Difference between revisions of "Template:Frieze group notations"
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imported>Lpetrich m (Turned the infinity symbols into Wikipedia markup: ∞ -- Help:Wiki markup) |
imported>Lpetrich (Simplified some of the description, stated abstract groups explicitly, and turned infinity and degree into Wiki markup) |
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!p1||∞∞||[∞]<sup>+</sup>||C<sub>∞</sub> | !p1||∞∞||[∞]<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. | + | ||(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||∞x||[2<sup>+</sup>,∞<sup>+</sup>]||S<sub>∞</sub> | !p11g||∞x||[2<sup>+</sup>,∞<sup>+</sup>]||S<sub>∞</sub> | ||
| − | ||(step): Glide-reflections and translations. This group is generated by a | + | ||(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 |
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!p11m||∞*||[2,∞<sup>+</sup>]||C<sub>∞h</sub> | !p11m||∞*||[2,∞<sup>+</sup>]||C<sub>∞h</sub> | ||
| − | ||(jump): Translations, the reflection in the horizontal axis and glide reflections. This group is | + | ||(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> |
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!p1m1||*∞∞||[∞]||C<sub>∞v</sub> | !p1m1||*∞∞||[∞]||C<sub>∞v</sub> | ||
| − | ||(sidle): Translations and reflections across certain vertical lines. 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 | + | ||(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]]. |
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!p211||22∞||[2,∞]<sup>+</sup>||D<sub>∞</sub> | !p211||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> |
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!p2mg||2*∞||[2<sup>+</sup>,∞]||D<sub>∞d</sub> | !p2mg||2*∞||[2<sup>+</sup>,∞]||D<sub>∞d</sub> | ||
| − | ||(spinning sidle): Reflections across certain vertical lines, glide | + | ||(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> |
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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 | + | ||(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> |
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|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 21:32, 25 June 2011
| Notations | Description | Examples | |||
|---|---|---|---|---|---|
| Intl | 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. |
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| 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 | *∞∞ | [∞] | 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. | |
| p211 | 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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