cubic-octahedral honeycomb#Truncated cubic-octahedral honeycomb

Wide-angle perspective views:

File:H3 4343-0010 center ultrawide.png|Centered on cube

File:H3 4343-1000 center ultrawide.png|Centered on octahedron

File:H3 4343-0001 center ultrawide.png|Centered on cuboctahedron

It contains a subgroup H2 tiling, the alternated order-4 hexagonal tiling, {{CDD|nodes_11|3a3b-cross|nodes}}, with vertex figure (3.4)⁴.

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A lower symmetry form, index 6, of this honeycomb can be constructed with [(4,3,4,3^*)] symmetry, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram {{CDD|node_1|splitplit1u|branch3u|3a3buc-cross|branch3u_11|splitplit2u|node}}. This lower symmetry can be extended by restoring one mirror as {{CDD|branchu_01r|3ab|branch_10lru|split2-44|node}}.

There are 5 related uniform honeycombs generated within the same family, generated with 2 or more rings of the Coxeter group {{CDD|label4|branch|3ab|branch|label4}}: {{CDD|label4|branch_10r|3ab|branch_10l|label4}}, {{CDD|label4|branch_01r|3ab|branch_10l|label4}}, {{CDD|label4|branch_11|3ab|branch|label4}}, {{CDD|label4|branch_11|3ab|branch_10l|label4}}, {{CDD|label4|branch_11|3ab|branch_11|label4}}.

The rectified cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from cuboctahedron and rhombicuboctahedron cells, in a cuboid vertex figure. It has a Coxeter diagram {{CDD|label4|branch_01r|3ab|branch_10l|label4}}.

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:Perspective view from center of rhombicuboctahedron

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The cyclotruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cube and octahedron cells, in a square antiprism vertex figure. It has a Coxeter diagram {{CDD|label4|branch_11|3ab|branch|label4}}.

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:Perspective view from center of octahedron

It can be seen as somewhat analogous to the trioctagonal tiling, which has truncated square and triangle facets:

: 160px

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|bgcolor=#e7dcc3|Cells||{4,3} 40px
t{3,4} 40px

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|bgcolor=#e7dcc3|Faces||square {4}
hexagon {6}

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|bgcolor=#e7dcc3|Vertex figure||80px
triangular antiprism

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|bgcolor=#e7dcc3|Coxeter group||(4,3)^[2], {{CDD|label4|branch_c1-2|3ab|branch_c1-2|label4}}

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|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive

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The cyclotruncated octahedral-cubic honeycomb is a compact uniform honeycomb, constructed from cube and truncated octahedron cells, in a triangular antiprism vertex figure. It has a Coxeter diagram {{CDD|label4|branch_10r|3ab|branch_10l|label4}}.

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:Perspective view from center of cube

It contains an H2 subgroup tetrahexagonal tiling alternating square and hexagonal faces, with Coxeter diagram {{CDD|branch_11|split2-44|node}} or half symmetry {{CDD|nodes_11|3a3b-cross|nodes_11}}:

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A radial subgroup symmetry, index 6, of this honeycomb can be constructed with [(4,3,4,3^*)], {{CDD|branch_11|4a4b|branch|labels}}, represented by a trigonal trapezohedron fundamental domain, and Coxeter diagram {{CDD|node_1|splitplit1u|branch3u_11|3a3buc-cross|branch3u_11|splitplit2u|node_1}}. This lower symmetry can be extended by restoring one mirror as {{CDD|branchu_11|3ab|branch_11|split2-44|node}}.

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The truncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated octahedron, truncated cube, rhombicuboctahedron, and truncated cuboctahedron cells, in a rectangular pyramid vertex figure. It has a Coxeter diagram {{CDD|label4|branch_11|3ab|branch_10l|label4}}.

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:Perspective view from center of rhombicuboctahedron

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The omnitruncated cubic-octahedral honeycomb is a compact uniform honeycomb, constructed from truncated cuboctahedron cells, in a rhombic disphenoid vertex figure. It has a Coxeter diagram {{CDD|label4|branch_11|3ab|branch_11|label4}} with [2,2]⁺ (order 4) extended symmetry in its rhombic disphenoid vertex figure.

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:Perspective view from center of truncated cuboctahedron

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• Convex uniform honeycombs in hyperbolic space
• List of regular polytopes

• 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)
• 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

{{DEFAULTSORT:Order-4 Dodecahedral Honeycomb}}

Category:3-honeycombs