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. |
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. |