List of spherical symmetry groups

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.

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).

Icosahedral symmetry [3,5]


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.

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

ee also

*Overview of point groups by crystal system
*Crystallographic point group
*List of planar symmetry groups
*Triangle group


* Peter R. Cromwell, "Polyhedra", (1997) "(See Appendix I.)"
* [ Finite spherical symmetry groups]

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