Frieze group

A frieze group is a discrete subgroup of the isometry group of the Euclidean plane whose translation subgroup is infinite cyclic and has finite index. It describes a pattern periodic in one planar direction. The planar hypothesis is part of the definition; analogous groups acting in three-dimensional Euclidean space need not be frieze groups.

Translations along a fixed line give $\mathbb{Z}$. Adding reflections in perpendicular parallel lines gives the infinite dihedral group; adding a reflection in the translation axis instead gives $\mathbb{Z}\times C_2$. A glide reflection supplies another way to combine the single periodic direction with orientation reversal.