View source for Cubic honeycomb

{{Short description|Only regular space-filling tessellation of the cube}}
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Cubic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Cubic honeycomb.png|150px]][[Image:Partial cubic honeycomb.png|150px]]
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of Euclidean 3-space|Regular honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Hypercube honeycomb]]
|-
|bgcolor=#e7dcc3|Indexing<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>
|J<sub>11,15</sub>, A<sub>1</sub><br />W<sub>1</sub>, G<sub>22</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node|3|node|4|node}}
|-
|bgcolor=#e7dcc3|Cell type||[[cube|{4,3}]] [[File:Uniform polyhedron-43-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Face type||[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Cubic honeycomb verf.svg|80px]]<BR>[[octahedron]]
|-
|bgcolor=#e7dcc3|[[Space group]]<br />[[Fibrifold notation]]||Pm{{overline|3}}m (221)<br />4<sup>−</sup>:2
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[Self-dual polytope|self-dual]]<br />Cell: [[File:Cubic full domain.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], [[Regular honeycomb|regular]]
|}
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'''.

{{Honeycomb}}

== Related honeycombs==
It is part of a multidimensional family of [[hypercube honeycomb]]s, with [[Schläfli symbol]]s of the form {4,3,...,3,4}, starting with the [[square tiling]], {4,4} in the plane.

It is one of 28 [[Convex uniform honeycomb|uniform honeycombs]] using [[uniform polyhedron|convex uniform polyhedral]] cells.

== Isometries of simple cubic lattices==
Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:
{| class=wikitable
![[Crystal system]]
![[Monoclinic crystal system|Monoclinic]]<br />[[Triclinic crystal system|Triclinic]]
![[Orthorhombic crystal system|Orthorhombic]]
![[Tetragonal crystal system|Tetragonal]]
![[rhombohedral lattice system|Rhombohedral]]
![[Cubic crystal system|Cubic]]
|- align=center
![[Unit cell]]
|[[Parallelepiped]]
|Rectangular [[cuboid]]
|Square [[cuboid]]
|Trigonal<br />[[trapezohedron]]
|[[Cube]]
|- valign=top align=center
![[Point group]]<br />Order<br />Rotation subgroup
|[ ], (*)<br />Order 2<br />[ ]<sup>+</sup>, (1)
|[2,2], (*222)<br />Order 8<br />[2,2]<sup>+</sup>, (222)
|[4,2], (*422)<br />Order 16<br />[4,2]<sup>+</sup>, (422)
|[3], (*33)<br />Order 6<br />[3]<sup>+</sup>, (33)
|[4,3], (*432)<br />Order 48<br />[4,3]<sup>+</sup>, (432)
|- align=center
!Diagram
|[[File:Monoclinic.svg|80px]]
|[[File:Orthorhombic.svg|80px]]
|[[File:Tetragonal.svg|60px]]
|[[File:Rhombohedral.svg|80px]]
|[[File:Cubic.svg|80px]]
|- align=center
![[Space group]]<br />Rotation subgroup
|Pm (6)<br />P1 (1)
|Pmmm (47)<br />P222 (16)
|P4/mmm (123)<br />P422 (89)
|R3m (160)<br />R3 (146)
|Pm{{overline|3}}m (221)<br />P432 (207)
|- align=center
![[Coxeter notation]]
| - 
| [&infin;]<sub>a</sub>×[&infin;]<sub>b</sub>×[&infin;]<sub>c</sub>
| [4,4]<sub>a</sub>×[&infin;]<sub>c</sub>
| -
| [4,3,4]<sub>a</sub>
|- align=center
![[Coxeter diagram]]
| - 
| {{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}
| {{CDD|node_1|4|node|4|node|2|node_1|infin|node}}
| -
| {{CDD|node_1|4|node|3|node|4|node}}
|}

== Uniform colorings ==

There is a large number of [[uniform coloring]]s, derived from different symmetries. These include:
{| class="wikitable"
![[Coxeter notation]]<br />[[Space group]]
![[Coxeter diagram]]
![[Schläfli symbol]]
!Partial<br />honeycomb
!Colors by letters
|- align=center
![4,3,4]<br />Pm{{overline|3}}m (221)
|{{CDD|node_1|4|node|3|node|4|node}}<br />{{CDD|labelinfin|branch_10|2|labelinfin|branch_10|2|labelinfin|branch_10}} = {{CDD|node_1|4|node_g|3sg|node_g|4|node}}
|{4,3,4}
| [[Image:Partial cubic honeycomb.png|50px]]
| 1: aaaa/aaaa
|- align=center
![4,3<sup>1,1</sup>] = [4,3,4,1<sup>+</sup>]<br />Fm{{overline|3}}m (225)
|{{CDD|node_1|4|node|split1|nodes}} = {{CDD|node_1|4|node|3|node|4|node_h0}}
|{4,3<sup>1,1</sup>}
| [[Image:Bicolor cubic honeycomb.png|50px]]
| 2: abba/baab
|- align=center
![4,3,4]<br />Pm{{overline|3}}m (221)
|{{CDD|node_1|4|node|3|node|4|node_1}}
|t<sub>0,3</sub>{4,3,4}
| [[Image:Runcinated cubic honeycomb.png|50px]]
| 4: abbc/bccd
|- align=center
!<nowiki>[[</nowiki>4,3,4]]<br />Pm{{overline|3}}m (229)
|{{CDD|branch|4a4b|nodes_11}}
|t<sub>0,3</sub>{4,3,4}
|
| 4: abbb/bbba
|- align=center
![4,3,4,2,&infin;]
|{{CDD|node_1|4|node|4|node|2|node_1|infin|node_1}}<br />or {{CDD|labelinfin|branch_11|2|labelinfin|branch_10|2|labelinfin|branch_10}}
|{4,4}×t{&infin;}
| [[Image:Square prismatic honeycomb.png|50px]]
| 2: aaaa/bbbb
|- align=center
![4,3,4,2,&infin;]
|{{CDD|node|4|node_1|4|node|2|node_1|infin|node}}
|t<sub>1</sub>{4,4}×{&infin;}
| [[Image:Square prismatic 2-color honeycomb.png|50px]]
| 2: abba/abba
|- align=center
![&infin;,2,&infin;,2,&infin;]
|{{CDD|labelinfin|branch_11|2|labelinfin|branch_11|2|labelinfin|branch_10}}
|t{&infin;}×t{&infin;}×{&infin;}
| [[Image:Square 4-color prismatic honeycomb.png|50px]]
| 4: abcd/abcd
|- align=center
![&infin;,2,&infin;,2,&infin;] = [4,(3,4)<sup>*</sup>]
|{{CDD|labelinfin|branch_11|2|labelinfin|branch_11|2|labelinfin|branch_11}} = {{CDD|node_1|4|node_g|3sg|node_g|4g|node_g}}
|t{&infin;}×t{&infin;}×t{&infin;}
| [[Image:Cubic 8-color honeycomb.png|50px]]
| 8: abcd/efgh
|}

