Cubic honeycomb

The cubic honeycomb is a space-filling or three-dimensional tessellation consisting of many cubes that attach each other to the faces; the cube is known as cell of a honeycomb. The parallelepiped is the member of a parallelohedron, generated from three line segments that are not all parallel to a common plane. The cube is the special case of a parallelepiped for having the most symmetric form, generated by three perpendicular unit-length line segments.{{citation

| last = Alexandrov | first = A. D. | author-link = Aleksandr Danilovich Aleksandrov

| contribution = 8.1 Parallelohedra

| pages = 349–359

| publisher = Springer

| title = Convex Polyhedra

| title-link = Convex Polyhedra (book)

| year = 2005

}} In three-dimensional space, the cubic honeycomb is the only proper regular space-filling tessellation.{{citation

| title = Geometry of Lie Groups

| first1 = B. | last1 = Rosenfeld

| first2 = Bill | last2 = Wiebe

| url = https://books.google.com/books?id=mIjSBwAAQBAJ&pg=PA185

| page = 185

| year = 1997

| publisher = Springer

| isbn = 978-1-4757-5325-7 }} It is self-dual.{{citation

| last1 = Nelson | first1 = Roice

| last2 = Segerman | first2 = Henry

| title = Visualizing hyperbolic honeycombs

| journal = Journal of Mathematics and the Arts

| year = 2017

| volume = 11 | issue = 1 | pages = 4–39

| doi = 10.1080/17513472.2016.1263789

| arxiv = 1511.02851

}}

{{more sources needed|date=April 2025}}

{{original research section|date=April 2025}}

{{multiple image

| image1 = Rectified cubic tiling.png

| caption1 = Rectified cubic honeycomb

| image2 = Truncated cubic tiling.png

| caption2 = Truncated cubic tiling

| total_width = 400

}}

The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1, with a square prism vertex figure. John Horton Conway calls this honeycomb a cuboctahedrille,{{r|conway}} and its dual an oblate octahedrille.

The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1, with an isosceles square pyramid vertex figure. John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.

File:Cubes-A4 ani.gif

The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has four truncated octahedra around each vertex, in a tetragonal disphenoid vertex figure. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs. John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list, with its dual called an oblate tetrahedrille, also called a disphenoid tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.

The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb is non-uniform, with the highest symmetry construction reflecting an alternation of the uniform bitruncated cubic honeycomb. A lower-symmetry construction involves regular icosahedra paired with golden icosahedra (with 8 equilateral triangles paired with 12 golden triangles). There are three constructions from three related Coxeter diagrams: {{CDD|node|4|node_h|3|node_h|4|node}}, {{CDD|node|4|node_h|split1|nodes_hh}}, and {{CDD|node_h|split1|nodes_hh|split2|node_h}}. These have symmetry [4,3⁺,4], [4,(3^1,1)⁺] and [3^[4]]⁺ respectively. The first and last symmetry can be doubled as 4,3⁺,4 and [[3^[4]]]⁺. This honeycomb is represented in the boron atoms of the α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.Williams, 1979, p 199, Figure 5-38.

File:Cantellated cubic tiling.png

The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3, with a wedge vertex figure. John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.

Image:Cantitruncated cubic tiling.png

The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3, with a mirrored sphenoid vertex figure. John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille. Its dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, {{CDD|node_f1|4|node_f1|3|node_f1|4|node}}. These honeycomb cells represent the fundamental domains of $\{\backslash tilde\{B\}\}\_3$ symmetry. A cell can be as 1/24 of a translational cube with vertices positioned: taking two corners, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.

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File:Alternated cantitruncated cubic honeycomb.png

The alternated cantitruncated cubic honeycomb or snub rectified cubic honeycomb contains three types of cells: snub cubes, icosahedra (with T_h symmetry), tetrahedra (as tetragonal disphenoids), and new tetrahedral cells created at the gaps.
Although it is not uniform, constructionally it can be given as Coxeter diagrams {{CDD|node_h|4|node_h|split1|nodes_hh}} or {{CDD|node_h|4|node_h|3|node_h|4|node}}. Despite being non-uniform, there is a near-miss version with two edge lengths shown below, one of which is around 4.3% greater than the other. The snub cubes in this case are uniform, but the rest of the cells are not.

