns based on function composition of the isometries described above. The seven patterns are:.
Pattern I: Translation only.
Pattern II: Glide-reflection.
Pattern III: Translation and vertical reflection.
Pattern IV: Translation and a Rotation of 180 degrees (half-turn).
Pattern V: Glide-reflection and 180 degree rotation.
Pattern VI: Trivial glide-reflection.
Pattern VII: Translation, horizontal reflection, and vertical reflection.
The Crystallographic Groups.
The crystallographic groups are seventeen additional kinds of planar symmetry that arise from repeating designs in a plane. The subgroups of translations associated with these groups are isomorphic to Z+Z. These groups are often called wallpaper groups, since wallpaper sometimes has a small image repeated throughout the pattern. If the wallpaper would be covering an entire plane, it would be a crystallographic group.