=== Projections ===
The ''cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a [[triangular tiling]]. A square symmetry projection forms a [[square tiling]]. 
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:cubic_honeycomb-2.svg|120px]]
|[[File:cubic_honeycomb-1.svg|120px]]
|[[File:cubic_honeycomb-3.svg|120px]]
|-
!Frame
|[[File:cubic_honeycomb-2b.svg|120px]]
|[[File:cubic_honeycomb-1b.svg|120px]]
|[[File:cubic_honeycomb-3b.svg|120px]]
|}

== Related polytopes and honeycombs==
It is related to the regular [[4-polytope]] [[tesseract]], [[Schläfli symbol]] {4,3,3}, which exists in 4-space, and only has ''3'' cubes around each edge. It's also related to the [[order-5 cubic honeycomb]], Schläfli symbol {4,3,5}, of [[hyperbolic space]] with 5 cubes around each edge.

It is in a sequence of polychora and honeycombs with [[octahedron|octahedral]] [[vertex figure]]s.
{{Octahedral_vertex_figure_tessellations}}

It in a sequence of [[regular polytopes]] and honeycombs with [[cube|cubic]] [[cell (geometry)|cells]].
{{Cubic cell tessellations}}

{{Symmetric2_tessellations}}

== Related polytopes ==
The cubic honeycomb has a lower symmetry as a runcinated cubic honeycomb, with two sizes of [[cube]]s. A double symmetry construction can be constructed by placing a small cube into each large cube, resulting in a nonuniform honeycomb with [[cube]]s, square prisms, and rectangular trapezoprisms (a cube with ''D<sub>2d</sub>'' symmetry). Its vertex figure is a triangular pyramid with its lateral faces augmented by tetrahedra.

[[File:Biruncinatocubic honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

The resulting honeycomb can be alternated to produce another nonuniform honeycomb with regular [[tetrahedron|tetrahedra]], two kinds of tetragonal disphenoids, triangular pyramids, and sphenoids. Its vertex figure has ''C<sub>3v</sub>'' symmetry and has 26 triangular faces, 39 edges, and 15 vertices.

== Related Euclidean tessellations ==
The [4,3,4], {{CDD|node|4|node|3|node|4|node}}, [[Coxeter group]] generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The [[Expansion (geometry)|expanded]] cubic honeycomb (also known as the runcinated cubic honeycomb) is geometrically identical to the cubic honeycomb.
{{C3 honeycombs}}

The [4,3<sup>1,1</sup>], {{CDD|node|4|node|split1|nodes}}, [[Coxeter group]] generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.
{{B3 honeycombs}}

This honeycomb is one of [[Uniform 5-polytope#Regular and uniform honeycombs|five distinct uniform honeycombs]]<ref>[http://mathworld.wolfram.com/Necklace.html], [http://oeis.org/A000029 A000029] 6-1 cases, skipping one with zero marks</ref> constructed by the <math>{\tilde{A}}_3</math> [[Coxeter group]]. The symmetry can be multiplied by the symmetry of rings in the [[Coxeter–Dynkin diagram]]s: 
{{A3 honeycombs}}
{{clear}}
----

=== Rectified cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Rectified cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Rectified cubic tiling.png|200px]][[File:HC A3-P3.png|110px]]-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||r{4,3,4} or t<sub>1</sub>{4,3,4}<br />r{4,3<sup>1,1</sup>}<br />2r{4,3<sup>1,1</sup>}<br />r{3<sup>[4]</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|4|node_1|3|node|4|node}}<br />{{CDD|node|4|node_1|split1|nodes}} = {{CDD|node|4|node_1|3|node|4|node_h0}}<br />{{CDD|node|4|node|split1|nodes_11}} = {{CDD|node|4|node|3|node_1|4|node_h0}}<br />{{CDD|node_1|split1|nodes|split2|node_1}} = {{CDD|node_h0|4|node_1|split1|nodes}} = {{CDD|node_h0|4|node|split1|nodes_11}} = {{CDD|node_h0|4|node_1|3|node|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[Cuboctahedron|r{4,3}]] [[Image:Uniform polyhedron-43-t1.png|40px]]<br />[[octahedron|{3,4}]] [[Image:Uniform polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified cubic honeycomb verf.png|80px]]<br />[[square prism]]
|-
|bgcolor=#e7dcc3|[[Space group]]<br />[[Fibrifold notation]]||Pm{{overline|3}}m (221)<br />4<sup>−</sup>:2
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
|-
|bgcolor=#e7dcc3|Dual||[[oblate octahedrille]]<br />Cell: [[File:Cubic square bipyramid.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], [[edge-transitive]]
|}
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''', and its dual an [[oblate octahedrille]].

[[Image:Rectified cubic tiling.png|300px]][[File:HC A3-P3.png|160px]]

==== Projections ====
The ''rectified cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements.
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:rectified cubic_honeycomb-2.png|120px]]
|[[File:rectified cubic_honeycomb-1.png|120px]]
|[[File:rectified cubic_honeycomb-3.png|150px]]
|-
!Frame
|[[File:rectified cubic_honeycomb-2b.png|120px]]
|[[File:rectified cubic_honeycomb-1b.png|120px]]
|[[File:rectified cubic_honeycomb-3b.png|150px]]
|}

