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List of symmetry groups on the sphere
Spherical symmetry groups are also called point groups in three dimensions. This article is about the finite ones.
There are four fundamental symmetry classes which have triangular fundamental domains: dihedral, tetrahedral, octahedral, icosahedral. There are infinitely many dihedral symmetry groups.
The final classes, under other have digonal or monogonal fundamental domains.
Dihedral symmetry [2,n]
There are an infinite set of dihedral symmetries. n can be any positive integer 2 or greater (n = 1 is also possible, but these three symmetries are equal to C2, C2v, and C2h).
| Name | Schönflies crystallographic notation |
Coxeter notation |
Orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Polyditropic | Dn | [2,n]+ | 22n | 2n |   |
| Polydiscopic | Dnh | [2,n] | *22n | 4n |   |
| Polydigyros | Dnd | [2+,2n] | 2*n | 4n |   |
Tetrahedral symmetry [3,3]
| Name | Schönflies crystallographic notation |
Coxeter notation |
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 |  |
Octahedral symmetry [3,4]
| Name | Schönflies crystallographic notation |
Coxeter notation |
Orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Chiral octahedral | O | [3,4]+ | 432 | 24 |  |
| Achiral octahedral | Oh | [3,4] | *432 | 48 |  |
Icosahedral symmetry [3,5]
| Name | Schönflies crystallographic notation |
Coxeter notation |
Orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| Chiral icosahedral | I | [3,5]+ | 532 | 60 |  |
| Achiral icosahedral | Ih | [3,5] | *532 | 120 |  |
Other
These final forms have digonal or monogonal fundamental regions with Cyclic symmetries and reflection symmetry. There are four infinite sets with index n being any positive integer; for n=1 two cases are equal, so there are three; they are separately named.
| Name | Schönflies crystallographic notation |
Coxeter notation |
Orbifold notation |
Order | Fundamental domain |
|---|---|---|---|---|---|
| no symmetry (monotropic) | C1 | [1]+ | 11 | 1 |  |
| discrete rotational symmetry (polytropic) | Cn | [n]+ | nn | n |  |
| reflection symmetry (monoscopic) | Cs = C1v = C1h | [1] | *11 | 2 |  |
| Polyscopic | Cnv | [n] | *nn | 2n |   |
| Polygyros | Cnh | [2,n+] | n* | 2n |  |
| inversion symmetry (monodromic) | Ci = S2 | [2+,2+] | 1× | 2 |  |
| Polydromic | S2n | [2+,2n+] | n× | 2n |
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
See also
- Overview of point groups by crystal system
- Crystallographic point group
- List of planar symmetry groups
- Triangle group
References
- Peter R. Cromwell, Polyhedra, (1997) (See Appendix I.)
- Finite spherical symmetry groups
- Eric W. Weisstein, Schoenflies symbol at MathWorld.
