Triangular tiling honeycomb: Difference between revisions

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|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|6|node|3|node}}<BR>{{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}<BR>{{CDD|node_h1|6|node|split1|branch}} ↔ {{CDD|node_1|splitsplit1|branch4|splitsplit2|node}} ↔ {{CDD|branch_10ru|split2|node|6|node_h0}}

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|bgcolor=#e7dcc3|Cells||[[Triangular tiling|{3,6}]] [[File:Uniform tiling 63-t2.pngsvg|40px]] [[File:Uniform tiling 333-t1.pngsvg|40px]]

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}

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|bgcolor=#e7dcc3|[[Edge figure]]||[[triangle]] {3}

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|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Uniform tiling 63-t0.pngsvg|40px]] [[File:Uniform tiling 63-t12.pngsvg|40px]] [[File:Uniform tiling 333-t012.pngsvg|40px]]<BR>[[hexagonal tiling]]

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|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[Self-dual polytope|Self-dual]]

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{{Honeycomb}}

== Symmetry ==

[[File:Hyperbolic subgroup tree 363.png|left|thumb|Subgroups of [3,6,3] and [6,3,6]]]

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It has two lower reflective symmetry constructions, as an [[Alternation (geometry)|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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== Related Tilings ==

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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|3|node_1|6|node|3|node}}<BR>{{CDD|node_h1|6|node|3|node_1|6|node}} ↔ {{CDD|branch_10ru|split2|node_1|6|node}}<BR>{{CDD|node|splitsplit1|branch4_11|splitsplit2|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|6|node_h0}} ↔ {{CDD|node|3|node_1|6|node_g|3sg|node_g}}

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|bgcolor=#e7dcc3|Cells||[[trihexagonal tiling|r{3,6}]] [[File:Uniform polyhedron-63-t1.pngsvg|40px]]<BR>[[hexagonal tiling|{6,3}]] [[File:Uniform polyhedron-63-t0.png|40px]]

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[hexagon]] {6}

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[[File:H3 363 boundary 0100.png|480px]]

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=== Truncated triangular tiling honeycomb===

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[[File:H3 363-1100.png|480px]]

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=== Bitruncated triangular tiling honeycomb===

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[[File:H3 363-0110.png|480px]]

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=== Cantellated triangular tiling honeycomb===

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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node|6|node_1|3|node}}<BR>{{CDD|node_h|3|node_h|6|node_1|3|node}}

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|bgcolor=#e7dcc3|Cells||[[rhombitrihexagonal tiling|rr{6,3}]] [[File:Uniform polyhedron-63-t02.png|40px]]<BR>[[trihexagonal tiling|r{6,3}]] [[File:Uniform polyhedron-63-t1.pngsvg|40px]]<BR>[[Triangular prism|{}×{3}]] [[File:Triangular prism.png|40px]]

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}

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[[File:H3 363-1010.png|480px]]

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=== Cantitruncated triangular tiling honeycomb===

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[[File:H3 363-1110.png|480px]]

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=== Runcinated triangular tiling honeycomb===

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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|3|node|6|node|3|node_1}}

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|bgcolor=#e7dcc3|Cells||[[triangular tiling|{3,6}]] [[File:Uniform polyhedron-63-t2.pngsvg|40px]]<BR>[[Triangular prism|{}×{3}]] [[File:Triangular prism.png|40px]]

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}

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[[File:H3 363-1001.png|480px]]

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=== Runcitruncated triangular tiling honeycomb===

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[[File:H3 363-1101.png|480px]]

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=== Omnitruncated triangular tiling honeycomb===

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[[File:H3 363-1111.png|480px]]

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=== Runcisnub triangular tiling honeycomb===

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|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h|3|node_h|6|node|3|node_1}}

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|bgcolor=#e7dcc3|Cells|||[[Trihexagonal tiling|r{6,3}]] [[File:Uniform tiling 333-t02.pngsvg|40px]]<BR>[[Triangular prism|{}x{3}]] [[File:triangular prism.png|40px]]<BR>[[triangular tiling|{3,6}]] [[File:Uniform tiling 333-t1.pngsvg|40px]]<BR>[[triangular cupola|tricup]] [[File:Triangular cupola.png|40px]]

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}

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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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== See also ==

* [[Convex uniform honeycombs in hyperbolic space]]

* [[List_of_regular_polytopesList of regular polytopes#Tessellations_of_hyperbolic_3Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]

* [[Paracompact uniform honeycomb]]s

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*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;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 Weeks (mathematician)|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 (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript

** [[Norman Johnson (mathematician)|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:HoneycombsRegular (geometry)3-honeycombs]]

[[Category:Self-dual tilings]]

[[Category:Triangular tilings]]