==== Symmetry ====
There are four [[uniform coloring]]s for the cells of this honeycomb with reflective symmetry, listed by their [[Coxeter group]], and [[Wythoff construction]] name, and the [[Coxeter diagram]] below.
{| class="wikitable" width=320
|- valign=top
![[Coxeter notation|Symmetry]]
![4,3,4]<br /><math>{\tilde{C}}_3</math>
![1<sup>+</sup>,4,3,4]<br />[4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
![4,3,4,1<sup>+</sup>]<br />[4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
![1<sup>+</sup>,4,3,4,1<sup>+</sup>]<br />[3<sup>[4]</sup>], <math>{\tilde{A}}_3</math>
|-
![[Space group]]||Pm{{overline|3}}m<br />(221)||Fm{{overline|3}}m<br />(225)||Fm{{overline|3}}m<br />(225)|| F{{overline|4}}3m<br />(216)
|- align=center
|Coloring
|[[Image:Rectified cubic honeycomb.png|80px]]
|[[Image:Rectified cubic honeycomb4.png|80px]]
|[[Image:Rectified cubic honeycomb3.png|80px]]
|[[Image:Rectified cubic honeycomb2.png|80px]]
|- align=center
!rowspan=2|[[Coxeter diagram|Coxeter<br />diagram]]
|rowspan=2|{{CDD|node|4|node_1|3|node|4|node}}
|{{CDD|node|4|node_1|3|node|4|node_h0}}
|{{CDD|node_h0|4|node_1|3|node|4|node}}
|{{CDD|node_h0|4|node_1|3|node|4|node_h0}}
|- align=center
|{{CDD|node|4|node_1|split1|nodes}}
|{{CDD|nodes_11|split2|node|4|node}}
|{{CDD|node_1|split1|nodes|split2|node_1}}
|- align=center
![[Vertex figure]]
||[[Image:Rectified cubic honeycomb verf.png|80px]]
||[[File:Rectified alternate cubic honeycomb verf.png|80px]]
||[[File:Cantellated alternate cubic honeycomb verf.png|80px]]
||[[File:T02 quarter cubic honeycomb verf.png|80px]]
|- align=center
!Vertex<br />figure<br />symmetry
|D<sub>4h</sub><br />[4,2]<br />(*224)<br />order 16
|D<sub>2h</sub><br />[2,2]<br />(*222)<br />order 8
|C<sub>4v</sub><br />[4]<br />(*44)<br />order 8
|C<sub>2v</sub><br />[2]<br />(*22)<br />order 4
|}

This honeycomb can be divided on [[trihexagonal tiling]] planes, using the [[hexagon]] centers of the cuboctahedra, creating two [[triangular cupola]]e. This [[scaliform honeycomb]] is represented by Coxeter diagram {{CDD|node_h|2x|node_h|6|node|3|node_1}}, and symbol s<sub>3</sub>{2,6,3}, with [[coxeter notation]] symmetry [2<sup>+</sup>,6,3].
: [[File:Runcic snub 263 honeycomb.png|320px]]. 

==== Related polytopes ====
A double symmetry construction can be made by placing octahedra on the cuboctahedra, resulting in a nonuniform honeycomb with two kinds of [[octahedron|octahedra]] (regular octahedra and triangular antiprisms). The vertex figure is a [[square bifrustum]]. The dual is composed of [[elongated square bipyramid]]s.

[[File:Biambocubic honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

{{clear}}
----

=== Truncated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Truncated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t{4,3,4} or t<sub>0,1</sub>{4,3,4}<br />t{4,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|4|node_1|3|node|4|node}}<br />{{CDD|node_1|4|node_1|split1|nodes}} = {{CDD|node_1|4|node_1|3|node|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cell type||[[truncated cube|t{4,3}]] [[File:Uniform polyhedron-43-t01.png|40px]]<BR>[[octahedron|{3,4}]] [[File:Uniform polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Face type||[[triangle]] {3}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated cubic honeycomb verf.png|80px]]<br />isosceles [[square pyramid]]
|-
|bgcolor=#e7dcc3|[[Space group]]<br />[[Fibrifold notation]]||Pm{{overline|3}}m (221)<br />4<sup>−</sup>:2
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
|-
|bgcolor=#e7dcc3|Dual||[[Pyramidille]]<br />Cell: [[File:Cubic square pyramid.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}
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:Truncated cubic tiling.png|300px]][[File:HC A2-P3.png|160px]]

==== Projections ====
The ''truncated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements.
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:truncated cubic_honeycomb-2.png|120px]]
|[[File:truncated cubic_honeycomb-1.png|120px]]
|[[File:truncated cubic_honeycomb-3.png|150px]]
|-
!Frame
|[[File:truncated cubic_honeycomb-2b.png|120px]]
|[[File:truncated cubic_honeycomb-1b.png|120px]]
|[[File:truncated cubic_honeycomb-3b.png|150px]]
|}

==== Symmetry ====

There is a second [[uniform coloring]] by reflectional symmetry of the [[Coxeter group]]s, the second seen with alternately colored truncated cubic cells.
{| class="wikitable" width=280
!Construction
!Bicantellated alternate cubic
!Truncated cubic honeycomb
|- valign=top
![[Coxeter group]]
![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
![4,3,4], <math>{\tilde{C}}_3</math><br />=<[4,3<sup>1,1</sup>]>
|-
![[Space group]]||Fm{{overline|3}}m||Pm{{overline|3}}m
|- align=center
!Coloring
|[[Image:Truncated cubic honeycomb2.png|120px]]
|[[Image:Truncated cubic honeycomb.png|120px]]
|- align=center
![[Coxeter diagram]]
!{{CDD|node_1|4|node_1|split1|nodes}} = {{CDD|node_1|4|node_1|3|node|4|node_h0}}
!{{CDD|node_1|4|node_1|3|node|4|node}}
|- align=center
![[Vertex figure]]
|[[File:Bicantellated alternate cubic honeycomb verf.png|80px]]
|[[Image:Truncated cubic honeycomb verf.png|80px]]
|}

==== Related polytopes ====
A double symmetry construction can be made by placing octahedra on the truncated cubes, resulting in a nonuniform honeycomb with two kinds of [[octahedron|octahedra]] (regular octahedra and triangular antiprisms) and two kinds of [[tetrahedron|tetrahedra]] (tetragonal disphenoids and digonal disphenoids). The vertex figure is an octakis square cupola.

[[File:Bitruncatocubic honeycomb vertex figure.png|120px]]<br />[[Vertex figure]]

[[File:Bitruncatocubic honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

{{clear}}
----

=== Bitruncated cubic honeycomb===

{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Bitruncated cubic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Bitruncated cubic tiling.png|180px]]&nbsp;[[File:HC-A4.png|120px]]
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2t{4,3,4}<br />t<sub>1,2</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD||node|4|node_1|3|node_1|4|node}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated octahedron|t{3,4}]] [[File:Uniform polyhedron-43-t12.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square (geometry)|square]] {4}<br />[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Edge figure]]||[[isosceles triangle]] {3}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Bitruncated cubic honeycomb verf2.png|80px]]<br />[[tetragonal disphenoid]]
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]<br />[[Fibrifold notation]]<br />[[Coxeter notation#Space groups|Coxeter notation]]||[[Cubic crystal system|Im{{overline|3}}m (229)]]<br />8<sup>o</sup>:2<br />[<nowiki/>[4,3,4]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
|-
|bgcolor=#e7dcc3|Dual||Oblate tetrahedrille<br />[[Disphenoid tetrahedral honeycomb]]<br />Cell: [[File:Oblate tetrahedrille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], [[edge-transitive]], cell-transitive
|}
[[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.

