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 and reflection in a vertical axis (ditto).

the group consisting of the identity and reflection in the horizontal axis (isomorphic to C 2, the cyclic group of order 2).

the group with the identity only (isomorphic to C 1, the trivial group of order 1).

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

by considering rational numbers of which the denominators are powers of a given prime number. Even apart from scaling and shifting, there are infinitely many cases, e.g. the group of horizontal translations by rational distances). The inclusion of the discrete condition is to exclude the group containing all translations, and groups containing arbitrarily small translations (e.g. Therefore, in a way, this frieze group contains the 'largest' symmetry groups, which consist of all such transformations. 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). This last frieze group contains the symmetry groups of the simplest periodic patterns in the strip (or the plane), a row of dots. A symmetry group in frieze group 4 or 6 is a subgroup of a symmetry group in the last frieze group with half the translational distance. A symmetry group in frieze group 1, 2, 3, or 5 is a subgroup of a symmetry group in the last frieze group with the same translational distance. 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.įor two of the seven frieze groups (groups 1 and 4) the symmetry groups are singly generated, for four (groups 2, 3, 5, and 6) they have a pair of generators, and for group 7 the symmetry groups require three generators. 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 actual symmetry groups within a frieze group are characterized by the smallest translation distance, and, for the frieze groups with vertical line reflection or 180° rotation (groups 2, 5, 6, and 7), by a shift parameter locating the reflection axis or point of rotation. Many authors present the frieze groups in a different order. There are seven frieze groups, listed in the summary table. A symmetry group of a frieze group necessarily contains translations and may contain glide reflections, reflections along the long axis of the strip, reflections along the narrow axis of the strip, and 180° rotations. p2mm: TRHVG (translation, 180° rotation, horizontal line reflection, vertical line reflection, and glide reflection)įormally, a frieze group is a class of infinite discrete symmetry groups of patterns on a strip (infinitely wide rectangle), hence a class of groups of isometries of the plane, or of a strip.p2mg: TRVG (translation, 180° rotation, vertical line reflection, and glide reflection).p11g: TG (translation and glide reflection).p11m: THG (translation, horizontal line reflection, and glide reflection).

p1m1: TV (translation and vertical line reflection).

p1: T (translation only, in the horizontal direction).

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. The set of symmetries of a frieze pattern is called a frieze group.įrieze groups are two-dimensional line groups, having repetition in only one direction. Frieze patterns can be classified into seven types according to their symmetries.

Such patterns occur frequently in architecture and decorative art. In mathematics, a frieze or frieze pattern is a two-dimensional design that repeats in one direction. Type of symmetry group Examples of frieze patterns

Glide reflection symmetry example