Frieze group

A frieze group is a discrete symmetry group for a pattern on a strip (infititely wide rectangle).

A frieze group contains either a translation or a glide reflection. It may optionally contain a reflection along the long axis of the strip and a reflection along the narrow axis of the strip. It may also contain a 180° rotation. (It must be able to be generated by either a single translation or a single glide reflection, along with rotations and reflections.)

Basic descriptions

There are seven distinct subgroups (up to scaling) in the discrete frieze group generated by a translation, reflection (along the same axis) and a 180° rotation. Each of these subgroups is the symmetry group of one of the frieze patterns. The seven different patterns can be described as follows:

1. Translations only
2. Glide-reflections only
3. Translation followed by a vertical reflection across a stationary line
4. Translation followed by a rotation of 180 degrees around a stationary point that is half way between the translated images
5. Glide-reflection followed by a rotation of 180 degrees around a stationary point that is half way between the glided images
6. Translation followed by a horizontal reflection
7. Translation followed by a horizontal reflection followed by a vertical reflection across a stationary line

In other words, a frieze group may be generated from any set of translations, as long as the result is a symmetry group of a strip, and there aren't infinitely many translations within a finite amount of "translation space"--the translation space expands in two directions; translations are only allowed to occur in one dimension, in a straight line as opposed to a plane. The trivial group (C₁) is also excluded.

There are exactly seven different frieze groups, up to translation.