Difference between revisions of "Template:Frieze group notations"

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!p2mm||*22∞||[∞,2]||D<sub>∞h</sub>
 
!p2mm||*22∞||[∞,2]||D<sub>∞h</sub>
||(spinning jump): 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. Abstract group: Dih<sub>∞</sub> &times; Z<sub>2</sub>
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||(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<sub>∞</sub> &times; Z<sub>2</sub>
 
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:<sup>*</sup>Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
 
:<sup>*</sup>Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
 
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Revision as of 20:15, 2 August 2012

Frieze groups
Notations Description Examples
IUC Orbifold Coxeter Schönflies*
p1 ∞∞ [∞,1]+ 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. Frieze2b.png
p11g ∞x [∞+,2+] 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 ∞* [∞+,2] 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 *∞∞ [∞,1] 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∞ [∞,2]+ D (spinning hop): Translations and 180° rotations. The group is generated by a translation and a 180° rotation. Abstract group: Dih
p2mg 2*∞ [∞,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∞ [∞,2] 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