View source for Cubic honeycomb

{{Short description|Only regular space-filling tessellation of the cube}}
{{Infobox
 | name = {{{name|{{PAGENAME}}}}}
 | image = [[Image:Partial cubic honeycomb.png|150px]]
 | caption = Partial view of a cubic honeycomb
 | label1 = Type | data1 = [[List of regular polytopes#Tessellations of Euclidean 3-space|Regular honeycomb]]
 | label2 = Indexing | data2 = J<sub>11,15</sub>, A<sub>1</sub><br />W<sub>1</sub>, G<sub>22</sub><ref>For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).</ref>
 | label3 = [[Schläfli symbol]] | data3 = <math> \{4,3,4\} </math>
 | label4 = [[Cell (geometry)|Cell type]] | data4 = [[cube]]
 | label5 = Duality | data5 = [[self-dual]]
}}

The '''cubic honeycomb''' or '''cubic cellulation''' is the only proper regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]] made up of [[cube|cubic]] cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its [[vertex figure]] is a regular [[octahedron]]. It is a [[Self-dual tessellation|self-dual]] tessellation with [[Schläfli symbol]] {4,3,4}. [[John Horton Conway]] called this honeycomb a '''cubille'''.<ref name=conway>{{citation
 | first1 = John H. | last1 = Conway | author1-link = John Horton Conway
 | first2 = Heidi | last2 = Burgiel
 | first3= Chaim | last3 = Goodman-Strauss
 | title = The Symmetries of Things
 | title-link = The Symmetries of Things
 | year = 2008
 | publisher = AK Peters
 | isbn = 978-1-56881-220-5
 | page = [https://books.google.com/books?id=Drj1CwAAQBAJ&pg=PA293 293]
}}</ref>

{{Honeycomb}}

== Description ==
The cubic honeycomb is a space-filling or three-dimensional [[tessellation]] consisting of many [[cube]]s that attach each other to the faces; the cube is known as [[Cell (geometry)|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.<ref name=alexandrov>{{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
}}</ref> In three-dimensional space, the cubic honeycomb is the only proper regular space-filling tessellation.<ref>{{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 }}</ref> It is [[self-dual]].<ref name=ns>{{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
}}</ref>

== Related honeycombs ==
{{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 (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[octahedron|octahedra]] and [[cuboctahedron|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 (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[truncated cube]]s and [[octahedron|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|thumb|The bitruncated cubic honeycomb shown here in relation to a cubic honeycomb]]
The [[bitruncated cubic honeycomb]] is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]] made up of [[truncated octahedron|truncated octahedra]] (or, equivalently, [[Bitruncation (geometry)|bitruncated]] cubes). It has four [[truncated octahedron|truncated octahedra]] around each vertex, in a [[tetragonal disphenoid]] [[vertex figure]]. Being composed entirely of [[truncated octahedron|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 [[Convex uniform honeycomb|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 diagram]]s: {{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<sup>+</sup>,4], [4,(3<sup>1,1</sup>)<sup>+</sup>] and [3<sup>[4]</sup>]<sup>+</sup> respectively. The first and last symmetry can be doubled as <nowiki>[[</nowiki>4,3<sup>+</sup>,4]] and [<nowiki/>[3<sup>[4]</sup>]]<sup>+</sup>. This honeycomb is represented in the boron atoms of the [[Allotropes of boron#.CE.B1-rhombohedral boron|&alpha;-rhombohedral crystal]]. The centers of the icosahedra are located at the fcc positions of the lattice.<ref>Williams, 1979, p 199, Figure 5-38.</ref>

[[File:Cantellated cubic tiling.png|thumb|Cantellated cubic tiling]]
The ''cantellated cubic honeycomb'' or ''cantellated cubic cellulation'' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[rhombicuboctahedron|rhombicuboctahedra]], [[cuboctahedron|cuboctahedra]], and [[cube]]s in a ratio of 1:1:3, with a [[wedge (geometry)|wedge]] [[vertex figure]]. [[John Horton Conway]] calls this honeycomb a '''2-RCO-trille''', and its dual [[quarter oblate octahedrille]].

