Difference between revisions of "Template:Frieze group notations"

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{| class=wikitable
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{| class="wikitable sortable"
 
|+ [[Frieze group]]s
 
|+ [[Frieze group]]s
 
|-
 
|-
!colspan=4|Notations
+
![[IUC notation|IUC]]
!rowspan=2|Description
+
![[Coxeter notation|Cox]]
!rowspan=2|Examples
+
![[Schoenflies_notation|Schön]]<sup>*</sup><BR>[[Group (mathematics)|Struct.]]
 +
!Diagram<sup>§</sup><BR>[[Orbifold notation|Orbifold]]
 +
!Examples<BR>and [[John Horton Conway|Conway]] nickname<ref>[https://www.maa.org/sites/default/files/images/upload_library/4/vol1/architecture/Math/seven.html Frieze Patterns] Mathematician John Conway created names that relate to footsteps for each of the frieze groups.</ref>
 +
!Description
 
|-
 
|-
![[Hermann-Mauguin notation|Intl]]  
+
!p1||[∞]<sup>+</sup><BR>{{CDD|node_h2|infin|node_h2}}||C<sub>∞</sub><BR>[[Infinite cyclic group|Z<sub>∞</sub>]]
![[Orbifold notation|Orbifold]]
+
| style="text-align:center;" |[[File:Frieze group 11.png|100px]]<BR>∞∞|| style="text-align:center;" |'''<big>F F F F F F F F</big>'''<BR>[[File:Frieze example p1.png|150px]]<BR>[[File:Frieze hop.png|150px]]<BR>hop
![[Coxeter notation|Coxeter]]
+
||(T) Translations only:<BR>This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
![[Schoenflies_notation|Schönflies]]<sup>*</sup>
 
 
|-
 
|-
!p1||∞∞||[∞]<sup>+</sup>||C<sub>∞</sub>
+
!p11g||[∞<sup>+</sup>,2<sup>+</sup>]<BR>{{CDD|node_h2|infin|node_h4|2x|node_h2}}||S<sub>∞</sub><BR>Z<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. Consequently the group is isomorphic to Z, the group of integers.
+
| style="text-align:center;" |[[File:Frieze group 1g.png|100px]]<BR>∞×|| style="text-align:center;" |'''<big>Γ L Γ L Γ L Γ L</big>'''<BR>[[File:Frieze example p11g.png|150px]]<BR>[[File:Frieze step.png|150px]]<BR>step
!rowspan=7 align=center|[[File:Frieze2b.png|200px]]
+
|(TG) Glide-reflections and Translations:<BR>This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
 
|-
 
|-
!p11g||∞x||[2<sup>+</sup>,∞<sup>+</sup>]||S<sub></sub>
+
!p1m1||[∞]<BR>{{CDD|node|infin|node}}||C<sub>∞v</sub><BR>[[infinite dihedral group|Dih<sub></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.  
+
| style="text-align:center;" |[[File:Frieze group m1.png|100px]]<BR>*∞∞|| style="text-align:center;" |'''<big>Λ Λ Λ Λ Λ Λ Λ Λ</big>'''<BR>[[File:Frieze example p1m1.png|150px]]<BR>[[File:Frieze sidle.png|150px]]<BR>sidle
 +
||(TV) Vertical reflection lines and Translations:<BR>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.
 
|-
 
|-
!p11m||∞*||[2,∞<sup>+</sup>]||C<sub>∞h</sub>
+
!p2||[,2]<sup>+</sup><BR>{{CDD|node_h2|infin|node_h2|2x|node_h2}}||D<sub>∞</sub><BR>Dih<sub></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.  
+
| style="text-align:center;" |[[File:Frieze group 12.png|100px]]<BR>22∞|| style="text-align:center;" |'''<big>S S S S S S S S</big>'''<BR>[[File:Frieze example p2.png|150px]]<BR>[[File:Frieze spinning hop.png|150px]]<BR>spinning hop
 +
|(TR) Translations and 180° Rotations:<BR>The group is generated by a translation and a 180° rotation.
 
|-
 
|-
!p1m1||*∞∞||[∞]||C<sub>∞v</sub>
+
!p2mg||[∞,2<sup>+</sup>]<BR>{{CDD|node|infin|node_h2|2x|node_h2}}||D<sub>∞d</sub><BR>Dih<sub>∞</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
+
| style="text-align:center;" |[[File:Frieze group mg.png|100px]]<BR>2*∞|| style="text-align:center;" |'''<big>V Λ V Λ V Λ V Λ</big>'''<BR>[[File:Frieze example p2mg.png|150px]]<BR>[[File:Frieze spinning sidle.png|150px]]<BR>spinning sidle
 +
|(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° Rotations:<BR>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.
 
