List of spherical symmetry groups

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Contents 1 List of symmetry groups on the sphere 1.1 Dihedral symmetry [2,n] 1.2 Tetrahedral symmetry [3,3] 1.3 Octahedral symmetry [3,4] 1.4 Icosahedral symmetry [3,5] 1.5 Other 2 Relation between orbifold notation and order 3 See also 4 References

[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