==== Projections ====
The ''bitruncated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform [[rhombitrihexagonal tiling]]. A square symmetry projection forms two overlapping [[truncated square tiling]], which combine together as a [[chamfered square tiling]]. 
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:Bitruncated_cubic_honeycomb_ortho2.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho4.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho1.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho3.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho5.png|120px]]
|-
!Frame
|[[File:Bitruncated_cubic_honeycomb_orthoframe2.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe4.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe1.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe3.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe5.png|120px]]
|}

==== Symmetry ====
The vertex figure for this honeycomb is a [[disphenoid tetrahedron]], and it is also the [[Goursat tetrahedron]] ([[fundamental domain]]) for the <math>{\tilde{A}}_3</math> [[Coxeter group]]. This honeycomb has four uniform constructions, with the truncated octahedral cells having different [[Coxeter group]]s and [[Wythoff construction]]s.  These uniform symmetries can be represented by coloring differently the cells in each construction.

{| class=wikitable
|+ Five uniform colorings by cell
|-
![[Space group]]||Im{{overline|3}}m (229)||Pm{{overline|3}}m (221)||Fm{{overline|3}}m (225)||F{{overline|4}}3m (216)||Fd{{overline|3}}m (227)
|-
![[Fibrifold]]||8<sup>o</sup>:2||4<sup>−</sup>:2||2<sup>−</sup>:2||1<sup>o</sup>:2||2<sup>+</sup>:2
|- valign=top
! valign=center|[[Coxeter group]]
! <math>{\tilde{C}}_3</math>×2<br /><nowiki>[[</nowiki>4,3,4]]<br />=[4[3<sup>[4]</sup>]]<br />{{CDD|node|4|node_c1|3|node_c1|4|node}} = {{CDD|branch_c1|3ab|branch_c1}}
! <math>{\tilde{C}}_3</math><br />[4,3,4]<br />=[2[3<sup>[4]</sup>]]<br />{{CDD|node|4|node_c1|3|node_c2|4|node}} = {{CDD|branch_c1-2|3ab|branch_c2-1}}
! <math>{\tilde{B}}_3</math><br />[4,3<sup>1,1</sup>]<br />=<[3<sup>[4]</sup>]><br />{{CDD|nodeab_c1-2|split2|node_c3|4|node}} = {{CDD|node_c3|split1|nodeab_c1-2|split2|node_c3}}
! <math>{\tilde{A}}_3</math><br />[3<sup>[4]</sup>]<br />&nbsp;<br />{{CDD|node_c3|split1|nodeab_c1-2|split2|node_c4}}
! <math>{\tilde{A}}_3</math>×2<br />[<nowiki/>[3<sup>[4]</sup>]]<br />=[<nowiki/>[3<sup>[4]</sup>]]<br />{{CDD|branch_c1|3ab|branch_c2}}
|-
![[Coxeter diagram]]
!{{CDD||branch_11|4a4b|nodes}}
!{{CDD||node|4|node_1|3|node_1|4|node}}
!{{CDD|nodes_11|split2|node_1|4|node}}
!{{CDD|node_1|split1|nodes_11|split2|node_1}}
!{{CDD|branch_11|3ab|branch_11}}
|- align=center
![[truncated octahedron|truncated octahedra]]
! 1<br />[[File:Uniform polyhedron-43-t12.svg|25px]]
! 1:1<br />[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-43-t12.svg|25px]]
! 2:1:1<br />[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
! 1:1:1:1<br />[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
! 1:1<br />[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
|- align=center
![[Vertex figure]]
|[[Image:Bitruncated cubic honeycomb verf2.png|80px]]
|[[Image:Bitruncated cubic honeycomb verf.png|80px]]
|[[File:Cantitruncated alternate cubic honeycomb verf.png|80px]]
|[[File:Omnitruncated 3-simplex honeycomb verf.png|80px]]
|[[File:Omnitruncated 3-simplex honeycomb verf2.png|80px]]
|- align=center
!Vertex<br />figure<br />symmetry
|[2<sup>+</sup>,4]<br />(order 8)
|[2]<br />(order 4)
|[ ]<br />(order 2)
|[ ]<sup>+</sup><br />(order 1)
|[2]<sup>+</sup><br />(order 2)
|-
!Image<br />Colored by<br />cell
|[[Image:Bitruncated Cubic Honeycomb1.svg|100px]]
|[[Image:Bitruncated Cubic Honeycomb.svg|100px]]
|[[Image:Bitruncated cubic honeycomb3.png|100px]]
|[[Image:Bitruncated cubic honeycomb2.png|100px]]
|[[Image:Bitruncated Cubic Honeycomb1.svg|100px]]
|}

==== Related polytopes ====
Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with [[truncated octahedron|truncated octahedra]] and [[hexagonal prism]]s (as ditrigonal trapezoprisms). Its vertex figure is a ''C<sub>2v</sub>''-symmetric [[triangular bipyramid]].

This honeycomb can then be alternated to produce another nonuniform honeycomb with [[icosahedron|pyritohedral icosahedra]], [[octahedron|octahedra]] (as triangular antiprisms), and [[tetrahedron|tetrahedra]] (as sphenoids). Its vertex figure has ''C<sub>2v</sub>'' symmetry and consists of 2 [[pentagon]]s, 4 [[rectangle]]s, 4 [[isosceles triangle]]s (divided into two sets of 2), and 4 [[triangle|scalene triangle]]s.
{{clear}}
----

=== Alternated bitruncated cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Alternated bitruncated cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2s{4,3,4}<br />2s{4,3<sup>1,1</sup>}<br />sr{3<sup>[4]</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|4|node_h|3|node_h|4|node}}<br />{{CDD|node|4|node_h|split1|nodes_hh}} = {{CDD|node|4|node_h|3|node_h|4|node_h0}}<br />{{CDD|nodes_hh|split2|node_h|4|node}} = {{CDD|node_h0|4|node_h|3|node_h|4|node}}<br />{{CDD|node_h|split1|nodes_hh|split2|node_h}} = {{CDD|node_h0|4|node_h|3|node_h|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]<br />[[icosahedron|s{3,3}]] [[File:Uniform polyhedron-33-s012.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Alternated bitruncated cubic honeycomb verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[4,3<sup>+</sup>,4]], <math>{\tilde{C}}_3</math>
|-
|bgcolor=#e7dcc3|Dual||[[Ten-of-diamonds decahedron|Ten-of-diamonds honeycomb]]<br />Cell: [[File:Alternated bitruncated cubic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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>

