Triangular tiling honeycomb: Difference between revisions

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{| class="wikitable" align="right" style="margin-left:10px" width="250320"

!bgcolor=#e7dcc3 colspan=2|Triangular tiling honeycomb

|-

|bgcolor=#ffffff align=center colspan=2|[[File:H3 363 FC boundary.png|300px320px]]

|-

|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.3B6.2C6.2C3.5D family|Paracompact uniform honeycomb]]

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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}}

|-

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

|-

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

|-

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

|-

|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]]

|-

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

|-

|bgcolor=#e7dcc3|[[Coxeter group]]s||{{<math>\overline|{Y}}<sub>3_3</submath>, [3,6,3]<BR><math>\overline{VP}_3</math>, [6,3^[3]]<BR><math>\baroverline{PP}}_3</math>, [3^[3,3]]

|-

|bgcolor=#e7dcc3|Properties||Regular

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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}}.

{{-Clear}}

== Related Tilings ==

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|width=100 bgcolor=#e7dcc3|[[Schläfli symbol]]||r{3,6,3}<BR>h₂{6,3,6}

|-

|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node|3|node_1|6|node|3|node}}<BR>{{CDD|nodenode_h1|splitsplit16|branch4_11node|splitsplit23|node_1|6|node}} ↔ {{CDD|branch_10ru|split2|node_1|6|node_h0node}} ↔ <BR>{{CDD|node|3splitsplit1|node_1branch4_11|6splitsplit2|node_g|3sg|node_gnode_1}}<BR> ↔ {{CDD|node_h1branch_10ru|6|node|3split2|node_1|6|nodenode_h0}} ↔ {{CDD|branch_10runode|split23|node_1|6|nodenode_g|3sg|node_g}}

|-

|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]]

|-

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

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified triangular tiling honeycomb verf.png|80px]]<BR>[[triangular prism]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||{{<math>\overline|{Y}_3</math>, [3,6,3]<BR><math>\overline{VP}_3<sub/math>, [6,3^[3]]<BR><math>\overline{PP}_3</submath>, [3,6^[3,3]]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive

|}

The '''rectified triangular tiling honeycomb''', {{CDD|node|3|node_1|6|node|3|node}}, has [[trihexagonal tiling]] and [[hexagonal tiling]] cells, with a [[triangular prism]] vertex figure.

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

{{-Clear}}

=== Truncated triangular tiling honeycomb===

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|bgcolor=#e7dcc3|Faces||[[hexagon]] {6}

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated triangular tiling honeycomb verf.png|80px]]<BR>[[tetrahedron]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||{{<math>\overline|{Y}}_3<sub/math>, [3,6,3]<BR><math>\overline{V}_3</submath>, [3,6,3,6]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitiveRegular

|}

The '''truncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node|3|node}}, is thea samelower-symmetry asform of the [[hexagonal tiling honeycomb]], {{CDD|node_1|6|node|3|node|3|node}}. It contains [[hexagonal tiling]] facets with a [[tetrahedron|tetrahedral]] vertex figure.

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

{{-Clear}}

=== Bitruncated triangular tiling honeycomb===

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[dodecagon]] {12}

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Bitruncated triangular tiling honeycomb verf.png|60px80px]]<BR>[[tetrahedrontetragonal disphenoid]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||<math>2\times\overline{{overline|Y}}<sub>3_3</submath>×2, <nowiki>[[</nowiki>3,6,3]]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive, cell-transitive

|}

The '''bitruncated triangular tiling honeycomb''', {{CDD|node|3|node_1|6|node_1|3|node}}, has [[truncated hexagonal tiling]] cells, with a [[tetrahedron|tetrahedraltetragonal disphenoid]] vertex figure.

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

{{-Clear}}

=== 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}}

|-

|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]]

|-

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

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantellated triangular tiling honeycomb verf.png|80px]]<BR>[[triangularwedge prism(geometry)|wedge]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{{overline|Y}}<sub>3_3</submath>, [3,6,3]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive

|}

The '''cantellated triangular tiling honeycomb''', {{CDD|node_1|3|node|6|node_1|3|node}}, has [[rhombitrihexagonal tiling]], [[trihexagonal tiling]], and [[triangular prism]] cells, with a [[triangularwedge prism(geometry)|wedge]] vertex figure.

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

==== Symmetry====

It can also be constructed as a '''cantic snub triangular tiling honeycomb''', {{CDD|node_h|3|node_h|6|node_1|3|node}}, a half-symmetry form with symmetry [3⁺,6,3].

{{-}}

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

{{-Clear}}

=== Cantitruncated triangular tiling honeycomb===

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|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[hexagon]] {6}<BR>[[dodecagon]] {12}

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Cantitruncated triangular tiling honeycomb verf.png|80px]]<BR>[[tetrahedronmirrored sphenoid]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{{overline|Y}}<sub>3_3</submath>, [3,6,3]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive

|}

The '''cantitruncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node_1|3|node}}, has [[truncated trihexagonal tiling]], [[truncated hexagonal tiling]], and [[triangular prism]] cells, with a [[tetrahedronmirrored sphenoid]] vertex figure.

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

{{-Clear}}

=== Runcinated triangular tiling honeycomb===

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

|-

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

|-

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

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcinated triangular tiling honeycomb verf.png|80px]]<BR>[[hexagonal antiprism]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||2×<math>2\times\overline{{overline|Y}}<sub>3_3</submath>, <nowiki>[[</nowiki>3,6,3]]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive

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

{{-Clear}}

=== Runcitruncated triangular tiling honeycomb===

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

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcitruncated triangular tiling honeycomb verf.png|80px]]<BR>quadrilateral[[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{{overline|Y}}<sub>3_3</submath>, [3,6,3]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive

|}

The '''runcitruncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node|3|node_1}}, has [[hexagonal tiling]], [[rhombitrihexagonal tiling]], [[triangular prism]], and [[hexagonal prism]] cells, with aan quadrilateral[[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]] [[vertex figure]].

==== Symmetry====

It can also be constructed as a '''runcicantic snub triangular tiling honeycomb''', {{CDD|node_h|3|node_h|6|node_1|3|node_1}}, a half-symmetry form with symmetry [3⁺,6,3].

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

{{-Clear}}

=== Omnitruncated triangular tiling honeycomb===

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|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[dodecagon]] {12}

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Omnitruncated triangular tiling honeycomb verf.png|80px]]<BR>[[Phyllicphyllic disphenoid]]

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||2×<math>2\times\overline{{overline|Y}}<sub>3_3</submath>, <nowiki>[[</nowiki>3,6,3]]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive

|}

The '''omnitruncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node_1|3|node_1}}, has [[truncated trihexagonal tiling]] and [[hexagonal prism]] cells, with a [[tetrahedronphyllic disphenoid]] vertex figure.

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

{{-Clear}}

=== Runcisnub triangular tiling honeycomb===

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

|-

|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]]

|-

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

|-

|bgcolor=#e7dcc3|[[Vertex figure]]||<!--[[File:Runcisnub_triangular tiling_honeycomb_verf.png|80px]]-->

|-

|bgcolor=#e7dcc3|[[Coxeter group]]||<math>\overline{{overline|Y}}<sub>3_3</submath>, [3⁺,6,3]

|-

|bgcolor=#e7dcc3|Properties||Vertex-transitive, non-uniform

|}

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.

{{-Clear}}

== See also ==

* [[Convex uniform honeycombs in hyperbolic space]]

* [[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]

* [[Paracompact uniform honeycomb]]s

== References ==

*[[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]]