List of spherical symmetry groups

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,^[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).

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)

Dihedral symmetry

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

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.

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.

See also

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

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

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

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