Difference between revisions of "Template:Frieze group notations"

From blackwiki
Jump to navigation Jump to search
imported>Tomruen
imported>Yobot
m (WP:CHECKWIKI error fixes + general fixes using AWB (7914))
Line 11: Line 11:
 
![[Schoenflies_notation|Schönflies]]<sup>*</sup>
 
![[Schoenflies_notation|Schönflies]]<sup>*</sup>
 
|-
 
|-
!p1||&infin;&infin;||[&infin;,1]<sup>+</sup>||C<sub>&infin;</sub>
+
!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]]
 
|-
 
|-
!p11g||&infin;x||[&infin;<sup>+</sup>,2<sup>+</sup>]||S<sub>&infin;</sub>
+
!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||&infin;*||[&infin;<sup>+</sup>,2]||C<sub>&infin;h</sub>
+
!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 &times; 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 &times; Z<sub>2</sub>
 
|-
 
|-
!p1m1||*&infin;&infin;||[&infin;,1]||C<sub>&infin;v</sub>
+
!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>&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></sub>, the [[infinite dihedral group]].
 
|-
 
|-
!p2||22&infin;||[&infin;,2]<sup>+</sup>||D<sub>&infin;</sub>
+
!p2||22∞||[,2]<sup>+</sup>||D<sub></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° rotations. The group is generated by a translation and a 180° rotation. Abstract group: Dih<sub></sub>
 
|-
 
|-
!p2mg||2*&infin;||[&infin;,2<sup>+</sup>]||D<sub>&infin;d</sub>
+
!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>&infin;</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||*22&infin;||[&infin;,2]||D<sub>&infin;h</sub>
+
!p2mm||*22∞||[,2]||D<sub>∞h</sub>
||(spinning jump): Translations, glide reflections, reflections in both axes and 180&deg; 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>&infin;</sub> &times; Z<sub>2</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>
 
|-
 
|-
 
|colspan=6|
 
|colspan=6|
:<sup>*</sup>Sch&ouml;nflies'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
 
|}
 
|}

Revision as of 15:18, 14 January 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: 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