{| class=wikitable
|+ Five uniform colorings
|-
![[Space group]]||I{{overline|3}} (204) ||Pm{{overline|3}} (200) ||Fm{{overline|3}} (202)||Fd{{overline|3}} (203) || F23 (196)
|-
![[Fibrifold]]||8<sup>−o</sup>||4<sup>−</sup>||2<sup>−</sup>||2<sup>o+</sup> ||1<sup>o</sup>
|-
![[Coxeter group]]||<nowiki>[[</nowiki>4,3<sup>+</sup>,4]]|| [4,3<sup>+</sup>,4]|| [4,(3<sup>1,1</sup>)<sup>+</sup>]|| [<nowiki/>[3<sup>[4]</sup>]]<sup>+</sup>||[3<sup>[4]</sup>]<sup>+</sup>
|- align=center
![[Coxeter diagram]]
|{{CDD||branch_hh|4a4b|nodes}}
|{{CDD||node|4|node_h|3|node_h|4|node}}
|{{CDD|node|4|node_h|split1|nodes_hh}}
|{{CDD|branch_hh|3ab|branch_hh}}
|{{CDD|node_h|split1|nodes_hh|split2|node_h}}
|- align=center
!Order
|double
|full
|half
|quarter<br />double
|quarter
|}
{{clear}}
----

=== Cantellated cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Cantellated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|[[File:Cantellated cubic tiling.png|190px]] [[File:HC A5-A3-P2.png|112px]]-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||rr{4,3,4} or t<sub>0,2</sub>{4,3,4}<br />rr{4,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node|3|node_1|4|node}}<br />{{CDD|node_1|4|node|split1|nodes_11}} = {{CDD|node_1|4|node|3|node_1|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|rr{4,3}]] [[File:Uniform polyhedron-43-t02.png|40px]]<br />[[cuboctahedron|r{4,3}]] [[File:Uniform polyhedron-43-t1.png|40px]]<br />[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Cantellated cubic honeycomb verf.png|80px]]<br />[[Wedge (geometry)|wedge]]
|-
|bgcolor=#e7dcc3|[[Space group]]<br />[[Fibrifold notation]]||Pm{{overline|3}}m (221)<br />4<sup>−</sup>:2
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
|-
|bgcolor=#e7dcc3|Dual|| [[quarter oblate octahedrille]]<br />Cell: [[File:Quarter oblate octahedrille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}
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]].

:[[File:Cantellated cubic tiling.png|190px]] [[File:HC A5-A3-P2.png|112px]]

==== Images ====

{| class=wikitable width=400
|- valign=top
|[[File:Cantellated cubic honeycomb.png|200px]]
|[[File:Perovskite.jpg|200px]]<br />It is closely related to the [[Perovskite (structure)|perovskite structure]], shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb.
|}

==== Projections ====
The ''cantellated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements.
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:cantellated cubic_honeycomb-2.png|120px]]
|[[File:cantellated cubic_honeycomb-1.png|120px]]
|[[File:cantellated cubic_honeycomb-3.png|150px]]
|-
!Frame
|[[File:cantellated cubic_honeycomb-2b.png|120px]]
|[[File:cantellated cubic_honeycomb-1b.png|120px]]
|[[File:cantellated cubic_honeycomb-3b.png|150px]]
|}

==== Symmetry ====

There is a second [[uniform coloring]]s by reflectional symmetry of the [[Coxeter group]]s, the second seen with alternately colored rhombicuboctahedral cells.
{| class="wikitable" width=360
|+ Vertex uniform colorings by cell
|-
!Construction
!Truncated cubic honeycomb
!Bicantellated alternate cubic
|- valign=top
![[Coxeter group]]
![4,3,4], <math>{\tilde{C}}_3</math><br />=<[4,3<sup>1,1</sup>]>
![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
|-
![[Space group]]||Pm{{overline|3}}m||Fm{{overline|3}}m
|-
![[Coxeter diagram]]
!{{CDD|node_1|4|node|3|node_1|4|node}}
!{{CDD|node_1|4|node|split1|nodes_11}}
|- align=center
!Coloring
|[[File:Cantellated cubic honeycomb.png|120px]]
|[[File:Cantellated cubic honeycomb2.png|120px]]
|-
![[Vertex figure]]
|[[File:Cantellated cubic honeycomb verf.png|120px]]
|[[File:Runcicantellated alternate cubic honeycomb verf.png|120px]]
|- align=center
!Vertex<br />figure<br />symmetry
|[ ]<br />order 2
|[ ]<sup>+</sup><br />order 1
|}

==== Related polytopes ====
A double symmetry construction can be made by placing cuboctahedra on the rhombicuboctahedra, which results in the [[Cubic honeycomb#Rectified cubic honeycomb|rectified cubic honeycomb]], by taking the triangular antiprism gaps as regular [[octahedron|octahedra]], square antiprism pairs and zero-height tetragonal disphenoids as components of the [[cuboctahedron]]. Other variants result in [[cuboctahedron|cuboctahedra]], [[square antiprism]]s, [[octahedron|octahedra]] (as triangular antipodiums), and [[tetrahedron|tetrahedra]] (as tetragonal disphenoids), with a vertex figure topologically equivalent to a [[cube]] with a [[triangular prism]] attached to one of its square faces.
{{clear}}
----

====Quarter oblate octahedrille====
The dual of the ''cantellated cubic honeycomb'' is called a '''quarter oblate octahedrille''', a [[catoptric tessellation]] with [[Coxeter diagram]] {{CDD|node_f1|4|node|3|node_f1|4|node}}, containing faces from two of four hyperplanes of the cubic [4,3,4] fundamental domain.

It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, and 2 vertices.
:[[File:Quarter oblate octahedrille cell.png|240px]]

=== Cantitruncated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{4,3,4} or t<sub>0,1,2</sub>{4,3,4}<br />tr{4,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node_1|3|node_1|4|node}}<br />{{CDD|node_1|4|node_1|split1|nodes_11}} = {{CDD|node_1|4|node_1|3|node_1|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated cuboctahedron|tr{4,3}]] [[File:Uniform polyhedron-43-t012.png|40px]]<BR>[[truncated octahedron|t{3,4}]] [[File:Uniform polyhedron-43-t12.png|40px]]<BR>[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Cantitruncated cubic honeycomb verf.png|80px]][[File:Omnitruncated alternated cubic honeycomb verf.png|80px]]<br />[[mirrored sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]<br />[[Fibrifold notation]]||Pm{{overline|3}}m (221)<br />4<sup>−</sup>:2
|-
|bgcolor=#e7dcc3|Dual||[[triangular pyramidille]]<br />Cells: [[File:Triangular pyramidille cell1.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}
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]].

