order-5 cubic honeycomb#Alternated order-5 cubic honeycomb

File:H2-5-4-primal.svg, {4,5}]]

It has a radical subgroup symmetry construction with dodecahedral fundamental domains: Coxeter notation: [4,(3,5)^*], index 120.

The order-5 cubic honeycomb has a related alternated honeycomb, {{CDD|node_h1|4|node|3|node|5|node}} ↔ {{CDD|nodes_10ru|split2|node|5|node}}, with icosahedron and tetrahedron cells.

The honeycomb is also one of four regular compact honeycombs in 3D hyperbolic space:

{{Regular compact H3 honeycombs}}

There are fifteen uniform honeycombs in the [5,3,4] Coxeter group family, including the order-5 cubic honeycomb as the regular form:

{{534 family}}

The order-5 cubic honeycomb is in a sequence of regular polychora and honeycombs with icosahedral vertex figures.

{{Icosahedral vertex figure tessellations}}

It is also in a sequence of regular polychora and honeycombs with cubic cells. The first polytope in the sequence is the tesseract, and the second is the Euclidean cubic honeycomb.

{{Cubic cell tessellations}}

In 3-dimensional hyperbolic geometry, the alternated order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb). With Schläfli symbol h{4,3,5}, it can be considered a quasiregular honeycomb, alternating icosahedra and tetrahedra around each vertex in an icosidodecahedron vertex figure.

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It has 3 related forms: the cantic order-5 cubic honeycomb, {{CDD|node_h1|4|node|3|node_1|5|node}}, the runcic order-5 cubic honeycomb, {{CDD|node_h1|4|node|3|node|5|node_1}}, and the runcicantic order-5 cubic honeycomb, {{CDD|node_h1|4|node|3|node_1|5|node_1}}.

The cantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₂{4,3,5}. It has icosidodecahedron, truncated icosahedron, and truncated tetrahedron cells, with a rectangular pyramid vertex figure.

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The runcic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h₃{4,3,5}. It has dodecahedron, rhombicosidodecahedron, and tetrahedron cells, with a triangular frustum vertex figure.

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The runcicantic order-5 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h_2,3{4,3,5}. It has truncated dodecahedron, truncated icosidodecahedron, and truncated tetrahedron cells, with an irregular tetrahedron vertex figure.

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

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• Coxeter, The Beauty of Geometry: Twelve Essays, Dover Publications, 1999 {{isbn|0-486-40919-8}} (Chapter 10: Regular honeycombs in hyperbolic space, Summary tables II, III, IV, V, p212-213)
• 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, (2015) Chapter 13: Hyperbolic Coxeter groups

Category:Regular 3-honeycombs