Triangular tiling honeycomb

File:Hyperbolic subgroup tree 363.png]

It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}, and as {{CDD|node_1|splitsplit1|branch4|splitsplit2|node}} from {{CDD|node_1|3|node|6|node_g|3sg|node_g}}, which alternates 3 types (colors) of triangular tilings around every edge. In Coxeter notation, the removal of the 3rd and 4th mirrors, [3,6,3^*] creates a new Coxeter group [3^[3,3]], {{CDD|node|splitsplit1|branch4|splitsplit2|node}}, subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: {{CDD|node_c2|3|node_c1|6|node|3|node}} ↔ {{CDD|node_c2|splitsplit1|branch4_c1|splitsplit2|node_c1}}.

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It is similar to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.

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The triangular tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs.

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There are nine uniform honeycombs in the [3,6,3] Coxeter group family, including this regular form as well as the bitruncated form, t_1,2{3,6,3}, {{CDD|node|3|node_1|6|node_1|3|node}} with all truncated hexagonal tiling facets.

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The honeycomb is also part of a series of polychora and honeycombs with triangular edge figures.

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The runcisnub triangular tiling honeycomb, {{CDD|node_h|3|node_h|6|node|3|node_1}}, has trihexagonal tiling, triangular tiling, triangular prism, and triangular cupola cells. It is vertex-transitive, but not uniform, since it contains Johnson solid triangular cupola cells.

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• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space]) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition {{isbn|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I, II)
• Norman Johnson Uniform Polytopes, Manuscript
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups

Category:Regular 3-honeycombs

Category:Self-dual tilings

Category:Triangular tilings