: [[Image:Cantitruncated cubic tiling.png|190px]]&nbsp;[[File:HC A6-A4-P2.png|110px]]

==== Images====
Four cells exist around each vertex:
:[[File:2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png|320px]]

==== Projections ====
The ''cantitruncated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements.
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:cantitruncated cubic_honeycomb-2.png|120px]]
|[[File:cantitruncated cubic_honeycomb-1.png|120px]]
|[[File:cantitruncated cubic_honeycomb-3.png|150px]]
|-
!Frame
|[[File:cantitruncated cubic_honeycomb-2b.png|120px]]
|[[File:cantitruncated cubic_honeycomb-1b.png|120px]]
|[[File:cantitruncated cubic_honeycomb-3b.png|150px]]
|}

==== Symmetry ====

Cells can be shown in two different symmetries. The linear [[Coxeter diagram]] form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of [[truncated cuboctahedron]] cells alternating.
{| class="wikitable" width=320
!Construction
!Cantitruncated cubic
!Omnitruncated alternate cubic
|- valign=top
![[Coxeter group]]
![4,3,4], <math>{\tilde{C}}_3</math><br />=<[4,3<sup>1,1</sup>]>
![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
|-
![[Space group]]||Pm{{overline|3}}m (221)||Fm{{overline|3}}m (225)
|-
![[Fibrifold]]||4<sup>−</sup>:2||2<sup>−</sup>:2
|- align=center
!Coloring
|[[Image:Cantitruncated Cubic Honeycomb.svg|80px]]
|[[Image:Cantitruncated Cubic Honeycomb2.svg|80px]]
|-
![[Coxeter diagram]]
!{{CDD|node_1|4|node_1|3|node_1|4|node}}
!{{CDD|node_1|4|node_1|split1|nodes_11}}
|-
![[Vertex figure]]
|[[Image:Cantitruncated cubic honeycomb verf.png|80px]]
|[[File:Omnitruncated alternated cubic honeycomb verf.png|80px]]
|- align=center
!Vertex<br />figure<br />symmetry
|[ ]<br />order 2
|[ ]<sup>+</sup><br />order 1
|}

==== Triangular pyramidille====

The 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}}. This honeycomb cells represents 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 corner, 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]]

==== Related polyhedra and honeycombs====
It is related to a [[skew apeirohedron]] with [[vertex configuration]] 4.4.6.6, with the octagons and some of the squares removed. It can be seen as constructed by augmenting truncated cuboctahedral cells, or by augmenting alternated truncated octahedra and cubes.
{| class=wikitable
|+ Two views
|[[File:Cantitruncated cubic honeycomb apeirohedron 4466.png|240px]]
|[[File:Omnitruncated_cubic_honeycomb_apeirohedron_4466.png|240px]]
|}

==== Related polytopes ====
A double symmetry construction can be made by placing truncated octahedra on the truncated cuboctahedra, resulting in a nonuniform honeycomb with [[truncated octahedron|truncated octahedra]], [[hexagonal prism]]s (as ditrigonal trapezoprisms), [[cube]]s (as square prisms), [[triangular prism]]s (as ''C<sub>2v</sub>''-symmetric wedges), and [[tetrahedron|tetrahedra]] (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the [[octahedron]].

[[File:Bicantitruncatocubic honeycomb vertex figure.png|120px]]<br />[[Vertex figure]]

[[File:Bicantitruncatocubic honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

{{clear}}
----

=== Alternated cantitruncated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Alternated cantitruncated cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||sr{4,3,4}<br />sr{4,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_h|4|node_h|3|node_h|4|node}}<br />{{CDD|node_h|4|node_h|split1|nodes_hh}} = {{CDD|node_h|4|node_h|3|node_h|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[snub cube|s{4,3}]] [[File:Uniform polyhedron-43-s012.png|40px]]<br />[[icosahedron|s{3,3}]] [[File:Uniform polyhedron-33-s012.png|40px]]<br />[[tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Alternated cantitruncated cubic honeycomb vertex figure.png|80px]][[File:Alternated_cantitruncated_cubic_honeycomb_verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[(4,3)<sup>+</sup>,4]
|-
|bgcolor=#e7dcc3|Dual||{{CDD|node_fh|4|node_fh|3|node_fh|4|node}}<br />Cell: [[File:Alternated cantitruncated cubic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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.

{| class=wikitable
|- align=center
|[[File:Alternated cantitruncated cubic honeycomb.png|240px]]<br />{{CDD|node_h|4|node_h|3|node_h|4|node}}
|[[File:Althalfcell-honeycomb-cube3x3x3.png|220px]]<br />{{CDD|node_fh|4|node_fh|3|node_fh|4|node}}
|}
{{clear}}
----

=== Cantic snub cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Orthosnub cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2s<sub>0</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|4|node_h|3|node_h|4|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Rhombicuboctahedron#Pyritohedral symmetry|s<sub>2</sub>{3,4}]] [[File:Uniform polyhedron-43-t02.png|40px]]<br />[[icosahedron|s{3,3}]] [[File:Uniform polyhedron-33-s012.png|40px]]<br />[[triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Orthosnub cubic honeycomb vertex figure.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4<sup>+</sup>,3,4]
|-
|bgcolor=#e7dcc3|Dual||Cell: [[File:Orthosnub cubic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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>

==== Related polytopes ====
A double symmetry construction can be made by placing icosahedra on the rhombicuboctahedra, resulting in a nonuniform honeycomb with [[icosahedron|icosahedra]], [[octahedron|octahedra]] (as triangular antiprisms), [[triangular prism]]s (as ''C<sub>2v</sub>''-symmetric wedges), and [[square pyramid]]s.

[[File:Biorthopyritohedral honeycomb vertex figure.png|120px]]<br />[[Vertex figure]]

[[File:Biorthopyritohedral honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

{{clear}}
----

=== Runcitruncated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Runcitruncated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|4|node_1|3|node|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|rr{4,3}]] [[File:Uniform polyhedron-43-t02.png|40px]]<br />[[truncated cube|t{4,3}]] [[File:Uniform polyhedron-43-t01.png|40px]]<br />[[octagonal prism|{}x{8}]] [[File:Octagonal prism.png|40px]]<br />[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Runcitruncated cubic honeycomb verf.png|80px]]<br />[[isosceles trapezoid|isosceles-trapezoidal]] [[pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
|-
|bgcolor=#e7dcc3|[[Space group]]<br />[[Fibrifold notation]]||Pm{{overline|3}}m (221)<br />4<sup>−</sup>:2
|-
|bgcolor=#e7dcc3|Dual|| [[square quarter pyramidille]]<br />Cell [[File:Square quarter pyramidille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}
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]].

