order-4 octahedral honeycomb

A half symmetry construction, [3,4,4,1⁺], exists as {3,4^1,1}, with two alternating types (colors) of octahedral cells: {{CDD|node_1|3|node|4|node|4|node_h0}} ↔ {{CDD|node_1|3|node|split1-44|nodes}}.

A second half symmetry is [3,4,1⁺,4]: {{CDD|node_1|3|node|4|node_h0|4|node}} ↔ {{CDD|node_1|split1|nodes|2a2b-cross|nodes}}.

A higher index sub-symmetry, [3,4,4^*], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: {{CDD|branchu|split2|node_1|split1|branchu}}.

This honeycomb contains {{CDD|node_1|split1|branchu}} and {{CDD|node_1|3|node|ultra|node}} that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings {{CDD|node_1|split1|branch|labelinfin}} and {{CDD|node_1|3|node|infin|node}}, respectively:

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The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs.

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There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form.

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It is a part of a sequence of honeycombs with a square tiling vertex figure:

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It a part of a sequence of regular polychora and honeycombs with octahedral cells:

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

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

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The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.

The cantellated order-4 octahedral honeycomb, t_0,2{3,4,4}, {{CDD|node_1|3|node|4|node_1|4|node}} has rhombicuboctahedron, cube, and square tiling facets, with a wedge vertex figure.

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The cantitruncated order-4 octahedral honeycomb, t_0,1,2{3,4,4}, {{CDD|node_1|3|node_1|4|node_1|4|node}} has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.

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The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.

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

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The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.

The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling honeycomb.

The snub order-4 octahedral honeycomb, s{3,4,4}, has Coxeter diagram {{CDD|node_h|3|node_h|4|node|4|node}}. It is a scaliform honeycomb, with square pyramid, square tiling, and icosahedron facets.

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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, (2015) Chapter 13: Hyperbolic Coxeter groups
• Norman W. Johnson and Asia Ivic Weiss [https://cms.math.ca/cjm/v51/weisscox8.pdf Quadratic Integers and Coxeter Groups] PDF Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336

Category:Regular 3-honeycombs