page_title: Hexagonal tiling honeycomb
section_title: Symmetry constructions
hierarchical_section_title: Hexagonal tiling honeycomb / Symmetry constructions
caption_attribution_description: English: Subset of File:Hyperbolic subgroup tree 36.png
context_page_description: In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.
  The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, six hexagonal tilings meet at each vertex of this honeycomb, and four edges meet at each vertex.
context_section_description: It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular:  [6,3,3],  [3,6,3],  [6,3,6],  [6,3] and [3] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)*] (remove 3 mirrors, index 24 subgroup); [3,6,3*] or [3*,6,3] (remove 2 mirrors, index 6 subgroup); [1⁺,6,3,6,1⁺] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3]. The ringed Coxeter diagrams are , , ,  and , representing different types (colors) of hexagonal tilings in the Wythoff construction.
caption: Subgroup relations
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