order-5 hexagonal tiling honeycomb#Alternated order-5 hexagonal tiling honeycomb

A lower-symmetry construction of index 120, [6,(3,5)^*], exists with regular dodecahedral fundamental domains, and an icosahedral Coxeter-Dynkin diagram with 6 axial infinite-order (ultraparallel) branches.

The order-5 hexagonal tiling honeycomb is similar to the 2D hyperbolic regular paracompact order-5 apeirogonal tiling, {∞,5}, with five apeirogonal faces meeting around every vertex.

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The order-5 hexagonal tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.

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There are 15 uniform honeycombs in the [6,3,5] Coxeter group family, including this regular form, and its regular dual, the order-6 dodecahedral honeycomb.

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The order-5 hexagonal tiling honeycomb has a related alternation honeycomb, represented by {{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}, with icosahedron and triangular tiling cells.

It is a part of sequence of regular hyperbolic honeycombs of the form {6,3,p}, with hexagonal tiling facets:

{{Hexagonal tiling cell tessellations}}

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

{{Icosahedral vertex figure tessellations}}

The rectified order-5 hexagonal tiling honeycomb, t₁{6,3,5}, {{CDD|node|6|node_1|3|node|5|node}} has icosahedron and trihexagonal tiling facets, with a pentagonal prism vertex figure.

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It is similar to the 2D hyperbolic infinite-order square tiling, r{∞,5} with pentagon and apeirogonal faces. All vertices are on the ideal surface.

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{{Pentagonal prism vertex figure tessellations}}

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The truncated order-5 hexagonal tiling honeycomb, t_0,1{6,3,5}, {{CDD|node_1|6|node_1|3|node|5|node}} has icosahedron and truncated hexagonal tiling facets, with a pentagonal pyramid vertex figure.

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The bitruncated order-5 hexagonal tiling honeycomb, t_1,2{6,3,5}, {{CDD|node|6|node_1|3|node_1|5|node}} has hexagonal tiling and truncated icosahedron facets, with a digonal disphenoid vertex figure.

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The cantellated order-5 hexagonal tiling honeycomb, t_0,2{6,3,5}, {{CDD|node_1|6|node|3|node_1|5|node}} has icosidodecahedron, rhombitrihexagonal tiling, and pentagonal prism facets, with a wedge vertex figure.

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The cantitruncated order-5 hexagonal tiling honeycomb, t_0,1,2{6,3,5}, {{CDD|node_1|6|node_1|3|node_1|5|node}} has truncated icosahedron, truncated trihexagonal tiling, and pentagonal prism facets, with a mirrored sphenoid vertex figure.

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The runcinated order-5 hexagonal tiling honeycomb, t_0,3{6,3,5}, {{CDD|node_1|6|node|3|node|5|node_1}} has dodecahedron, hexagonal tiling, pentagonal prism, and hexagonal prism facets, with an irregular triangular antiprism vertex figure.

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The runcitruncated order-5 hexagonal tiling honeycomb, t_0,1,3{6,3,5}, {{CDD|node_1|6|node_1|3|node|5|node_1}} has truncated hexagonal tiling, rhombicosidodecahedron, pentagonal prism, and dodecagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure.

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The runcicantellated order-5 hexagonal tiling honeycomb is the same as the runcitruncated order-6 dodecahedral honeycomb.

The omnitruncated order-5 hexagonal tiling honeycomb, t_0,1,2,3{6,3,5}, {{CDD|node_1|6|node_1|3|node_1|5|node_1}} has truncated trihexagonal tiling, truncated icosidodecahedron, decagonal prism, and dodecagonal prism facets, with an irregular tetrahedral vertex figure.

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The alternated order-5 hexagonal tiling honeycomb, h{6,3,5}, {{CDD|node_h1|6|node|3|node|5|node}} ↔ {{CDD|branch_10ru|split2|node|5|node}}, has triangular tiling and icosahedron facets, with a truncated icosahedron vertex figure. It is a quasiregular honeycomb.

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The cantic order-5 hexagonal tiling honeycomb, h₂{6,3,5}, {{CDD|node_h1|6|node|3|node_1|5|node}} ↔ {{CDD|branch_10ru|split2|node_1|5|node}}, has trihexagonal tiling, truncated icosahedron, and icosidodecahedron facets, with a triangular prism vertex figure.

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The runcic order-5 hexagonal tiling honeycomb, h₃{6,3,5}, {{CDD|node_h1|6|node|3|node|5|node_1}} ↔ {{CDD|branch_10ru|split2|node|5|node_1}}, has triangular tiling, rhombicosidodecahedron, dodecahedron, and triangular prism facets, with a triangular cupola vertex figure.

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The runcicantic order-5 hexagonal tiling honeycomb, h_2,3{6,3,5}, {{CDD|node_h1|6|node|3|node_1|5|node_1}} ↔ {{CDD|branch_10ru|split2|node_1|5|node_1}}, has trihexagonal tiling, truncated icosidodecahedron, truncated dodecahedron, 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