Template:Frieze group notations

Frieze groups

IUC	Cox.	Schön.*	Orbifold	Diagram§	Examples and Conway nickname		Description
p1	[∞]+ File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.png	C∞ Z∞	∞∞	File:Frieze group 11.png	File:Frieze example p1.png	File: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+] File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h4.pngFile:CDel 2x.pngFile:CDel node h2.png	S∞ Z∞	∞×	File:Frieze group 1g.png	File:Frieze example p11g.png	File: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	[∞] File:CDel node.pngFile:CDel infin.pngFile:CDel node.png	C∞v Dih∞	*∞∞	File:Frieze group m1.png	File:Frieze example p1m1.png	File: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]+ File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 2x.pngFile:CDel node h2.png	D∞ Dih∞	22∞	File:Frieze group 12.png	File:Frieze example p2.png	File:Frieze spinning hop.png 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∞	2*∞	File:Frieze group mg.png	File:Frieze example p2mg.png	File: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] File:CDel node h2.pngFile:CDel infin.pngFile:CDel node h2.pngFile:CDel 2.pngFile:CDel node.png	C∞h Z∞×Dih1	∞*	File:Frieze group 1m.png	File:Frieze example p11m.png	File: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] File:CDel node.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node.png	D∞h Dih∞×Dih1	*22∞	File:Frieze group mm.png	File:Frieze example p2mm.png	File: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.