page_title: Coxeter notation
section_title: Trionic subgroups
hierarchical_section_title: Coxeter notation / Subgroups / Trionic subgroups
caption_attribution_description: English: en:Coxeter notation, trionic subgroup example, [8,3] --&gt;[4,8]
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: A trionic subgroup is an index 3 subgroups. There are many
  Johnson defines a trionic subgroup with operator ⅄, index 3. For rank 2 Coxeter groups, [3], the trionic subgroup, [3⅄] is [ ], a single mirror. And for [3p], the trionic subgroup is [3p]⅄ ≅ [p]. Given , with generators {0,1}, has 3 trionic subgroups. They can be differentiated by putting the ⅄ symbol next to the mirror generator to be removed, or on a branch for both: [3p,1⅄] =  = ,   = , and [3p⅄] =  =  with generators {0,10101}, {01010,1}, or {101,010}.
  Trionic subgroups of tetrahedral symmetry: [3,3]⅄ ≅ [2⁺,4], relating the symmetry of the regular tetrahedron and tetragonal disphenoid.
  For rank 3 Coxeter groups, [p,3], there is a trionic subgroup [p,3⅄] ≅ [p/2,p], or  = . For example, the finite group [4,3⅄] ≅ [2,4], and Euclidean group [6,3⅄] ≅ [3,6], and hyperbolic group [8,3⅄] ≅ [4,8].
  An odd-order adjacent branch, p, will not lower the group order, but create overlapping fundamental domains. The group order stays the same, while the density increases. For example, the icosahedral symmetry, [5,3], of the regular polyhedra icosahedron becomes [5/2,5], the symmetry of 2 regular star polyhedra. It also relates the hyperbolic tilings {p,3}, and star hyperbolic tilings {p/2,p}
  For rank 4, [q,2p,3⅄] = [2p,((p,q,q))],  = .
  For example, [3,4,3⅄] = [4,3,3], or  = , generators {0,1,2,3} in [3,4,3] with the trionic subgroup [4,3,3] generators {0,1,2,32123}. For hyperbolic groups, [3,6,3⅄] = [6,3], and [4,4,3⅄] = [4,4,4].
caption: Example trionic subgroups on octagonal symmetry [8,3] maps onto larger [4,8] symmetries.
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