List of spherical symmetry groups
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[edit] List of symmetry groups on the sphere
Spherical symmetry groups are also called point groups (in 3D).
There are four fundamental symmetry classes: dihedral, tetrahedral, octahedral, icosahedral which have triangular fundamental domains. The dihedral symmetry groups are an infinite set.
The final classes, under other have digonal or monogonal fundamental domains.
[edit] Dihedral symmetry [2,n]
There are an infinite set of dihedral symmetries. n can be any positive integer 2 or greater.
| Name | Schönflies crystallographic notation |
Coxeter notation |
Conway's orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Polyditropic | Dn | [2,n]+ | 22n | 2n | |
| Polydiscopic | Dnh | [2,n] | *22n | 4n |   |
| Polydigyros | Dnd | [2+,2n] | 2*n | 4n |   |
[edit] Tetrahedral symmetry [3,3]
| Name | Schönflies crystallographic notation |
Coxeter notation |
Conway's orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Chiral tetrahedral | T | [3,3]+ | 332 | 12 |  |
| Achiral tetrahedral | Td | [3,3] | *332 | 24 |  |
| Pyritohedral | Th | [3+,4] | 3*2 | 24 |  |
[edit] Octahedral symmetry [3,4]
| Name | Schönflies crystallographic notation |
Coxeter notation |
Conway's orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Chiral octahedral | O | [3,4]+ | 432 | 24 |  |
| Achiral octahedral | Oh | [3,4] | *432 | 48 |  |
[edit] Icosahedral symmetry [3,5]
| Name | Schönflies crystallographic notation |
Coxeter notation |
Conway's orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Chiral icosahedral | I | [3,5]+ | 532 | 60 |  |
| Achiral icosahedral | Ih | [3,5] | *532 | 120 |  |
[edit] Other
These final forms have digonal or monogonal fundamental regions with Cyclic symmetries and reflection symmetry. All form infinite sets n as any positive integer, and with 1 being named as a special case.
| Name | Schönflies crystallographic notation |
Coxeter notation |
Conway's orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| no symmetry (monotropic) | C1 | [1]+ | 11 | 1 |  |
| discrete rotational symmetry (polytropic) | Cn | [n]+ | nn | n |  |
| reflection symmetry (monoscopic) | Cs | [1] | *11 | 2 |  |
| Polyscopic | Cnv | [n] | *nn | 2n |   |
| Polygyros | Cnh | [2,n+] | n* | 2n |  |
| inversion symmetry (monodromic) | Ci | [2+,2+] | 1x | 2 |  |
| Polydromic | S2n | [2+,2n+] | nx | 2n |
[edit] Relation between orbifold notation and order
The order of each group is 2 divided by the orbifold Euler characteristic; the latter is 2 minus the sum of the feature values, assigned as follows:
- n without or before * counts as (n−1)/n
- n after * counts as (n−1)/(2n)
- * and x count as 1
This can also be applied for wallpaper groups: for them, the sum of the feature values is 2, giving an infinite order; see orbifold Euler characteristic for wallpaper groups
[edit] See also
- Point groups in three dimensions
- Overview of point groups by crystal system
- Crystallographic point group
- List of planar symmetry groups
- Triangle group
[edit] References
- Peter R. Cromwell, Polyhedra, (1997) (See Appendix I.)
- Finite spherical symmetry groups
