Order-6 hexagonal tiling honeycomb

The order-6 hexagonal tiling honeycomb is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞}, with infinite apeirogonal faces, and with all vertices on the ideal surface.

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It contains {{CDD|node_1|3|node_1|ultra|node}} and {{CDD|node|3|node|ultra|node_1}} that tile 2-hypercycle surfaces, which are similar to the paracompact tilings {{CDD|node_1|3|node_1|infin|node}} and {{CDD|node|3|node|infin|node_1}} (the truncated infinite-order triangular tiling and order-3 apeirogonal tiling, respectively):

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File:Hyperbolic subgroup tree 636.png

The order-6 hexagonal tiling honeycomb has a half-symmetry construction: {{CDD|node_1|6|node|split1|branch}}.

It also has an index-6 subgroup, [6,3^*,6], with a non-simplex fundamental domain. This subgroup corresponds to a Coxeter diagram with six order-3 branches and three infinite-order branches in the shape of a triangular prism: {{CDD|node_1|splitplit1u|branch4u_11|uabc|branch4u|splitplit2u|node}}.

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The order-6 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs in 3-space.

{{Regular_paracompact_H3_honeycombs}}

There are nine uniform honeycombs in the [6,3,6] Coxeter group family, including this regular form.

{{636 family}}

This honeycomb has a related alternated honeycomb, the triangular tiling honeycomb, but with a lower symmetry: {{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}.

The order-6 hexagonal tiling honeycomb is part of a sequence of regular polychora and honeycombs with triangular tiling vertex figures:

{{Triangular tiling vertex figure tessellations small}}

It is also part of a sequence of regular polychora and honeycombs with hexagonal tiling cells:

{{Hexagonal tiling cell tessellations}}

It is also part of a sequence of regular polychora and honeycombs with regular deltahedral vertex figures:

{{Symmetric2_tessellations}}

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

it can also be seen as a quarter order-6 hexagonal tiling honeycomb, q{6,3,6}, {{CDD|node_h1|6|node|3|node|6|node_h1}} ↔ {{CDD|branch_11|splitcross|branch}}.

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It is analogous to 2D hyperbolic order-4 apeirogonal tiling, r{∞,∞} with infinite apeirogonal faces, and with all vertices on the ideal surface.

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The order-6 hexagonal tiling honeycomb is part of a series of honeycombs with hexagonal prism vertex figures:

{{Hexagonal tiling vertex figure tessellations}}

It is also part of a matrix of 3-dimensional quarter honeycombs: q{2p,4,2q}

{{Quarter hyperbolic honeycomb table}}

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The truncated order-6 hexagonal tiling honeycomb, t_0,1{6,3,6}, {{CDD|node_1|6|node_1|3|node|6|node}} has triangular tiling and truncated hexagonal tiling facets, with a hexagonal pyramid vertex figure.[https://twitter.com/roice713/status/1111121384875483136 Twitter] Rotation around 3 fold axis

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The bitruncated order-6 hexagonal tiling honeycomb is a lower symmetry construction of the regular hexagonal tiling honeycomb, {{CDD|node|6|node_1|3|node_1|6|node}} ↔ {{CDD|node_1|6|node|3|node|3|node}}. It contains hexagonal tiling facets, with a tetrahedron vertex figure.

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

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

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

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It is analogous to the 2D hyperbolic rhombihexahexagonal tiling, rr{6,6}, {{CDD|node_1|6|node|6|node_1}} with square and hexagonal faces:

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

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The omnitruncated order-6 hexagonal tiling honeycomb, t_0,1,2,3{6,3,6}, {{CDD|node_1|6|node_1|3|node_1|6|node_1}} has truncated trihexagonal tiling and dodecagonal prism cells, with a phyllic disphenoid vertex figure.

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The alternated order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the regular triangular tiling honeycomb, {{CDD|node_h1|6|node|3|node|6|node}} ↔ {{CDD|branch_10ru|split2|node|6|node}}. It contains triangular tiling facets in a hexagonal tiling vertex figure.

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The cantic order-6 hexagonal tiling honeycomb is a lower-symmetry construction of the rectified triangular tiling honeycomb, {{CDD|node_h1|6|node|3|node_1|6|node}} ↔ {{CDD|branch_10ru|split2|node_1|6|node}}, with trihexagonal tiling and hexagonal tiling facets in a triangular prism vertex figure.

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The runcic hexagonal tiling honeycomb, h₃{6,3,6}, {{CDD|node_h1|6|node|3|node|6|node_1}}, or {{CDD|branch_10ru|split2|node|6|node_1}}, has hexagonal tiling, rhombitrihexagonal tiling, triangular tiling, and triangular prism facets, with a triangular cupola vertex figure.

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The runcicantic order-6 hexagonal tiling honeycomb, h_2,3{6,3,6}, {{CDD|node_h1|6|node|3|node_1|6|node_1}}, or {{CDD|branch_10ru|split2|node_1|6|node_1}}, contains truncated trihexagonal tiling, truncated hexagonal tiling, trihexagonal tiling, and triangular prism facets, with a rectangular pyramid vertex figure.

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

• 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

Category:Self-dual tilings