page_title: Coxeter–Dynkin diagram
section_title: Lorentzian groups
hierarchical_section_title: Coxeter–Dynkin diagram / Lorentzian groups
caption_attribution_description: English: en:Coxeter-Dynkin diagram subgroup tree for Lorentzian groups from [6,3,3,3] and [6,3,6,3]
context_page_description: In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors. It describes a kaleidoscopic construction: each graph "node" represents a mirror and the label attached to a branch encodes the dihedral angle order between two mirrors, that is, the amount by which the angle between the reflective planes can be multiplied by to get 180 degrees. An unlabeled branch implicitly represents order-3.
  Each diagram represents a Coxeter group, and Coxeter groups are classified by their associated diagrams.
  Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed, while Coxeter diagrams are undirected; secondly, Dynkin diagrams must satisfy an additional restriction, namely that the only allowed branch labels are 2, 3, 4, and 6. Dynkin diagrams correspond to and are used to classify root systems and therefore semisimple Lie algebras.
context_section_description: Lorentzian groups for simplex domains can be defined as graphs beyond the paracompact hyperbolic forms. These are sometimes called super-ideal simplices and are also related to a Lorentzian geometry, named after Hendrik Lorentz in the field of special and general relativity space-time, containing one (or more) time-like dimensional components whose self dot products are negative. Danny Calegari calls these convex cocompact Coxeter groups in n-dimensional hyperbolic space.
  A 1982 paper by George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, enumerates the finite list of Lorentzian of rank 5 to 11. He calls them level 2, meaning removal any permutation of 2 nodes leaves a finite or Euclidean graph. His enumeration is complete, but didn't list graphs that are a subgroup of another. All higher-order branch Coxeter groups of rank-4 are Lorentzian, ending in the limit as a complete graph 3-simplex Coxeter-Dynkin diagram with 6 infinite order branches, which can be expressed as [∞]. Rank 5-11 have a finite number of groups 186, 66, 36, 13, 10, 8, and 4 Lorentzian groups respectively. A 2013 paper by H. Chen and J.-P. Labbé, Lorentzian Coxeter groups and Boyd--Maxwell ball packings, recomputed and published the complete list.
  For the highest ranks 8-11, the complete lists are:
caption: This shows rank 5 Lorentzian groups arranged as subgroups from [6,3,3,3], and [6,3,6,3]. The highly symmetric group , [3[3,3,3]] is an index 120 subgroup of [6,3,3,3].
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