- List of spherical symmetry groups
= 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.
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).
These final forms have digonal or monogonal fundamental regions with
Cyclic symmetriesand 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.
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)/(2"n")
* * and x count as 1This 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
*Overview of point groups by crystal system
Crystallographic point group
List of planar symmetry groups
* Peter R. Cromwell, "Polyhedra", (1997) "(See Appendix I.)"
* [http://www.geom.uiuc.edu/~math5337/Orbifolds/costs.html Finite spherical symmetry groups]
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