List of symmetry groups on the sphere

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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.

Dihedral

There are an infinite set of dihedral symmetries. n can be any positive integer 2 or greater.

Sphere_symmetry_group_d3h.pngSphere symmetry group d3n.png

Name	Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order	Fundamental domain
Polyditropic	Dn	*+	22n	2n
Polydiscopic	Dnh	*	*22n	4n
Polydigyros	Dnd	*	2*n	4n

Tetrahedral

Name	Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order	Fundamental domain
Chiral tetrahedral	T	*+	332	12
Achiral tetrahedral	Td	*	*332	24
Pyritohedral	Th	*	3*2	24

Octahedral

Name	Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order	Fundamental domain
Chiral octahedral	O	*+	432	24
Achiral octahedral	Oh	*	*432	48

Icosahedral

Name	Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order	Fundamental domain
Chiral icosahedral	I	*+	532	60
Achiral icosahedral	Ih	*	*532	120

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.

Sphere_symmetry_group_c3v.png

Name	Schönflies crystallographic notation	Coxeter notation	Conway's orbifold notation	Order	Fundamental domain
no symmetry (monotropic)	C1	*+	11	1
discrete rotational symmetry (polytropic)	Cn	*+	nn	n
reflection symmetry (monoscopic)	Cs	*	*11	2
Polyscopic	Cnv	*	*nn	2n
Polygyros	Cnh	*	n*	2n
inversion symmetry (monodromic)	Ci	*	1x	2
Polydromic	S2n	*	nx	2n

Relation between orbifold notation and order

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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

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• Point groups in three dimensions
• Overview of point groups by crystal system
• Crystallographic point group
• List of planar symmetry groups
• Triangle group

References

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• Peter R. Cromwell, Polyhedra, (1997) (See Appendix I.)
• Finite spherical symmetry groups

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