The cantic snub cubic honeycomb is constructed by snubbing the truncated octahedra in a way that leaves only rectangles from the cubes (square prisms). It is not uniform but it can be represented as Coxeter diagram {{CDD|node_1|4|node_h|3|node_h|4|node}}. It has rhombicuboctahedra (with T_h symmetry), icosahedra (with T_h symmetry), and triangular prisms (as C_2v-symmetry wedges) filling the gaps.[https://bendwavy.org/klitzing/incmats/x4s3s4o.htm cantic snub cubic honeycomb]

The runcitruncated cubic honeycomb or runcitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3, with an isosceles-trapezoidal pyramid vertex figure. Its name is derived from its Coxeter diagram, {{CDD|node_1|4|node_1|3|node|4|node_1}} with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb. John Horton Conway calls this honeycomb a 1-RCO-trille, and its dual square quarter pyramidille. Its dual is square quarter pyramidille, with Coxeter diagram {{CDD|node_f1|4|node_f1|3|node|4|node_f1}}. Faces exist in 3 of 4 hyperplanes of the [4,3,4], $\{\backslash tilde\{C\}\}\_3$ Coxeter group. Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center.

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An alternated omnitruncated cubic honeycomb or omnisnub cubic honeycomb can be constructed by alternation of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram: {{CDD|node_h|4|node_h|3|node_h|4|node_h}} and has symmetry [[4,3,4]]⁺. It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms, and creates new tetrahedral cells from the gaps. Its dual is a space-filling honeycomb constructed as the dual of the alternated omnitruncated cubic honeycomb. The 24 cells fit around a vertex, making a chiral octahedral symmetry that can be stacked in all 3-dimensions:

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The runcic cantitruncated cubic honeycomb or runcic cantitruncated cubic cellulation is constructed by removing alternating long rectangles from the octagons and is not uniform, but it can be represented as Coxeter diagram {{CDD|node_h|4|node_h|3|node_h|4|node_1}}. It has rhombicuboctahedra (with T_h symmetry), snub cubes, two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D_2d symmetry), and triangular prisms (as C_2v-symmetry wedges) filling the gaps.

The biorthosnub cubic honeycomb is constructed by removing alternating long rectangles from the octagons orthogonally and is not uniform, but it can be represented as Coxeter diagram {{CDD|node_h|4|node_1|3|node_1|4|node_h}}. It has rhombicuboctahedra (with T_h symmetry) and two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D_2d symmetry).

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Image:Truncated square prismatic honeycomb.png

The truncated square prismatic honeycomb or tomo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octagonal prisms and cubes in a ratio of 1:1. It is constructed from a truncated square tiling extruded into prisms. It is one of 28 convex uniform honeycombs.

Image:Snub square prismatic honeycomb.png

The snub square prismatic honeycomb or simo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2. It is constructed from a snub square tiling extruded into prisms. It is one of 28 convex uniform honeycombs.

A snub square antiprismatic honeycomb can be constructed by alternation of the truncated square prismatic honeycomb, although it can not be made uniform, but it can be given Coxeter diagram: {{CDD|node|4|node_h|4|node_h|2x|node_h|infin|node}} and has symmetry [4,4,2,∞]⁺. It makes square antiprisms from the octagonal prisms, tetrahedra (as tetragonal disphenoids) from the cubes, and two tetrahedra from the triangular bipyramids.

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{{Commons category|Cubic honeycomb}}

• Architectonic and catoptric tessellation
• Alternated cubic honeycomb
• List of regular polytopes
• Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge
• Snub (geometry)
• Voxel

{{reflist}}

• Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p. 296, Table II: Regular honeycombs
• George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)
• Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
• A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
• {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x4o3o4o - chon - O1}}
• [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 01-Chon]

{{Honeycombs}}

Category:Regular 3-honeycombs

Category:Regular 4-polytopes

Category:Self-dual tilings

Category:Regular tessellations