{|
| [[File:Runcitruncated cubic tiling.png|x140px]]
| [[File:HC A5-A2-P2-Pr8.png|x140px]]
| [[Image:Runcitruncated cubic honeycomb.jpg|x140px]]
| [[File:Runcitruncated cubic honeycomb (Schoute 1911).jpg|x140px]]
|}

==== Projections ====
The ''runcitruncated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements.
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:runcitruncated cubic_honeycomb-2.png|120px]]
|[[File:runcitruncated cubic_honeycomb-1.png|120px]]
|[[File:runcitruncated cubic_honeycomb-3.png|150px]]
|-
!Frame
|[[File:runcitruncated cubic_honeycomb-2b.png|120px]]
|[[File:runcitruncated cubic_honeycomb-1b.png|120px]]
|[[File:runcitruncated cubic_honeycomb-3b.png|150px]]
|}

==== Related skew apeirohedron====
Two related uniform [[skew apeirohedron]]s exists with the same [[vertex arrangement]], seen as boundary cells from a subset of cells. One has triangles and squares, and the other triangles, squares, and octagons.
:[[File:Skew polyhedron 34444.png|200px]][[File:Skew polyhedron 3448.png|200px]]

==== Square quarter pyramidille ====
The dual to the ''runcitruncated cubic honeycomb'' is called a '''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]]

==== Related polytopes ====
A double symmetry construction can be made by placing rhombicuboctahedra on the truncated cubes, resulting in a nonuniform honeycomb with [[rhombicuboctahedron|rhombicuboctahedra]], [[octahedron|octahedra]] (as triangular antiprisms), [[cube]]s (as square prisms), two kinds of [[triangular prism]]s (both ''C<sub>2v</sub>''-symmetric wedges), and [[tetrahedron|tetrahedra]] (as digonal disphenoids). Its vertex figure is topologically equivalent to the [[augmented triangular prism]].

[[File:Biruncitruncatocubic honeycomb vertex figure.png|120px]]<br />[[Vertex figure]]

[[File:Biruncitruncatocubic honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

{{clear}}
----

===Omnitruncated cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_1|4|node_1|3|node_1|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated cuboctahedron|tr{4,3}]] [[File:Uniform polyhedron-43-t012.png|40px]]<BR>[[octagonal prism|{}x{8}]] [[File:Octagonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Omnitruncated cubic honeycomb verf.png|80px]]<br />[[phyllic disphenoid]]
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]<br />[[Fibrifold notation]]<br />[[Coxeter notation#Space groups|Coxeter notation]]||[[Cubic crystal system|Im{{overline|3}}m (229)]]<br />8<sup>o</sup>:2<br />[<nowiki/>[4,3,4]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
|-
|bgcolor=#e7dcc3|Dual||[[eighth pyramidille]]<br />Cell [[File:Fundamental tetrahedron1.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}
The '''omnitruncated cubic honeycomb''' or '''omnitruncated cubic cellulation''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[truncated cuboctahedron|truncated cuboctahedra]] and [[octagonal prism]]s in a ratio of 1:3, with a [[phyllic disphenoid]] [[vertex figure]].

[[John Horton Conway]] calls this honeycomb a '''b-tCO-trille''', and its dual [[eighth pyramidille]].

: [[Image:Omnitruncated cubic tiling.png|320px]]&nbsp;[[File:HC A6-Pr8.png|194px]]

==== Projections ====
The ''omnitruncated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements.
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:omnitruncated cubic_honeycomb-2.png|120px]]
|[[File:omnitruncated cubic_honeycomb-1.png|120px]]
|[[File:omnitruncated cubic_honeycomb-3.png|150px]]
|-
!Frame
|[[File:omnitruncated cubic_honeycomb-2b.png|120px]]
|[[File:omnitruncated cubic_honeycomb-1b.png|120px]]
|[[File:omnitruncated cubic_honeycomb-3b.png|150px]]
|}

==== Symmetry ====

Cells can be shown in two different symmetries. The [[Coxeter diagram]] form has two colors of [[truncated cuboctahedron|truncated cuboctahedra]] and [[octagonal prism]]s. The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octagonal prism cells.
{| class="wikitable"
|+ Two uniform colorings
|-
![[Coxeter notation|Symmetry]]
!<math>{\tilde{C}}_3</math>, [4,3,4]
!<math>{\tilde{C}}_3</math>×2, {{brackets|4,3,4}}
|-
![[Space group]]||Pm{{overline|3}}m (221)||Im{{overline|3}}m (229)
|-
![[Fibrifold]]||4<sup>−</sup>:2||8<sup>o</sup>:2
|- align=center
!Coloring
|[[Image:Omnitruncated cubic honeycomb1.png|150px]]
|[[Image:Omnitruncated cubic honeycomb2.png|150px]]
|-
![[Coxeter diagram]]
!{{CDD|node_1|4|node_1|3|node_1|4|node_1}}
!{{CDD|branch_11|4a4b|nodes_11}}
|- align=center
![[Vertex figure]]
|[[File:Omnitruncated cubic honeycomb verf.png|100px]]
|[[Image:Omnitruncated cubic honeycomb verf2.png|100px]]
|}

==== Related polyhedra ====
Two related uniform [[skew apeirohedron]] exist with the same [[vertex arrangement]]. The first has octagons removed, and vertex configuration 4.4.4.6. It can be seen as truncated cuboctahedra and octagonal prisms augmented together. The second can be seen as augmented octagonal prisms, vertex configuration 4.8.4.8.
{| class=wikitable
!4.4.4.6<br />[[File:Omnitruncated_cubic_honeycomb_skew1_verf.png|120px]]
!4.8.4.8<br />[[File:Omnitruncated_cubic_honeycomb_skew2_verf.png|120px]]
|-
|[[File:Omnitruncated cubic honeycomb apeirohedron 4446.png|320px]]
|[[File:Skew polyhedron 4848.png|320px]]
|}

==== Related polytopes ====
Nonuniform variants with [4,3,4] symmetry and two types of truncated cuboctahedra can be doubled by placing the two types of truncated cuboctahedra on each other to produce a nonuniform honeycomb with [[truncated cuboctahedron|truncated cuboctahedra]], [[octagonal prism]]s, [[hexagonal prism]]s (as ditrigonal trapezoprisms), and two kinds of [[cube]]s (as rectangular trapezoprisms and their ''C<sub>2v</sub>''-symmetric variants). Its vertex figure is an irregular [[triangular bipyramid]].

[[File:Biomnitruncatocubic honeycomb vertex figure.png|120px]]<br />[[Vertex figure]]

[[File:Biomnitruncatocubic honeycomb dual cell.png|120px]]<br />[[Dual polyhedron|Dual cell]]

This honeycomb can then be alternated to produce another nonuniform honeycomb with [[snub cube]]s, [[square antiprism]]s, [[octahedron|octahedra]] (as triangular antiprisms), and three kinds of [[tetrahedron|tetrahedra]] (as tetragonal disphenoids, phyllic disphenoids, and irregular tetrahedra).

