List of spherical symmetry groups

Spherical symmetry groups are also called point groups in three dimensions, however this article is limitied to the finite symmetries. This article lists the common name and associated Schoenflies notation, Coxeter notation, Orbifold notation, and order to describe three dimensional symmetries.

List of symmetry groups on the sphere

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, C_2v, and C_2h).

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
  • triclinic
  • monclinic
  • orthorhombic
  • tetragonal
  • rhombohedral
  • hexagonal
  • cubic
• Crystallographic point group
• List of planar symmetry groups
• Triangle group

Schönflies Notation

• Schönflies Notation page describing the Schönflies system
• Arthur Moritz Schönflies (1853-1928) German mathematician.

Coxeter notation

• Coxeter number
• Harold Scott MacDonald Coxeter (1907-2003) Canadian mathematician

Orbital notation

• Orbifold notation
• John Horton Conway (1937-) UK mathematician

References

• Peter R. Cromwell, Polyhedra (1997), Appendix I
• Finite spherical symmetry groups
• Weisstein, Eric W., "Schoenflies symbol" from MathWorld.
• Simplest Canonical Polyhedra of Each Symmetry Type, by David I. McCooey