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| !bgcolor=#e7dcc3 colspan=2|Order-6 tetrahedral honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[File:H3 336 CC center.png|320px]]<BR>[[Perspective projection]] view<BR>within [[Poincaré disk model]]
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| |bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]s||{3,3,6}<BR>{3,3<sup>[3]</sup>}
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| |bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|3|node|6|node}}<BR>{{CDD|node_1|3|node|3|node|6|node_h0}} = {{CDD|node_1|3|node|split1|branch}}
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| |bgcolor=#e7dcc3|Cells||[[Tetrahedron]] {3,3}<BR>[[File:Uniform_polyhedron-33-t0.png|40px]]
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| |bgcolor=#e7dcc3|Faces||[[Triangle]] {3}
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| |bgcolor=#e7dcc3|Edge figure||[[Hexagon]] {6}
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| |bgcolor=#e7dcc3|Vertex figure||[[Triangular tiling]] {3,6}<BR>[[File:Uniform_tiling_63-t2.png|80px]] [[File:Uniform tiling 333-t1.png|80px]]
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| |bgcolor=#e7dcc3|Dual||[[Hexagonal tiling honeycomb]], {6,3,3}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin_diagram#Ranks_4.E2.80.9310|Coxeter groups]]||<math>{\bar{V}}_3</math>, [6,3,3]<BR><math>{\bar{P}}_3</math>, [3,3<sup>[3]</sup>]
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| |bgcolor=#e7dcc3|Properties||Regular
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| In the [[geometry]] of [[Hyperbolic space|hyperbolic 3-space]], the '''order-6 tetrahedral honeycomb''' a regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]). With [[Schläfli symbol]] {3,3,6}. It has six [[tetrahedra]] {3,3} around each edge. All vertices are [[ideal vertex|ideal vertices]] with infinitely many tetrahedra existing around each ideal vertex in an [[triangular tiling]] [[vertex arrangement]]. <ref>Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III</ref>
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| == Symmetry constructions ==
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| [[File:Hyperbolic subgroup tree 336-direct.png|200px|thumb|left|[[Coxeter_diagram#Subgroup_relations_of_paracompact_hyperbolic_groups|Subgroup relations]]]]
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| It has a second construction as a uniform honeycomb, [[Schläfli symbol]] {3,3<sup>[3]</sup>}, with alternating types or colors of tetrahedral cells.
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| == Related polytopes and honeycombs ==
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| It is [[List of regular polytopes#Tessellations of hyperbolic 3-space|one of 15 regular hyperbolic honeycombs]] in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.
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| It is similar to the 2-dimensional hyperbolic tiling, [[infinite-order triangular tiling]], {3,∞}, for having all ideal vertices made of regular simplices.
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| :[[File:H2checkers iii.png|120px]]
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| It is [[Paracompact_uniform_honeycomb#.5B6.2C3.2C3.5D_family|one of 15 uniform paracompact honeycombs]] in the [6,3,3] Coxeter group, along with its dual [[hexagonal tiling honeycomb]], {6,3,3}.
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| The ''rectified order-6 tetrahedral honeycomb'', t<sub>1</sub>{3,3,6} has tetrahedron and triangular tiling cells connected in a hexagonal prism vertex figure:
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| : [[File:Hyperbolic rectified order-6 tetrahedral honeycomb.png|240px]]
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| It a part of a sequence of [[regular polychora]] and honeycombs with [[tetrahedron|tetrahedral]] [[cell (geometry)|cells]].
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| {{Tetrahedral cell tessellations}}
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| It a part of a sequence of honeycombs with [[triangular tiling]] [[vertex figure]]s.
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| {{Hexagonal tiling vertex figure tessellations}}
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| == See also ==
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| * [[Convex uniform honeycombs in hyperbolic space]]
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| * [[List of regular polytopes]]
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| == References ==
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| {{reflist}}
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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. 294–296)
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| * ''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
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| * [[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)
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| [[Category:Honeycombs (geometry)]]
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