[[Image:Cantitruncated cubic tiling.png|thumb|Cantitruncated cubic tiling]]
The ''cantitruncated cubic honeycomb'' or ''cantitruncated cubic cellulation'' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space, made up of [[truncated cuboctahedron|truncated cuboctahedra]], [[truncated octahedron|truncated octahedra]], and [[cube]]s 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  <math>{\tilde{B}}_3</math> 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.
:[[File:triangular pyramidille cell1.png|160px]]

[[File:Alternated cantitruncated cubic honeycomb.png|thumb|The alternated cantitruncated cubic honeycomb]]
The ''alternated cantitruncated cubic honeycomb'' or ''snub rectified cubic honeycomb'' contains three types of cells: [[snub cube]]s, [[icosahedron|icosahedra]] (with ''T<sub>h</sub>'' symmetry), [[tetrahedron|tetrahedra]] (as tetragonal disphenoids), and new tetrahedral cells created at the gaps.<br />Although it is not uniform, constructionally it can be given as [[Coxeter diagram]]s {{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 octahedron|truncated octahedra]] in a way that leaves only [[rectangle]]s from the [[cube]]s (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 [[rhombicuboctahedron|rhombicuboctahedra]] (with ''T<sub>h</sub>'' symmetry), [[icosahedron|icosahedra]] (with ''T<sub>h</sub>'' symmetry), and [[triangular prism]]s (as ''C<sub>2v</sub>''-symmetry wedges) filling the gaps.<ref>[https://bendwavy.org/klitzing/incmats/x4s3s4o.htm cantic snub cubic honeycomb]</ref>

The ''runcitruncated cubic honeycomb'' or ''runcitruncated cubic cellulation'' is a uniform [[Tessellation of space|space-filling tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[rhombicuboctahedron|rhombicuboctahedra]], [[truncated cube]]s, [[octagonal prism]]s, and [[cube]]s in a ratio of 1:1:3:3, with an [[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|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 [[List of regular polytopes#Tessellations of Euclidean 3-space|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], <math>{\tilde{C}}_3</math> 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.
:[[File:Square quarter pyramidille cell.png|160px]]

An ''alternated omnitruncated cubic honeycomb'' or ''omnisnub cubic honeycomb'' can be constructed by [[Alternation (geometry)|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 [<nowiki/>[4,3,4]]<sup>+</sup>. It makes [[snub cube]]s from the [[truncated cuboctahedron|truncated cuboctahedra]], [[square antiprism]]s from the [[octagonal prism]]s, and creates new [[tetrahedron|tetrahedral]] cells from the gaps. Its dual is a space-filling honeycomb constructed as the dual of the [[#alternated omnitruncated cubic honeycomb|alternated omnitruncated cubic honeycomb]]. The 24 cells fit around a vertex, making a chiral [[octahedral symmetry]] that can be stacked in all 3-dimensions:
:[[File:Altbasetet-24-in-cube.png|240px]]

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 [[rhombicuboctahedron|rhombicuboctahedra]] (with ''T<sub>h</sub>'' symmetry), [[snub cube]]s, two kinds of [[cube]]s: square prisms and rectangular trapezoprisms (topologically equivalent to a [[cube]] but with ''D<sub>2d</sub>'' symmetry), and [[triangular prism]]s (as ''C<sub>2v</sub>''-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 [[rhombicuboctahedron|rhombicuboctahedra]] (with ''T<sub>h</sub>'' symmetry) and two kinds of [[cube]]s: square prisms and rectangular trapezoprisms (topologically equivalent to a [[cube]] but with ''D<sub>2d</sub>'' symmetry).
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[[Image:Truncated square prismatic honeycomb.png|thumb|Truncated square prismatic honeycomb]]
The ''truncated square prismatic honeycomb'' or ''tomo-square prismatic cellulation'' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]].  It is composed of [[octagonal prism]]s and [[cube]]s in a ratio of 1:1. It is constructed from a [[truncated square tiling]] extruded into prisms. It is one of 28 [[convex uniform honeycomb]]s.

[[Image:Snub square prismatic honeycomb.png|thumb|Snub square prismatic honeycomb]]
The ''snub square prismatic honeycomb'' or ''simo-square prismatic cellulation'' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]].  It is composed of [[cube]]s and [[triangular prism]]s in a ratio of 1:2. It is constructed from a [[snub square tiling]] extruded into prisms. It is one of 28 [[convex uniform honeycomb]]s.

A ''snub square antiprismatic honeycomb'' can be constructed by [[Alternation (geometry)|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,∞]<sup>+</sup>. It makes [[square antiprism]]s from the [[octagonal prism]]s, [[tetrahedron|tetrahedra]] (as tetragonal disphenoids) from the [[cube]]s, and two tetrahedra from the [[triangular bipyramid]]s.
{{clear}}

==See also ==
{{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]]

== References ==
{{reflist}}
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p.&nbsp;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)
* [[Alfredo Andreini|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]]