Frieze Group - Descriptions of The Seven Frieze Groups

Descriptions of The Seven 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 Fig. 1. The seven different groups correspond to the 7 infinite series of axial point groups in three dimensions, with n = ∞. They are identified using Hermann-Mauguin notation or IUC notation, orbifold notation, Coxeter notation, and Schönflies notation:

Frieze groups
Notations Description Examples
IUC Orbifold Coxeter Schönflies*
p1 ∞∞ C∞ (hop): Translations only. This group is singly generated, with a generator being a translation by the smallest distance over which the pattern is periodic. Abstract group: Z, the group of integers under addition.
p11g ∞× S∞ (step): Glide-reflections and translations. This group is generated by a glide reflection, with translations being obtained by combining two glide reflections. Abstract group: Z
p11m ∞* C∞h (jump): Translations, the reflection in the horizontal axis and glide reflections. This group is generated by a translation and the reflection in the horizontal axis. Abstract group: Z × Z2
p1m1 *∞∞ C∞v (sidle): Translations and reflections across certain vertical lines. The group is the same as the non-trivial group in the one-dimensional case; it is generated by a translation and a reflection in the vertical axis. 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 Z2. Abstract group: Dih∞, the infinite dihedral group.
p2 22∞ + D∞ (spinning hop): Translations and 180° rotations. The group is generated by a translation and a 180° rotation. Abstract group: Dih∞
p2mg 2*∞ D∞d (spinning sidle): 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 either a rotation or a vertical reflection. Abstract group: Dih∞
p2mm *22∞ D∞h (spinning jump): Translations, glide reflections, reflections in both axes and 180° rotations. This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis. Abstract group: Dih∞ × Z2
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries

Summarized:

  1. p1: T (translation only, in the horizontal direction)
  2. p11g: TG (translation and glide reflection)
  3. p11m: THG (translation, horizontal line reflection, and glide reflection)
  4. p2m1: TV (translation and vertical line reflection)
  5. p2: TR (translation and 180° rotation)
  6. p2mg: TRVG (translation, 180° rotation, vertical line reflection, and glide reflection)
  7. p2mm: 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.

The groups can be classified by their type of two-dimensional grid or lattice:

Lattice type # Groups
Oblique 1-2 p1, p211
Rectangular 3-7 p1m1, p11m, p11g, p2mm, p2mg

The lattice being oblique means that the second direction need not be orthogonal to the direction of repeat. The groups' order in this table is their order in the International Tables for Crystallography, which differs from orders given elsewhere.

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