Frieze group

A frieze group is a mathematical concept to classify designs on two-dimensional surfaces which are repetitive in one direction, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art. The mathematical study of such patterns reveals that exactly 7 different types of patterns can occur.

Frieze groups are related to the more complex wallpaper groups, which classify patterns which are repetitive in two directions.

As with wallpaper groups, a frieze group is often visualised by a simple periodic pattern in the category concerned.

General

Formally, a frieze group is a class of infinite discrete symmetry groups for patterns on a strip (infinitely wide rectangle), hence a class of groups of isometries of the plane, or of a strip. There are seven different frieze groups. The actual symmetry groups within a frieze group are characterized by the smallest translation distance, and, for the frieze groups 4-7, by a shifting parameter. In the case of symmetry groups in the plane, additional parameters are the direction of the translation vector, and, for the frieze groups 2, 3, 5, 6, and 7, the positioning perpendicular to the translation vector. Thus there are two degrees of freedom for group 1, three for groups 2, 3, and 4, and four for groups 5, 6, and 7.

A symmetry group of a frieze group necessarily contains translations and may contain glide reflections. Other possible group elements are reflections along the long axis of the strip, reflections along the narrow axis of the strip and 180° rotations. For two of the seven frieze groups (numbers 1 and 2 below) the symmetry groups are singly-generated, for four (numbers 3–6) they have a pair of generators, and for number 7 the symmetry groups require three generators.

A symmetry group in frieze group 1, 3, 4, or 5 is a subgroup of a symmetry group in the last frieze group with the same translational distance. A symmetry group in frieze group 2 or 6 is a subgroup of a symmetry group in the last frieze group with half the translational distance. This last freeze group contains the symmetry groups of the simplest periodic patterns in the strip (or the plane), a row of dots. Any transformation of the plane leaving this pattern invariant can be decomposed into a translation, (x,y) → (n+x,y), optionally followed by a reflection in either the horizontal axis, (x,y) → (x,−y), or the vertical axis, (x,y) → (−x,y), provided that this axis is chosen through or midway between two dots, or a rotation by 180°, (x,y) → (−x,−y) (ditto). Therefore, in a way, this frieze group contains the "largest" symmetry groups, which consist of all such transformations.

The inclusion of the discrete condition is to exclude the group containing all translations, and groups containing arbitrarily small translations (e.g. the group of horizontal translations by rational distances). Even apart from scaling and shifting, there are infinitely many cases, e.g. by considering rational numbers of which the denominators are powers of a given prime number.

The inclusion of the infinite condition is to exclude groups that have no translations:

• the group with the identity only (isomorphic to C1, the trivial group of order 1).
• the group consisting of the identity and reflection in the horizontal axis (isomorphic to C2, the cyclic group of order 2).
• the groups each consisting of the identity and reflection in a vertical axis (ditto)
• the groups each consisting of the identity and 180° rotation about a point on the horizontal axis (ditto)
• the groups each consisting of the identity, reflection in a vertical axis, reflection in a vertical axis, and 180° rotation about the point of intersection (isomorphic to the Klein four-group)

Descriptions of the seven frieze groups

patterns for the 7 frieze groups
Figure 1. Patterns corresponding to the 7 frieze groups

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. Each of these subgroups is the symmetry group of a frieze pattern, and sample patterns are shown in the Fig. 1. The seven different groups correspond to the 7 infinite series of point groups in three dimensions, with n = $\infty$. They are, with Conway's orbifold notation in parentheses:

1. ( $\infty \infty$) Translations only. This group is singly-generated, with a generator being a translation by the distance over which the pattern is periodic. Consequently the group is isomorphic to Z, the group of integers.
2. ( $\infty x$) Glide-reflections and translations. This group is generated by a single glide reflection, with translations being obtained by combining two glide reflections. Consequently, this group is also isomorphic to Z.
3. ( $\infty *$) Translations, the reflection in the horizontal axis and glide reflections. This group is isomorphic to the direct product Z × C2, and is generated by a translation and the reflection in the horizontal axis.
4. ( $*\infty \infty$) Translations and reflections across certain vertical lines. The elements in this group correspond to isometries (or equivalently, bijective affine transformations) of the set of integers, and so it is isomorphic to a semidirect product of the integers with C2. The group is generated by a translation and a reflection in a vertical axis. It is the same as the non-trivial group in the one-dimensional case
5. ( $22\infty$) Translations and 180° rotations. Again, the transformations in this group correspond to isometries of the set of integers, and so the group is isomorphic to a semidirect product of Z and C2. The group is generated by a translation and a 180° rotation.
6. ( $2*\infty$) Reflections across certain vertical lines, glide-reflections, translations and rotations. The translations here arise from the glide reflections, so this group is generated by a glide reflection and a rotation. It is isomorphic to a semi-direct product of Z and C2.
7. ( $*22\infty$) Translations, glide reflections, reflections in both axes and 180° rotations. This group is the "largest" frieze group and requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis. It is isomorphic to a semidirect product of Z × C2 with C2.
File:Frieze.png
An example for each frieze group (in order 1, 4, 7, 5, 3, 6, 2)

Summarized:

1. T (translation only)
2. TG (translation and glide reflection)
3. THG (translation, horizontal line reflection, and glide reflection)
4. TV (translation and vertical line reflection)
5. TR (translation and 180° rotation)
6. TRVG (translation, 180° rotation, vertical line reflection, and glide reflection)
7. TRHVG (translation, 180° rotation, horizontal line reflection, vertical line reflection, and glide reflection)

As we have seen, up to isomorphism, there are four groups, two abelian, and two non-abelian.

Hierarchy

With the same translation distance, sequences of increasing symmetry are 137, 147, 157, and 126; with halving of the translation distance we also have 23 and 67.

The symmetries of groups 1,3,4,5, and 7 with translation distance t imply those of the same group and translation distance nt, for an integer n. For groups 2 and 6 this is only true if n is odd.