Frieze group

The term is derived from architecture and decorative arts, where such repeating patterns are often used.

They are related to the more complex wallpaper groups, which classify patterns that are repetitive in two directions, and crystallographic groups, which classify patterns that are repetitive in three directions.

In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups with horizontal line reflection, glide reflection, or 180° rotation (groups 3–7), the position of the reflection axis or rotation point in the direction perpendicular to the translation vector.

The inclusion of the infinite condition is to exclude groups that have no translations: There are seven distinct subgroups (up to scaling and shifting of patterns) in the discrete frieze group generated by a translation, reflection (along the same axis) and a 180° rotation.

[5] The groups can be classified by their type of two-dimensional grid or lattice.

There exist software graphic tools that create 2D patterns using frieze groups.

Usually, the entire pattern is updated automatically in response to edits of the original strip.