List of spherical symmetry groups

Point groups in three dimensions

Involutional symmetry Cs, (*) [ ] =	Cyclic symmetry Cnv, (*nn) [n] =	Dihedral symmetry Dnh, (*n22) [n,2] =
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetry Td, (*332) [3,3] =	Octahedral symmetry Oh, (*432) [4,3] =	Icosahedral symmetry Ih, (*532) [5,3] =

Spherical symmetry groups are also called point groups in three dimensions; however, this article is limited to the finite symmetries. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry.

This article lists the groups by Schoenflies notation, Coxeter notation,^[1] orbifold notation,^[2] and order. John Conway uses a variation of the Schoenflies notation, based on the groups' quaternion algebraic structure, labeled by one or two upper case letters, and whole number subscripts. The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion.^[3]

Hermann–Mauguin notation (International notation) is also given. The crystallography groups, 32 in total, are a subset with element orders 2, 3, 4 and 6.^[4]

Involutional symmetry

There are four involutional groups: no symmetry (C₁), reflection symmetry (C_s), 2-fold rotational symmetry (C₂), and central point symmetry (C_i).

Intl Geo [5] Orb. Schön. Con. Cox. Ord. Fund. domain 1 1 11 C1 C1 ][ [ ]^+ 1 2 2 22 D1 = C2 D2 = C2 [2]^+ 2

Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 1 22 × Ci = S2 CC2 [2^+,2^+] 2 2 = m 1 * Cs = C1v = C1h ±C1 = CD2 [ ] 2

Cyclic symmetry

There are four infinite cyclic symmetry families, with n=2 or higher. (n may be 1 as a special case as no symmetry)

Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 2 2 22 C2 = D1 C2 = D2 [2]^+ [2,1]^+ 2 mm2 2 *22 C2v = D1h CD4 = DD4 [2] [2,1] 4 4 42 2× S4 CC4 [2^+,4^+] 4 2/m 22 2* C2h = D1d ±C2 = ±D2 [2,2^+] [2^+,2] 4

Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 3 4 5 6 n 3 4 5 6 n 33 44 55 66 nn C3 C4 C5 C6 Cn C3 C4 C5 C6 Cn [3]^+ [4]^+ [5]^+ [6]^+ [n]^+ 3 4 5 6 n 3m 4mm 5m 6mm - 3 4 5 6 n *33 *44 *55 *66 *nn C3v C4v C5v C6v Cnv CD6 CD8 CD10 CD12 CD2n [3] [4] [5] [6] [n] 6 8 10 12 2n 3 8 5 12 - 62 82 10.2 12.2 2n.2 3× 4× 5× 6× n× S6 S8 S10 S12 S2n ±C3 CC8 ±C5 CC12 CC2n / ±Cn [2^+,6^+] [2^+,8^+] [2^+,10^+] [2^+,12^+] [2^+,2n^+] 6 8 10 12 2n 3/m=6 4/m 5/m=10 6/m n/m 32 42 52 62 n2 3* 4* 5* 6* n* C3h C4h C5h C6h Cnh CC6 ±C4 CC10 ±C6 ±Cn / CC2n [2,3^+] [2,4^+] [2,5^+] [2,6^+] [2,n^+] 6 8 10 12 2n

Dihedral symmetry

There are three infinite dihedral symmetry families, with n as 2 or higher. (n may be 1 as a special case)

Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 222 2.2 222 D2 D4 [2,2]^+ 4 42m 42 2*2 D2d DD8 [2^+,4] 8 mmm 22 *222 D2h ±D4 [2,2] 8

Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 32 422 52 622 3.2 4.2 5.2 6.2 n.2 223 224 225 226 22n D3 D4 D5 D6 Dn D6 D8 D10 D12 D2n [2,3]^+ [2,4]^+ [2,5]^+ [2,6]^+ [2,n]^+ 6 8 10 12 2n 3m 82m 5m 12.2m 62 82 10.2 12.2 n2 2*3 2*4 2*5 2*6 2*n D3d D4d D5d D6d Dnd ±D6 DD16 ±D10 DD24 DD4n / ±D2n [2^+,6] [2^+,8] [2^+,10] [2^+,12] [2^+,2n] 12 16 20 24 4n 6m2 4/mmm 10m2 6/mmm 32 42 52 62 n2 *223 *224 *225 *226 *22n D3h D4h D5h D6h Dnh DD12 ±D8 DD20 ±D12 ±D2n / DD4n [2,3] [2,4] [2,5] [2,6] [2,n] 12 16 20 24 4n

Polyhedral symmetry

There are three types of polyhedral symmetry: tetrahedral symmetry, octahedral symmetry, and icosahedral symmetry, named after the triangle-faced regular polyhedra with these symmetries.

Tetrahedral symmetry Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 23 3.3 332 T T [3,3]^+ = [4,3^+]^+ 12 m3 43 3*2 Th ±T [4,3^+] 24 43m 33 *332 Td TO [3,3] = [1^+,4,3] 24

Octahedral symmetry Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 432 4.3 432 O O [4,3]^+ = [[3,3]]^+ 24 m3m 43 *432 Oh ±O [4,3] = [[3,3]] 48 Icosahedral symmetry Intl Geo Orb. Schön. Con. Cox. Ord. Fund. domain 532 5.3 532 I I [5,3]^+ 60 532/m 53 *532 Ih ±I [5,3] 120

See also

• Crystallographic point group
• Triangle group
• List of planar symmetry groups
• Point groups in two dimensions

Notes

1. Johnson, 2015
2. Conway, 2008
3. Conway, 2003
4. Sands, 1993
5. The Crystallographic Space groups in Geometric algebra, D. Hestenes and J. Holt, Journal of Mathematical Physics. 48, 023514 (2007) (22 pages) PDF

References

• Peter R. Cromwell, Polyhedra (1997), Appendix I
• Sands, Donald E. (1993). "Crystal Systems and Geometry". Introduction to Crystallography. Mineola, New York: Dover Publications, Inc. p. 165. ISBN 0-486-67839-3.
• On Quaternions and Octonions, 2003, John Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
  • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
  • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
  • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• N.W. Johnson: Geometries and Transformations, (2015) Chapter 11: Finite symmetry groups

External links

• Finite spherical symmetry groups
• Weisstein, Eric W., "Schoenflies symbol", MathWorld.
• Weisstein, Eric W., "Crystallographic point groups", MathWorld.
• Simplest Canonical Polyhedra of Each Symmetry Type, by David I. McCooey