Order-4 hexagonal tiling honeycomb

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The order-4 hexagonal tiling honeycomb has three reflective simplex symmetry constructions.

The half-symmetry uniform construction {6,3^1,1} has two types (colors) of hexagonal tilings, with Coxeter diagram {{CDD|node_1|6|node|3|node|4|node_h0}} ↔ {{CDD|node_1|6|node|split1|nodes}}. A quarter-symmetry construction also exists, with four colors of hexagonal tilings: {{CDD|label6|branch_10r|3ab|branch_10l|label6}}.

An additional two reflective symmetries exist with non-simplectic fundamental domains: [6,3^*,4], which is index 6, with Coxeter diagram {{CDD|node|ultra|node_1|split1|branch_11|uaub|nodes}}; and [6,(3,4)^*], which is index 48. The latter has a cubic fundamental domain, and an octahedral Coxeter diagram with three axial infinite branches: File:CDel K6 636 11.png. It can be seen as using eight colors to color the hexagonal tilings of the honeycomb.

The order-4 hexagonal tiling honeycomb contains {{CDD|node_1|3|node_1|ultra|node}}, which tile 2-hypercycle surfaces and are similar to the truncated infinite-order triangular tiling, {{CDD|node_1|3|node_1|infin|node}}:

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The order-4 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

{{Regular_paracompact_H3_honeycombs}}

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form, and its dual, the order-6 cubic honeycomb.

{{634 family}}

The order-4 hexagonal tiling honeycomb has a related alternated honeycomb, {{CDD|node_h1|6|node|3|node|4|node}} ↔ {{CDD|branch_10ru|split2|node|4|node}}, with triangular tiling and octahedron cells.

It is a part of sequence of regular honeycombs of the form {6,3,p}, all of which are composed of hexagonal tiling cells:

{{Hexagonal tiling cell tessellations}}

This honeycomb is also related to the 16-cell, cubic honeycomb and order-4 dodecahedral honeycomb, all of which have octahedral vertex figures.

{{Octahedral_vertex_figure_tessellations}}

The aforementioned honeycombs are also quasiregular:

{{Regular and Quasiregular honeycombs}}

The rectified order-4 hexagonal tiling honeycomb, t₁{6,3,4}, {{CDD|node|6|node_1|3|node|4|node}} has octahedral and trihexagonal tiling facets, with a square prism vertex figure.

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It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{∞,4}, {{CDD|node|infin|node_1|4|node}} which alternates apeirogonal and square faces:

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The truncated order-4 hexagonal tiling honeycomb, t_0,1{6,3,4}, {{CDD|node_1|6|node_1|3|node|4|node}} has octahedron and truncated hexagonal tiling facets, with a square pyramid vertex figure.

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It is similar to the 2D hyperbolic truncated order-4 apeirogonal tiling, t{∞,4}, {{CDD|node_1|infin|node_1|4|node}} with apeirogonal and square faces:

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The bitruncated order-4 hexagonal tiling honeycomb, t_1,2{6,3,4}, {{CDD|node|6|node_1|3|node_1|4|node}} has truncated octahedron and hexagonal tiling cells, with a digonal disphenoid vertex figure.

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The cantellated order-4 hexagonal tiling honeycomb, t_0,2{6,3,4}, {{CDD|node_1|6|node|3|node_1|4|node}} has cuboctahedron, cube, and rhombitrihexagonal tiling cells, with a wedge vertex figure.

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The cantitruncated order-4 hexagonal tiling honeycomb, t_0,1,2{6,3,4}, {{CDD|node_1|6|node_1|3|node_1|4|node}} has truncated octahedron, cube, and truncated trihexagonal tiling cells, with a mirrored sphenoid vertex figure.

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The runcinated order-4 hexagonal tiling honeycomb, t_0,3{6,3,4}, {{CDD|node_1|6|node|3|node|4|node_1}} has cube, hexagonal tiling and hexagonal prism cells, with an irregular triangular antiprism vertex figure.

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It contains the 2D hyperbolic rhombitetrahexagonal tiling, rr{4,6}, {{CDD|node_1|6|node|4|node_1}} with square and hexagonal faces. The tiling also has a half symmetry construction {{CDD|branch_11|2a2b-cross|nodes_11}}.

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The runcitruncated order-4 hexagonal tiling honeycomb, t_0,1,3{6,3,4}, {{CDD|node_1|6|node_1|3|node|4|node_1}} has rhombicuboctahedron, cube, dodecagonal prism, and truncated hexagonal tiling cells, with an isosceles-trapezoidal pyramid vertex figure.

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The runcicantellated order-4 hexagonal tiling honeycomb is the same as the runcitruncated order-6 cubic honeycomb.

The omnitruncated order-4 hexagonal tiling honeycomb, t_0,1,2,3{6,3,4}, {{CDD|node_1|6|node_1|3|node_1|4|node_1}} has truncated cuboctahedron, truncated trihexagonal tiling, dodecagonal prism, and octagonal prism cells, with an irregular tetrahedron vertex figure.

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The alternated order-4 hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node|4|node}} ↔ {{CDD|branch_10ru|split2|node|4|node}}, is composed of triangular tiling and octahedron cells, in a truncated octahedron vertex figure.

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The cantic order-4 hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|4|node}} ↔ {{CDD|branch_10ru|split2|node_1|4|node}}, is composed of trihexagonal tiling, truncated octahedron, and cuboctahedron cells, with a wedge vertex figure.

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The runcic order-4 hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node|4|node_1}} ↔ {{CDD|branch_10ru|split2|node|4|node_1}}, is composed of triangular tiling, rhombicuboctahedron, cube, and triangular prism cells, with a triangular cupola vertex figure.

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The runcicantic order-4 hexagonal tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|4|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|4|node_1}}, is composed of trihexagonal tiling, truncated cuboctahedron, truncated cube, and triangular prism cells, with a rectangular pyramid vertex figure.

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The quarter order-4 hexagonal tiling honeycomb, q{6,3,4}, {{CDD|node_h1|6|node|3|node|4|node_h1}} or {{CDD|node_1|split1|branch_10luru|split2|node}}, is composed of triangular tiling, trihexagonal tiling, tetrahedron, and truncated tetrahedron cells, with a triangular cupola vertex figure.

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• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs

{{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]) 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:Hexagonal tilings

Category:Regular 3-honeycombs