[[File:Alternated biomnitruncatocubic honeycomb vertex figure.png|120px]]<br />[[Vertex figure]]
{{clear}}
----

=== Alternated omnitruncated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Alternated omnitruncated cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||ht<sub>0,1,2,3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_h|4|node_h|3|node_h|4|node_h}}
|-
|bgcolor=#e7dcc3|Cells||[[snub cube|s{4,3}]] [[File:Uniform polyhedron-43-s012.png|40px]]<br />[[square antiprism|s{2,4}]] [[File:Square antiprism.png|40px]]<br />[[tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Omnisnub cubic honeycomb vertex figure.png|80px]][[File:Snub cubic honeycomb verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||[<nowiki/>[4,3,4]]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Dual||[[Cubic honeycomb#Dual alternated omnitruncated cubic honeycomb|Dual alternated omnitruncated cubic honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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.

{{clear}}

==== Dual alternated omnitruncated cubic honeycomb ====
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Dual alternated omnitruncated cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||Dual alternated uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||dht<sub>0,1,2,3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]||{{CDD|node_fh|4|node_fh|3|node_fh|4|node_fh}}
|-
|bgcolor=#e7dcc3|Cell||[[File:Omnisnub cubic honeycomb dual cell.png|100px]]
|-
|bgcolor=#e7dcc3|Vertex figures||[[pentagonal icositetrahedron]]<br />[[tetragonal trapezohedron]]<br />[[tetrahedron]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||[<nowiki/>[4,3,4]]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Dual||[[Cubic honeycomb#Alternated omnitruncated cubic honeycomb|Alternated omnitruncated cubic honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||[[Cell-transitive]]
|}
A '''dual alternated omnitruncated cubic honeycomb''' is a space-filling honeycomb constructed as the dual of the [[#alternated omnitruncated cubic honeycomb|alternated omnitruncated cubic honeycomb]].

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]]

Individual cells have 2-fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.
{| class=wikitable
|+ Cell views
|- align=center
|[[File:Altbasetet_net.png|100px]]<br />Net
|[[File:Altbasetet.png|100px]]
|[[File:Altbasetet-frame1.png|100px]]
|[[File:Altbasetet-frame2.png|100px]]
|-
|[[File:Altbasetet-frame3.png|100px]]
|[[File:Altbasetet-frame4.png|100px]]
|[[File:Altbasetet-frame5.png|100px]]
|[[File:Altbasetet-frame7.png|100px]]
|}
{{clear}}
----

=== Runcic cantitruncated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Runcic cantitruncated cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||sr<sub>3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_h|4|node_h|3|node_h|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|s<sub>2</sub>{3,4}]] [[File:Uniform polyhedron-43-t02.png|40px]]<br />[[snub cube|s{4,3}]] [[File:Uniform polyhedron-43-s012.png|40px]]<br />[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]<br />[[triangular prism|{}x{3}]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Bialternatosnub cubic honeycomb vertex figure.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4,3<sup>+</sup>,4]
|-
|bgcolor=#e7dcc3|Dual||Cell: [[File:Bialternatosnub cubic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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.
{{clear}}
----

=== Biorthosnub cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Biorthosnub cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2s<sub>0,3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_h|4|node_1|3|node_1|4|node_h}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|s<sub>2</sub>{3,4}]] [[File:Uniform polyhedron-43-t02.png|40px]]<br />[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Biorthosnub cubic honeycomb vertex figure.png|80px]]<br />([[Hexahedron|Tetragonal antiwedge]])
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[4,3<sup>+</sup>,4]]
|-
|bgcolor=#e7dcc3|Dual||Cell: [[File:Biorthosnub cubic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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).
{{clear}}
----

=== Truncated square prismatic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Truncated square prismatic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t{4,4}×{∞} or t<sub>0,1,3</sub>{4,4,2,∞}<br />tr{4,4}×{∞} or t<sub>0,1,2,3</sub>{4,4,∞}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node_1|4|node|2|node_1|infin|node}}<br />{{CDD|node_1|4|node_1|4|node_1|2|node_1|infin|node}}
|-
|bgcolor=#e7dcc3|Cells||[[Octagonal prism|{}x{8}]] [[File:Octagonal prism.png|40px]]<BR>[[Cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4,4,2,∞]
|-
|bgcolor=#e7dcc3|Dual||[[Tetrakis square prismatic tiling]]<br />Cell: [[File:Cubic half domain.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}
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.

[[Image:Truncated square prismatic honeycomb.png|320px]]

It is constructed from a [[truncated square tiling]] extruded into prisms.

It is one of 28 [[convex uniform honeycomb]]s.
{{clear}}
----

=== Snub square prismatic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Snub square prismatic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||s{4,4}×{∞}<br />sr{4,4}×{∞}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_h|4|node_h|4|node|2|node_1|infin|node}}<br />{{CDD|node_h|4|node_h|4|node_h|2|node_1|infin|node}}
|-
|bgcolor=#e7dcc3|Cells||[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]<BR>[[triangular prism|{}x{3}]] [[File:Triangular prism.png|30px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[4<sup>+</sup>,4,2,∞]<br />[(4,4)<sup>+</sup>,2,∞]
|-
|bgcolor=#e7dcc3|Dual||[[Cairo pentagonal prismatic honeycomb]]<br />Cell: [[File:Snub square prismatic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]]
|}

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.

[[Image:Snub square prismatic honeycomb.png|320px]]

It is constructed from a [[snub square tiling]] extruded into prisms.

It is one of 28 [[convex uniform honeycomb]]s.
{{clear}}

----

=== Snub square antiprismatic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Snub square antiprismatic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||ht<sub>1,2,3</sub>{4,4,2,∞}<br />ht<sub>0,1,2,3</sub>{4,4,∞}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node|4|node_h|4|node_h|2x|node_h|infin|node}}<BR>{{CDD|node_h|4|node_h|4|node_h|2x|node_h|infin|node}}
|-
|bgcolor=#e7dcc3|Cells||[[square antiprism|s{2,4}]] [[File:Square antiprism.png|40px]]<br />[[tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Snub square antiprismatic honeycomb vertex figure.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||[4,4,2,∞]<sup>+</sup>
|-
|bgcolor=#e7dcc3|Properties||[[Vertex-transitive]], non-uniform
|}
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}}

<br>
==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}}
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) ''The Symmetries of Things'', {{ISBN|978-1-56881-220-5}} (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
* [[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]]