# Coxeter notation

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], Dp, of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram . The cyclic subgroup [p]+, Cp, 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.