square tiling honeycomb

It is also seen as a rectified order-4 square tiling honeycomb, r{4,4,4}:

The square tiling honeycomb has three reflective symmetry constructions: {{CDD|node_1|4|node|4|node|3|node}} as a regular honeycomb, a half symmetry construction {{CDD|node_1|4|node_h0|4|node|3|node}} ↔ {{CDD|nodes_11|2a2b-cross|nodes|split2|node}}, and lastly a construction with three types (colors) of checkered square tilings {{CDD|node_1|4|node|4|node_g|3sg|node_g}} ↔ {{CDD|node|4|node_1|split1-44|nodes}}.

It also contains an index 6 subgroup [4,4,3^*] ↔ [4^1,1,1], and a radial subgroup [4,(4,3)^*] of index 48, with a right dihedral-angled octahedral fundamental domain, and four pairs of ultraparallel mirrors: {{CDD|branchu_11|2|branchu_11|2|branchu_11|2|branchu_11}}.

This honeycomb contains {{CDD|node|3|node|ultra|node_1}} that tile 2-hypercycle surfaces, which are similar to the paracompact order-3 apeirogonal tiling {{CDD|node|3|node|infin|node_1}}:

: 120px

The square tiling honeycomb is a regular hyperbolic honeycomb in 3-space. It is one of eleven regular paracompact honeycombs.

{{Regular_paracompact_H3_honeycombs}}

There are fifteen uniform honeycombs in the [4,4,3] Coxeter group family, including this regular form, and its dual, the order-4 octahedral honeycomb, {3,4,4}.

{{443_family}}

The square tiling honeycomb is part of the order-4 square tiling honeycomb family, as it can be seen as a rectified order-4 square tiling honeycomb.

{{444_family}}

It is related to the 24-cell, {3,4,3}, which also has a cubic vertex figure.

It is also part of a sequence of honeycombs with square tiling cells:

{{Square tiling tessellations}}

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

480px

It is similar to the 2D hyperbolic uniform triapeirogonal tiling, r{∞,3}, with triangle and apeirogonal faces.

: 240px

{{Clear}}

The truncated square tiling honeycomb, t{4,4,3}, {{CDD|node_1|4|node_1|4|node|3|node}} has cube and truncated square tiling facets, with a triangular pyramid vertex figure. It is the same as the cantitruncated order-4 square tiling honeycomb, tr{4,4,4}, {{CDD|node_1|4|node_1|4|node_1|4|node}}.

480px

{{Clear}}

The bitruncated square tiling honeycomb, 2t{4,4,3}, {{CDD|node|4|node_1|4|node_1|3|node}} has truncated cube and truncated square tiling facets, with a digonal disphenoid vertex figure.

480px

{{Clear}}

The cantellated square tiling honeycomb, rr{4,4,3}, {{CDD|node_1|4|node|4|node_1|3|node}} has cuboctahedron, square tiling, and triangular prism facets, with an isosceles triangular prism vertex figure.

480px

{{Clear}}

The cantitruncated square tiling honeycomb, tr{4,4,3}, {{CDD|node_1|4|node_1|4|node_1|3|node}} has truncated cube, truncated square tiling, and triangular prism facets, with an isosceles triangular pyramid vertex figure.

480px

{{Clear}}

The runcinated square tiling honeycomb, t_0,3{4,4,3}, {{CDD|node_1|4|node|4|node|3|node_1}} has octahedron, triangular prism, cube, and square tiling facets, with an irregular triangular antiprism vertex figure.

480px

{{Clear}}

The runcitruncated square tiling honeycomb, t_0,1,3{4,4,3}, {{CDD|node_1|4|node_1|4|node|3|node_1}} has rhombicuboctahedron, octagonal prism, triangular prism and truncated square tiling facets, with an isosceles-trapezoidal pyramid vertex figure.

480px

{{Clear}}

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

The omnitruncated square tiling honeycomb, t_0,1,2,3{4,4,3}, {{CDD|node_1|4|node_1|4|node_1|3|node_1}} has truncated square tiling, truncated cuboctahedron, hexagonal prism, and octagonal prism facets, with an irregular tetrahedron vertex figure.

480px

{{Clear}}

The alternated omnitruncated square tiling honeycomb (or omnisnub square tiling honeycomb), h(t_0,1,2,3{4,4,3}), {{CDD|node_h|4|node_h|4|node_h|3|node_h}} has snub square tiling, snub cube, triangular antiprism, square antiprism, and tetrahedron cells, with an irregular tetrahedron vertex figure.

{{Clear}}

The alternated square tiling honeycomb, h{4,4,3}, {{CDD|nodes_10ru|split2-44|node|3|node}} is a quasiregular paracompact uniform honeycomb in hyperbolic 3-space. It has cube and square tiling facets in a cuboctahedron vertex figure.

{{Clear}}

The cantic square tiling honeycomb, h₂{4,4,3}, {{CDD|nodes_10ru|split2-44|node_1|3|node}} is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cube, and cuboctahedron facets, with a rectangular pyramid vertex figure.

{{Clear}}

The runcic square tiling honeycomb, h₃{4,4,3}, {{CDD|nodes_10ru|split2-44|node|3|node_1}} is a paracompact uniform honeycomb in hyperbolic 3-space. It has square tiling, rhombicuboctahedron, and octahedron facets in a square frustum vertex figure.

{{Clear}}

The runcicantic square tiling honeycomb, h_2,3{4,4,3}, {{CDD|nodes_10ru|split2-44|node_1|3|node_1}} ↔ {{CDD|node_h1|4|node|4|node_1|3|node_1}}, is a paracompact uniform honeycomb in hyperbolic 3-space. It has truncated square tiling, truncated cuboctahedron, and truncated octahedron facets in a mirrored sphenoid vertex figure.

{{Clear}}

The alternated rectified square tiling honeycomb is a paracompact uniform honeycomb in hyperbolic 3-space.

{{Clear}}

• 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, (2018) 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

Category:Square tilings