page_title: Coxeter notation
section_title: Rank two groups
hierarchical_section_title: Coxeter notation / Rank two groups
caption_attribution_description: English: en:Coxeter notation subgroups of hexagon
context_page_description: In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
context_section_description: In two dimensions, the rectangular group [2], abstract D₁² or D₂, also can be represented as a direct product [ ]×[ ], being the product of two bilateral groups, represents two orthogonal mirrors, with Coxeter diagram, , with order 4. The 2 in [2] comes from linearization of the orthogonal subgraphs in the Coxeter diagram, as  with explicit branch order 2. The rhombic group, [2]⁺ ( or ), half of the rectangular group, the point reflection symmetry, Z₂, order 2.
  Coxeter notation to allow a 1 place-holder for lower rank groups, so [1] is the same as [ ], and [1⁺] or [1]⁺ is the same as [ ]⁺ and Coxeter diagram .
  The full p-gonal group [p], abstract dihedral group Dₚ, (nonabelian for p>2), of order 2p, is generated by two mirrors at angle π/p, represented by Coxeter diagram . The p-gonal subgroup [p]⁺, cyclic group Zₚ, of order p, generated by a rotation angle of π/p.
  Coxeter notation uses double-bracking to represent an automorphic doubling of symmetry by adding a bisecting mirror to the fundamental domain. For example, [[p]] adds a bisecting mirror to [p], and is isomorphic to [2p].
  In the limit, going down to one dimensions, the full apeirogonal group is obtained when the angle goes to zero, so [∞], abstractly the infinite dihedral group D∞, represents two parallel mirrors and has a Coxeter diagram . The apeirogonal group [∞]⁺, , abstractly the infinite cyclic group Z∞, isomorphic to the additive group of the integers, is generated by a single nonzero translation.
  In the hyperbolic plane, there is a full pseudogonal group [iπ/λ], and pseudogonal subgroup [iπ/λ]⁺, . These groups exist in regular infinite-sided polygons, with edge length λ. The mirrors are all orthogonal to a single line.
caption: A regular hexagon, with markings on edges and vertices has 8 symmetries: [6], [3], [2], [1], [6]+, [3]+, [2]+, [1]+, with [3] and [1] existing in two forms, depending whether the mirrors are on the edges or vertices.
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