Coxeter notation

In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter.

In one dimension or higher, the bilateral group [ ] represents a single mirror symmetry, D₁, symmetry order 2. It is represented as a Coxeter–Dynkin diagram with a single node, . The identity group is the direct subgroup [ ]⁺, C₁, symmetry order 1.

In two dimensions or higher, the rectangular group [2], D₂, represented as a direct product [ ]x[ ], the product of two bilateral groups, represents two orthogonal mirrors, and Coxeter diagram, . The rhombic group, [2]⁺, half of the rectangular group, C₂, symmetry order 2.

The nonabelian dihedral group [p], D_p, of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram . The cyclic subgroup [p]⁺, C_p, of order p, generated by a rotation angle of π/p.

The infinite dihedral group is obtained when the angle goes to zero, so [∞], D_∞ represents two parallel mirrors and has a Coxeter diagram . The apeirogonal group [∞]⁺, isomorphic to the additive group of the integers, is generated by a single nonzero translation.

In three or higher dimension, the full orthorhombic group [2,2], D₁xD₂, order 8, represents three orthogonal mirrors, and also can be represented by Coxeter diagram as three separate dots . There is a semidirect subgroup, the orthorhombic group, [2,2⁺], D₁xC₂, of order 4. Others are the pararhombic group [2,2]⁺, also order 4, and finally the central group [2⁺,2⁺] of order 2.

See also

• List of spherical symmetry groups‎
• List of planar symmetry groups

References

• H.S.M. Coxeter:
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied 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]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • N.W. Johnson: Geometries and Transformations, Manuscript, (2011) Chapter 11: Finite symmetry groups

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