Difference between revisions of "Template:Frieze group notations"
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imported>Tomruen |
imported>Yobot m (WP:CHECKWIKI error fixes + general fixes using AWB (7914)) |
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![[Schoenflies_notation|Schönflies]]<sup>*</sup> | ![[Schoenflies_notation|Schönflies]]<sup>*</sup> | ||
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| − | !p1|| | + | !p1||∞∞||[∞,1]<sup>+</sup>||C<sub>∞</sub> |
||(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. | ||(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. | ||
!rowspan=7 align=center|[[File:Frieze2b.png|200px]] | !rowspan=7 align=center|[[File:Frieze2b.png|200px]] | ||
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| − | !p11g|| | + | !p11g||∞x||[∞<sup>+</sup>,2<sup>+</sup>]||S<sub>∞</sub> |
||(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 | ||(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|| | + | !p11m||∞*||[∞<sup>+</sup>,2]||C<sub>∞h</sub> |
||(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 × Z<sub>2</sub> | ||(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 × Z<sub>2</sub> | ||
|- | |- | ||
| − | !p1m1||* | + | !p1m1||*∞∞||[∞,1]||C<sub>∞v</sub> |
| − | ||(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 Z<sub>2</sub>. Abstract group: Dih<sub> | + | ||(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 Z<sub>2</sub>. Abstract group: Dih<sub>∞</sub>, the [[infinite dihedral group]]. |
|- | |- | ||
| − | !p2|| | + | !p2||22∞||[∞,2]<sup>+</sup>||D<sub>∞</sub> |
| − | ||(spinning hop): Translations and | + | ||(spinning hop): Translations and 180° rotations. The group is generated by a translation and a 180° rotation. Abstract group: Dih<sub>∞</sub> |
|- | |- | ||
| − | !p2mg||2* | + | !p2mg||2*∞||[∞,2<sup>+</sup>]||D<sub>∞d</sub> |
| − | ||(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: Abstract group: Dih<sub> | + | ||(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: Abstract group: Dih<sub>∞</sub> |
|- | |- | ||
| − | !p2mm||* | + | !p2mm||*22∞||[∞,2]||D<sub>∞h</sub> |
| − | ||(spinning jump): Translations, glide reflections, reflections in both axes and | + | ||(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> × Z<sub>2</sub> |
|- | |- | ||
|colspan=6| | |colspan=6| | ||
| − | :<sup>*</sup> | + | :<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 15:18, 14 January 2012
| 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. |
|
| 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: Abstract group: Dih∞ | |
| p2mm | *22∞ | [∞,2] | D∞h | (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∞ × Z2 | |
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