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Template:Frieze group notations

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Frieze groups
IUC Cox. Schön.* Orbifold Diagram§ Examples and
Conway nickname[1]
Description
p1 [∞]+
C
Z
∞∞
hop
(T) Translations only:
This group is singly generated, by a translation by the smallest distance over which the pattern is periodic.
p11g [∞+,2+]
S
Z
∞×
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 [∞]
C∞v
Dih
*∞∞
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]+
D
Dih
22∞
spinning hop
(TR) Translations and 180° Rotations:
The group is generated by a translation and a 180° rotation.
p2mg [∞,2+]
D∞d
Dih
2*∞
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]
C∞h
Z×Dih1
∞*
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]
D∞h
Dih×Dih1
*22∞
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.