Triangular tiling honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Triangular tiling honeycomb |- |bgcolor=#ffffff align=center colspan=2|320px |- |bgcolor=#e7dcc3|Type||Hyperbolic regular honeycomb<BR>Paracompact uniform honeycomb |- |bgcolor=#e7dcc3|Schläfli symbol||{3,6,3}<BR>h{6,3,6}<BR>h{6,3^[3]} ↔ {3^[3,3]} |- |bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||{{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||{3,6} 40px 40px |- |bgcolor=#e7dcc3|Faces||triangle {3} |- |bgcolor=#e7dcc3|Edge figure||triangle {3} |- |bgcolor=#e7dcc3|Vertex figure||40px 40px 40px<BR>hexagonal tiling |- |bgcolor=#e7dcc3|Dual||Self-dual |- |bgcolor=#e7dcc3|Coxeter groups||<math>\overline{Y}_3</math>, [3,6,3]<BR><math>\overline{VP}_3</math>, [6,3^[3]]<BR><math>\overline{PP}_3</math>, [3^[3,3]] |- |bgcolor=#e7dcc3|Properties||Regular |} The '''triangular tiling honeycomb''' is one of 11 paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called ''paracompact'' because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

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

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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== Related Tilings ==

It is similar to the 2D hyperbolic infinite-order apeirogonal tiling, {&infin;,&infin;}, with infinite apeirogonal faces, and with all vertices on the ideal surface. : 240px

== Related honeycombs == The triangular tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs. {{Regular_paracompact_H3_honeycombs}}

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. {{363_family}}

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

=== Rectified triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Rectified triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |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|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}} |- |bgcolor=#e7dcc3|Cells||r{3,6} 40px<BR>{6,3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>triangular prism |- |bgcolor=#e7dcc3|Coxeter group||<math>\overline{Y}_3</math>, [3,6,3]<BR><math>\overline{VP}_3</math>, [6,3^[3]]<BR><math>\overline{PP}_3</math>, [3^[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.

==== Symmetry==== A lower symmetry of this honeycomb can be constructed as a ''cantic order-6 hexagonal tiling honeycomb'', {{CDD|branch_10ru|split2|node_1|6|node}} ↔ {{CDD|node_h1|6|node|3|node_1|6|node}}. A second lower-index construction is {{CDD|node|3|node_1|6|node_g|3sg|node_g}} ↔ {{CDD|node|splitsplit1|branch4_11|splitsplit2|node_1}}.

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=== Truncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Truncated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||t{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|3|node_1|6|node|3|node}}<BR>{{CDD|node_1|6|node|3|node|3|node}} |- |bgcolor=#e7dcc3|Cells||t{3,6} 40px<BR>{6,3} 40px |- |bgcolor=#e7dcc3|Faces||hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>tetrahedron |- |bgcolor=#e7dcc3|Coxeter group||<math>\overline{Y}_3</math>, [3,6,3]<BR><math>\overline{V}_3</math>, [3,3,6] |- |bgcolor=#e7dcc3|Properties||Regular |}

The '''truncated triangular tiling honeycomb''', {{CDD|node_1|3|node_1|6|node|3|node}}, is a lower-symmetry form of the hexagonal tiling honeycomb, {{CDD|node_1|6|node|3|node|3|node}}. It contains hexagonal tiling facets with a tetrahedral vertex figure.

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=== Bitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Bitruncated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||2t{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node|3|node_1|6|node_1|3|node}} |- |bgcolor=#e7dcc3|Cells||t{6,3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>dodecagon {12} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>tetragonal disphenoid |- |bgcolor=#e7dcc3|Coxeter group||<math>2\times\overline{Y}_3</math>, <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 tetragonal disphenoid vertex figure.

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=== Cantellated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Cantellated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||rr{3,6,3} or t_0,2{3,6,3}<BR>s₂{3,6,3} |- |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||rr{6,3} 40px<BR>r{6,3} 40px<BR>{}×{3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>wedge |- |bgcolor=#e7dcc3|Coxeter group||<math>\overline{Y}_3</math>, [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 wedge vertex figure.

==== 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].

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=== Cantitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Cantitruncated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||tr{3,6,3} or t_0,1,2{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|3|node_1|6|node_1|3|node}} |- |bgcolor=#e7dcc3|Cells||tr{6,3} 40px<BR>t{6,3} 40px<BR>{}×{3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4}<BR>hexagon {6}<BR>dodecagon {12} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>mirrored sphenoid |- |bgcolor=#e7dcc3|Coxeter group||<math>\overline{Y}_3</math>, [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 mirrored sphenoid vertex figure.

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=== Runcinated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcinated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||t_0,3{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|3|node|6|node|3|node_1}} |- |bgcolor=#e7dcc3|Cells||{3,6} 40px<BR>{}×{3} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>hexagonal antiprism |- |bgcolor=#e7dcc3|Coxeter group||<math>2\times\overline{Y}_3</math>, <nowiki></nowiki>3,6,3 |- |bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive |} The '''runcinated triangular tiling honeycomb''', {{CDD|node_1|3|node|6|node|3|node_1}}, has triangular tiling and triangular prism cells, with a hexagonal antiprism vertex figure.

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=== Runcitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcitruncated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbols||t_0,1,3{3,6,3}<BR>s_2,3{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node_1|3|node_1|6|node|3|node_1}}<BR>{{CDD|node_h|3|node_h|6|node_1|3|node_1}} |- |bgcolor=#e7dcc3|Cells||t{3,6} 40px<BR>rr{3,6} 40px<BR>{}×{3} 40px<BR>{}×{6} 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>isosceles-trapezoidal pyramid |- |bgcolor=#e7dcc3|Coxeter group||<math>\overline{Y}_3</math>, [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 an isosceles-trapezoidal 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].

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=== Omnitruncated triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Omnitruncated triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact uniform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||t_0,1,2,3{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_1|3|node_1|6|node_1|3|node_1}} |- |bgcolor=#e7dcc3|Cells||tr{3,6} 40px<BR>{}×{6} 40px |- |bgcolor=#e7dcc3|Faces||square {4}<BR>hexagon {6}<BR>dodecagon {12} |- |bgcolor=#e7dcc3|Vertex figure||80px<BR>phyllic disphenoid |- |bgcolor=#e7dcc3|Coxeter group||<math>2\times\overline{Y}_3</math>, <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 phyllic disphenoid vertex figure.

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=== Runcisnub triangular tiling honeycomb=== {| class="wikitable" align="right" style="margin-left:10px" width="320" !bgcolor=#e7dcc3 colspan=2|Runcisnub triangular tiling honeycomb <!--|- |bgcolor=#ffffff align=center colspan=2|--> |- |bgcolor=#e7dcc3|Type||Paracompact scaliform honeycomb |- |width=100 bgcolor=#e7dcc3|Schläfli symbol||s₃{3,6,3} |- |bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node_h|3|node_h|6|node|3|node_1}} |- |bgcolor=#e7dcc3|Cells|||r{6,3} 40px<BR>{}x{3} 40px<BR>{3,6} 40px<BR>tricup 40px |- |bgcolor=#e7dcc3|Faces||triangle {3}<BR>square {4}<BR>hexagon {6} |- |bgcolor=#e7dcc3|Vertex figure||<!--80px--> |- |bgcolor=#e7dcc3|Coxeter group||<math>\overline{Y}_3</math>, [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.

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== See also == * Convex uniform honeycombs in hyperbolic space * Regular tessellations of hyperbolic 3-space * Paracompact uniform honeycombs

== References == *Coxeter, ''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 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