Frieze Group - General

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. Many authors present the frieze groups in a different order.

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 frieze 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 the horizontal axis, and 180° rotation about the point of intersection (isomorphic to the Klein four-group)