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List of spherical symmetry groups
List article
Selected point groups in three dimensions
Involutional symmetryCs, (*)[ ] = Cyclic symmetryCnv, (*nn)[n] = Dihedral symmetryDnh, (*n22)[n,2] =
Polyhedral group, [n,3], (*n32)
Tetrahedral symmetryTd, (*332)[3,3] = Octahedral symmetryOh, (*432)[4,3] = Icosahedral symmetryIh, (*532)[5,3] =

Finite spherical symmetry groups are also called point groups in three dimensions. 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, orbifold notation, 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.

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.

Involutional symmetry

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

IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
1111C1C1][[ ]+1Z1
2222D1= C2D2= C2[2]+2Z2
122×Ci= S2CC2[2+,2+]2Z2
2= m1*Cs= C1v= C1h±C1= CD2[ ]2Z2

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)

IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
4422×S4CC4[2+,4+]4Z4
2/m222*C2h= D1d±C2= ±D2[2,2+][2+,2]4Z4
IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
23456n23456n2233445566nnC2C3C4C5C6CnC2C3C4C5C6Cn[2]+[3]+[4]+[5]+[6]+[n]+23456nZ2Z3Z4Z5Z6Zn
2mm3m4mm5m6mmnm (n is odd)nmm (n is even)23456n*22*33*44*55*66*nnC2vC3vC4vC5vC6vCnvCD4CD6CD8CD10CD12CD2n[2][3][4][5][6][n]46810122nD4D6D8D10D12D2n
38512-628210.212.22n.23×4×5×6×n×S6S8S10S12S2n±C3CC8±C5CC12CC2n / ±Cn[2+,6+][2+,8+][2+,10+][2+,12+][2+,2n+]6810122nZ6Z8Z10Z12Z2n
3/m=64/m5/m=106/mn/m32425262n23*4*5*6*n*C3hC4hC5hC6hCnhCC6±C4CC10±C6±Cn / CC2n[2,3+][2,4+][2,5+][2,6+][2,n+]6810122nZ6Z2×Z4Z10Z2×Z6Z2×Zn≅Z2n (odd n)

Dihedral symmetry

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

IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
2222.2222D2D4[2,2]+4D4
42m422*2D2dDD8[2+,4]8D4
mmm22*222D2h±D4[2,2]8Z2×D4
IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
32422526223.24.25.26.2n.222322422522622nD3D4D5D6DnD6D8D10D12D2n[2,3]+[2,4]+[2,5]+[2,6]+[2,n]+6810122nD6D8D10D12D2n
3m82m5m12.2m628210.212.2n22*32*42*52*62*nD3dD4dD5dD6dDnd±D6DD16±D10DD24DD4n / ±D2n[2+,6][2+,8][2+,10][2+,12][2+,2n]121620244nD12D16D20D24D4n
6m24/mmm10m26/mmm32425262n2*223*224*225*226*22nD3hD4hD5hD6hDnhDD12±D8DD20±D12±D2n / DD4n[2,3][2,4][2,5][2,6][2,n]121620244nD12Z2×D8D20Z2×D12Z2×D2n≅D4n (odd n)

Polyhedral symmetry

Further information: Polyhedral groups

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
IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
233.3332TT[3,3]+12A4
m3433*2Th±T[4,3+]242×A4
43m33*332TdTO[3,3]24S4
Octahedral symmetry
IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
4324.3432OO[4,3]+24S4
m3m43*432Oh±O[4,3]482×S4
Icosahedral symmetry
IntlGeoOrbifoldSchönfliesConwayCoxeterOrderAbstractFund.domain
5325.3532II[5,3]+60A5
532/m53*532Ih±I[5,3]1202×A5

Continuous symmetries

All of the discrete point symmetries are subgroups of certain continuous symmetries. They can be classified as products of orthogonal groups O(n) or special orthogonal groups SO(n). O(1) is a single orthogonal reflection, dihedral symmetry order 2, Dih1. SO(1) is just the identity. Half turns, C2, are needed to complete.

Rank 3 groupsOther namesExample geometryExample finite subgroups
O(3)Full symmetry of the sphere[3,3] = , [4,3] = , [5,3] = [4,3+] =
SO(3)Sphere groupRotational symmetry[3,3]+ = , [4,3]+ = , [5,3]+ =
O(2)×O(1)O(2)⋊C2Dih∞×Dih1Dih∞⋊C2Full symmetry of a spheroid, torus, cylinder, bicone or hyperboloidFull circular symmetry with half turn[p,2] = [p]×[ ] = [2p,2+] = , [2p+,2+] =
SO(2)×O(1)C∞×Dih1Rotational symmetry with reflection[p+,2] = [p]+×[ ] =
SO(2)⋊C2C∞⋊C2Rotational symmetry with half turn[p,2]+ =
O(2)×SO(1)Dih∞Circular symmetryFull symmetry of a hemisphere, cone, paraboloidor any surface of revolution[p,1] = [p] =
SO(2)×SO(1)C∞Circle groupRotational symmetry[p,1]+ = [p]+ =

See also

Further reading

  • 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-Strauss, 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 [1]
    • (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, (2018) ISBN 978-1-107-10340-5 Chapter 11: Finite symmetry groups, Table 11.4 Finite Groups of Isometries in 3-space

References

  1. Johnson, 2015 ↩

  2. Conway, John H. (2008). The symmetries of things. Wellesley, Mass: A.K. Peters. ISBN 978-1-56881-220-5. OCLC 181862605. 978-1-56881-220-5 ↩

  3. Conway, John; Smith, Derek A. (2003). On quaternions and octonions: their geometry, arithmetic, and symmetry. Natick, Mass: A.K. Peters. ISBN 978-1-56881-134-5. OCLC 560284450. 978-1-56881-134-5 ↩

  4. Sands, "Introduction to Crystallography", 1993 ↩