Difference between revisions of "Template:Frieze group notations"

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imported>Lpetrich
(Simplified some of the description, stated abstract groups explicitly, and turned infinity and degree into Wiki markup)
imported>Tomruen
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!rowspan=2|Examples
 
!rowspan=2|Examples
 
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![[Hermann-Mauguin notation|Intl]]  
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![[IUC notation|IUC]]  
 
![[Orbifold notation|Orbifold]]
 
![[Orbifold notation|Orbifold]]
 
![[Coxeter notation|Coxeter]]
 
![[Coxeter notation|Coxeter]]
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||(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>&infin;</sub>, the [[infinite dihedral group]].
 
||(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>&infin;</sub>, the [[infinite dihedral group]].
 
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!p211||22&infin;||[2,&infin;]<sup>+</sup>||D<sub>&infin;</sub>
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!p2||22&infin;||[2,&infin;]<sup>+</sup>||D<sub>&infin;</sub>
 
||(spinning hop): Translations and 180&deg; rotations. The group is generated by a translation and a 180&deg; rotation. Abstract group: Dih<sub>&infin;</sub>
 
||(spinning hop): Translations and 180&deg; rotations. The group is generated by a translation and a 180&deg; rotation. Abstract group: Dih<sub>&infin;</sub>
 
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Revision as of 20:34, 30 June 2011

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. 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 *∞∞ [∞] 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
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries