Alternated hexagonal tiling honeycomb

File:Hyperbolic subgroup tree 336-direct.png]]

It has five alternated constructions from reflectional Coxeter groups all with four mirrors and only the first being regular: {{CDD|node_c1|6|node|3|node|3|node}} [6,3,3], {{CDD|node_c1|3|node_c1|6|node|3|node}} [3,6,3], {{CDD|node|6|node_c1|3|node_c1|6|node}} [6,3,6], {{CDD|branch_c1|split2|node_c1|6|node}} [6,3^[3]] and [3^[3,3]] {{CDD|branch_c1|splitcross|branch_c1}}, 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^[3,3]]. The ringed Coxeter diagrams are {{CDD|node_h|6|node|3|node|3|node}}, {{CDD|node_h|3|node_h|6|node|3|node}}, {{CDD|node|6|node_h|3|node_h|6|node}}, {{CDD|branch_hh|split2|node_h|6|node}} and {{CDD|branch_hh|splitcross|branch_hh}}, representing different types (colors) of hexagonal tilings in the Wythoff construction.

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The alternated hexagonal tiling honeycomb has 3 related forms: the cantic hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|3|node}}; the runcic hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node|3|node_1}}; and the runcicantic hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|3|node_1}}.

The cantic hexagonal tiling honeycomb, h₂{6,3,3}, {{CDD||node_h1|6|node|3|node_1|3|node}} or {{CDD|branch_10ru|split2|node_1|3|node}}, is composed of octahedron, truncated tetrahedron, and trihexagonal tiling facets, with a wedge vertex figure.

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The runcic hexagonal tiling honeycomb, h₃{6,3,3}, {{CDD||node_h1|6|node|3|node|3|node_1}} or {{CDD|branch_10ru|split2|node|3|node_1}}, has tetrahedron, triangular prism, cuboctahedron, and triangular tiling facets, with a triangular cupola vertex figure.

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The runcicantic hexagonal tiling honeycomb, h_2,3{6,3,3}, {{CDD||node_h1|6|node|3|node_1|3|node_1}} or {{CDD|branch_10ru|split2|node_1|3|node_1}}, has truncated tetrahedron, triangular prism, truncated octahedron, and trihexagonal tiling facets, with a rectangular pyramid vertex figure.

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• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs
• Semiregular honeycomb
• Hexagonal tiling honeycomb

{{reflist}}

• 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] {{Webarchive|url=https://web.archive.org/web/20160610043106/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |date=2016-06-10 }}) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition {{ISBN|0-8247-0709-5}} (Chapters 16–17: Geometries on Three-manifolds I,II)
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, The size of a hyperbolic Coxeter simplex, Transformation Groups (1999), Volume 4, Issue 4, pp 329–353 [https://link.springer.com/article/10.1007%2FBF01238563] [https://web.archive.org/web/20140223225217/http://homeweb1.unifr.ch/kellerha/pub/TGarticle.pdf]
• N. W. Johnson, R. Kellerhals, J. G. Ratcliffe, S. T. Tschantz, Commensurability classes of hyperbolic Coxeter groups, (2002) H³: p130. [http://www.sciencedirect.com/science/article/pii/S0024379501004773]

Category:Hexagonal tilings

Category:3-honeycombs