Difference between revisions of "Template:Frieze group notations"

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!p1||&infin;&infin;||[&infin;]<sup>+</sup>||C<sub>&infin;</sub>
 
!p1||&infin;&infin;||[&infin;]<sup>+</sup>||C<sub>&infin;</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. Consequently the group is isomorphic to Z, the group of integers.  
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||(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||&infin;x||[2<sup>+</sup>,&infin;<sup>+</sup>]||S<sub>&infin;</sub>
 
!p11g||&infin;x||[2<sup>+</sup>,&infin;<sup>+</sup>]||S<sub>&infin;</sub>
||(step): Glide-reflections and translations. This group is generated by a single glide reflection, with translations being obtained by combining two glide reflections. Consequently, this group is also isomorphic to Z.
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||(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||&infin;*||[2,&infin;<sup>+</sup>]||C<sub>&infin;h</sub>
 
!p11m||&infin;*||[2,&infin;<sup>+</sup>]||C<sub>&infin;h</sub>
||(jump): Translations, the reflection in the horizontal axis and glide reflections. This group is isomorphic to the direct product Z × C2, and is generated by a translation and the reflection in the horizontal axis.  
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||(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>
 
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!p1m1||*&infin;&infin;||[&infin;]||C<sub>&infin;v</sub>
 
!p1m1||*&infin;&infin;||[&infin;]||C<sub>&infin;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 C2, and isomorphic to the [[infinite dihedral group]]. The group is generated by a translation and a reflection in a vertical axis. It is the same as the non-trivial group in the one-dimensional case
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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]].
 
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!p211||22&infin;||[2,&infin;]<sup>+</sup>||D<sub>&infin;</sub>
 
!p211||22&infin;||[2,&infin;]<sup>+</sup>||D<sub>&infin;</sub>
||(spinning hop): Translations and 180° rotations. Again, the transformations in this group correspond to isometries of the set of integers, and so the group is isomorphic to a semidirect product of Z and C2. The group is generated by a translation and a 180° rotation.  
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||(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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!p2mg||2*&infin;||[2<sup>+</sup>,&infin;]||D<sub>&infin;d</sub>
 
!p2mg||2*&infin;||[2<sup>+</sup>,&infin;]||D<sub>&infin;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 a rotation. It is isomorphic to a semi-direct product of Z and C2.  
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||(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>
 
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!p2mm||*22&infin;||[2,&infin;]||D<sub>&infin;h</sub>
 
!p2mm||*22&infin;||[2,&infin;]||D<sub>&infin;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. It is isomorphic to C2 × (a semidirect product of Z and C2).
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||(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>
 
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|colspan=6|
 
|colspan=6|
:<sup>*</sup>Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
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:<sup>*</sup>Sch&ouml;nflies'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

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