|-
 
|-
!p211||22∞||[2,]<sup>+</sup>||D<sub>∞</sub>
+
!p11m||[∞<sup>+</sup>,2]<BR>{{CDD|node_h2|infin|node_h2|2|node}}||C<sub>∞h</sub><BR>Z<sub>∞</sub>×Dih<sub>1</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.
+
| style="text-align:center;" |[[File:Frieze group 1m.png|100px]]<BR>∞*|| style="text-align:center;" |'''<big>B B B B B B B B</big>'''<BR>[[File:Frieze example p11m.png|150px]]<BR>[[File:Frieze jump.png|150px]]<BR>jump
 +
|(THG) Translations, Horizontal reflections, Glide reflections:<BR>This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
 
|-
 
|-
!p2mg||2*∞||[2<sup>+</sup>,]||D<sub>∞d</sub>
+
!p2mm||[∞,2]<BR>{{CDD|node|infin|node|2|node}}||D<sub>∞h</sub><BR>Dih<sub>∞</sub>×Dih<sub>1</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.
+
| style="text-align:center;" |[[File:Frieze group mm.png|100px]]<BR>*22∞|| style="text-align:center;" |<big>'''H H H H H H H H'''</big><BR>[[File:Frieze example p2mm.png|150px]]<BR>[[File:Frieze spinning jump.png|150px]]<BR>spinning jump
|-
+
|(TRHVG) Horizontal and Vertical reflection lines, Translations and 180° Rotations:<BR>This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis.
!p2mm||*22∞||[2,∞]||D<sub>∞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).
 
|-
 
|colspan=6|
 
 
:<sup>*</sup>Schönflies'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
|}
+
:<sup>§</sup>The diagram shows one [[fundamental domain]] in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.

Latest revision as of 11:23, 26 October 2020

Frieze groups
IUC Cox Schön*
Struct.
Diagram§
Orbifold
Examples
and Conway nickname[1]
Description
p1 [∞]+
CDel node h2.pngCDel infin.pngCDel node h2.png
C
Z
Frieze group 11.png
∞∞
F F F F F F F F
Frieze example p1.png
Frieze hop.png
hop
(T) Translations only:
This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
p11g [∞+,2+]
CDel node h2.pngCDel infin.pngCDel node h4.pngCDel 2x.pngCDel node h2.png
S
Z
Frieze group 1g.png
∞×
Γ L Γ L Γ L Γ L
Frieze example p11g.png
Frieze step.png
step
(TG) Glide-reflections and Translations:
This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections.
p1m1 [∞]
CDel node.pngCDel infin.pngCDel node.png
C∞v
Dih
Frieze group m1.png
*∞∞
Λ Λ Λ Λ Λ Λ Λ Λ
Frieze example p1m1.png
Frieze sidle.png
sidle
(TV) Vertical reflection lines and Translations:
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.
p2 [∞,2]+
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
D
Dih
Frieze group 12.png
22∞
S S S S S S S S
Frieze example p2.png
Frieze spinning hop.png
spinning hop
(TR) Translations and 180° Rotations:
The group is generated by a translation and a 180° rotation.
p2mg [∞,2+]
CDel node.pngCDel infin.pngCDel node h2.pngCDel 2x.pngCDel node h2.png
D∞d
Dih
Frieze group mg.png
2*∞
V Λ V Λ V Λ V Λ
Frieze example p2mg.png
Frieze spinning sidle.png
spinning sidle
(TRVG) Vertical reflection lines, Glide reflections, Translations and 180° 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.
p11m [∞+,2]
CDel node h2.pngCDel infin.pngCDel node h2.pngCDel 2.pngCDel node.png
C∞h
Z×Dih1
Frieze group 1m.png
∞*
B B B B B B B B
Frieze example p11m.png
Frieze jump.png
jump
(THG) Translations, Horizontal reflections, Glide reflections:
This group is generated by a translation and the reflection in the horizontal axis. The glide reflection here arises as the composition of translation and horizontal reflection
p2mm [∞,2]
CDel node.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node.png
D∞h
Dih×Dih1
Frieze group mm.png
*22∞
H H H H H H H H
Frieze example p2mm.png
Frieze spinning jump.png
spinning jump
(TRHVG) Horizontal and Vertical reflection lines, Translations and 180° Rotations:
This group requires three generators, with one generating set consisting of a translation, the reflection in the horizontal axis and a reflection across a vertical axis.
*Schönflies's point group notation is extended here as infinite cases of the equivalent dihedral points symmetries
§The diagram shows one fundamental domain in yellow, with reflection lines in blue, glide reflection lines in dashed green, translation normals in red, and 2-fold gyration points as small green squares.
  1. Frieze Patterns Mathematician John Conway created names that relate to footsteps for each of the frieze groups.