Difference between revisions of "Template:Frieze group notations"
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↑ Frieze Patterns Mathematician John Conway created names that relate to footsteps for each of the frieze groups.
imported>Newone |
imported>Parcly Taxel m (no need to rely on Canadian syllabics?) |
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!p11g||[∞<sup>+</sup>,2<sup>+</sup>]<BR>{{CDD|node_h2|infin|node_h4|2x|node_h2}}||S<sub>∞</sub><BR>Z<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> | ||
| − | | style="text-align:center;" |[[File:Frieze group 1g.png|100px]]<BR>∞×|| style="text-align:center;" |'''<big> | + | | 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 |
|(TG) Glide-reflections and Translations:<BR>This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections. | |(TG) Glide-reflections and Translations:<BR>This group is singly generated, by a glide reflection, with translations being obtained by combining two glide reflections. | ||
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Latest revision as of 11:23, 26 October 2020
| IUC | Cox | Schön* Struct. |
Diagram§ Orbifold |
Examples and Conway nickname[1] |
Description |
|---|---|---|---|---|---|
| p1 | [∞]+ File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.png |
C∞ Z∞ |
100px ∞∞ |
F F F F F F F F 150px 150px hop |
(T) Translations only: This group is singly generated, by a translation by the smallest distance over which the pattern is periodic. |
| p11g | [∞+,2+] File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h4.pngFile:CDel 2x.pngFile:CDel node h2.png |
S∞ Z∞ |
100px ∞× |
Γ L Γ L Γ L Γ L 150px 150px 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 | [∞] File:CDel node.pngFile:CDel infin.pngFile:CDel node.png |
C∞v Dih∞ |
100px *∞∞ |
Λ Λ Λ Λ Λ Λ Λ Λ 150px 150px 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]+ File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png |
D∞ Dih∞ |
100px 22∞ |
S S S S S S S S 150px 150px spinning hop |
(TR) Translations and 180° Rotations: The group is generated by a translation and a 180° rotation. |
| p2mg | [∞,2+] File:CDel node.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png |
D∞d Dih∞ |
100px 2*∞ |
V Λ V Λ V Λ V Λ 150px 150px 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] File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 2.pngFile:CDel node.png |
C∞h Z∞×Dih1 |
100px ∞* |
B B B B B B B B 150px 150px 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] File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png |
D∞h Dih∞×Dih1 |
100px *22∞ |
H H H H H H H H 150px 150px 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.
