Coxeter notation
Encyclopedia
In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, Coxeter notation is a system of classifying symmetry group
Symmetry group
The symmetry group of an object is the group of all isometries under which it is invariant with composition as the operation...

s, describing the angles between with fundamental reflections of a Coxeter group
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

. 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.