Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular spacefilling tessellation (or honeycomb) in Euclidean 3space, made up of 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 selfdual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.
Cubic honeycomb  

Type  Regular honeycomb 
Family  Hypercube honeycomb 
Indexing[1]  J_{11,15}, A_{1} W_{1}, G_{22} 
Schläfli symbol  {4,3,4} 
Coxeter diagram  
Cell type  {4,3} 
Face type  {4} 
Vertex figure  (octahedron) 
Space group Fibrifold notation  Pm3m (221) 4^{−}:2 
Coxeter group  , [4,3,4] 
Dual  selfdual Cell: 
Properties  vertextransitive, regular 
A geometric honeycomb is a spacefilling of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in nonEuclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
Cartesian coordinates
The Cartesian coordinates of the vertices are:
 (i, j, k)
 for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1.
Related honeycombs
It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.
It is one of 28 uniform honeycombs using convex uniform polyhedral cells.
Isometries of simple cubic lattices
Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:
Crystal system  Monoclinic Triclinic 
Orthorhombic  Tetragonal  Rhombohedral  Cubic 

Unit cell  Parallelepiped  Rectangular cuboid  Square cuboid  Trigonal trapezohedron 
Cube 
Point group Order Rotation subgroup 
[ ], (*) Order 2 [ ]^{+}, (1) 
[2,2], (*222) Order 8 [2,2]^{+}, (222) 
[4,2], (*422) Order 16 [4,2]^{+}, (422) 
[3], (*33) Order 6 [3]^{+}, (33) 
[4,3], (*432) Order 48 [4,3]^{+}, (432) 
Diagram  
Space group Rotation subgroup 
Pm (6) P1 (1) 
Pmmm (47) P222 (16) 
P4/mmm (123) P422 (89) 
R3m (160) R3 (146) 
Pm3m (221) P432 (207) 
Coxeter notation    [∞]_{a}×[∞]_{b}×[∞]_{c}  [4,4]_{a}×[∞]_{c}    [4,3,4]_{a} 
Coxeter diagram     
Uniform colorings
There is a large number of uniform colorings, derived from different symmetries. These include:
Coxeter notation Space group 
Coxeter diagram  Schläfli symbol  Partial honeycomb 
Colors by letters 

[4,3,4] Pm3m (221) 
{4,3,4}  1: aaaa/aaaa  
[4,3^{1,1}] = [4,3,4,1^{+}] Fm3m (225) 
{4,3^{1,1}}  2: abba/baab  
[4,3,4] Pm3m (221) 
t_{0,3}{4,3,4}  4: abbc/bccd  
[[4,3,4]] Pm3m (229) 
t_{0,3}{4,3,4}  4: abbb/bbba  
[4,3,4,2,∞]  or 
{4,4}×t{∞}  2: aaaa/bbbb  
[4,3,4,2,∞]  t_{1}{4,4}×{∞}  2: abba/abba  
[∞,2,∞,2,∞]  t{∞}×t{∞}×{∞}  4: abcd/abcd  
[∞,2,∞,2,∞] = [4,(3,4)^{*}]  t{∞}×t{∞}×t{∞}  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.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Related polytopes and honeycombs
It is related to the regular 4polytope tesseract, Schläfli symbol {4,3,3}, which exists in 4space, and only has 3 cubes around each edge. It's also related to the order5 cubic honeycomb, Schläfli symbol {4,3,5}, of hyperbolic space with 5 cubes around each edge.
It is in a sequence of polychora and honeycomb with octahedral vertex figures.
{p,3,4} regular honeycombs  

Space  S^{3}  E^{3}  H^{3}  
Form  Finite  Affine  Compact  Paracompact  Noncompact  
Name  {3,3,4} 
{4,3,4} 
{5,3,4} 
{6,3,4} 
{7,3,4} 
{8,3,4} 
... {∞,3,4}  
Image  
Cells  {3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3} 
It in a sequence of regular polytopes and honeycombs with cubic cells.
{4,3,p} regular honeycombs  

Space  S^{3}  E^{3}  H^{3}  
Form  Finite  Affine  Compact  Paracompact  Noncompact  
Name 
{4,3,3} 
{4,3,4} 
{4,3,5} 
{4,3,6} 
{4,3,7} 
{4,3,8} 
... {4,3,∞}  
Image  
Vertex figure 
{3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞} 
{p,3,p} regular honeycombs  

Space  S^{3}  Euclidean E^{3}  H^{3}  
Form  Finite  Affine  Compact  Paracompact  Noncompact  
Name  {3,3,3}  {4,3,4}  {5,3,5}  {6,3,6}  {7,3,7}  {8,3,8}  ...{∞,3,∞}  
Image  
Cells  {3,3} 
{4,3} 
{5,3} 
{6,3} 
{7,3} 
{8,3} 
{∞,3}  
Vertex figure 
{3,3} 
{3,4} 
{3,5} 
{3,6} 
{3,7} 
{3,8} 
{3,∞} 
Related polytopes
The cubic honeycomb has a lower symmetry as a runcinated cubic honeycomb, with two sizes of cubes. A double symmetry construction can be constructed by placing a small cube into each large cube, resulting in a nonuniform honeycomb with cubes, square prisms, and rectangular trapezoprisms (a cube with D_{2d} symmetry). Its vertex figure is a triangular pyramid with its lateral faces augmented by tetrahedra.
The resulting honeycomb can be alternated to produce another nonuniform honeycomb with regular tetrahedra, two kinds of tetragonal disphenoids, triangular pyramids, and sphenoids. Its vertex figure has C_{3v} symmetry and has 26 triangular faces, 39 edges, and 15 vertices.
Related Euclidean tessellations
The [4,3,4],
C3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Pm3m (221) 
4^{−}:2  [4,3,4]  ×1  
Fm3m (225) 
2^{−}:2  [1^{+},4,3,4] ↔ [4,3^{1,1}] 
↔ 
Half  
I43m (217) 
4^{o}:2  [[(4,3,4,2^{+})]]  Half × 2  
Fd3m (227) 
2^{+}:2  [[1^{+},4,3,4,1^{+}]] ↔ [[3^{[4]}]] 
↔ 
Quarter × 2  
Im3m (229) 
8^{o}:2  [[4,3,4]]  ×2 
The [4,3^{1,1}],
B3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Fm3m (225) 
2^{−}:2  [4,3^{1,1}] ↔ [4,3,4,1^{+}] 
↔ 
×1  
Fm3m (225) 
2^{−}:2  <[1^{+},4,3^{1,1}]> ↔ <[3^{[4]}]> 
↔ 
×2  
Pm3m (221) 
4^{−}:2  <[4,3^{1,1}]>  ×2 
This honeycomb is one of five distinct uniform honeycombs[2] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A3 honeycombs  

Space group 
Fibrifold  Square symmetry 
Extended symmetry 
Extended diagram 
Extended group 
Honeycomb diagrams 
F43m (216) 
1^{o}:2  a1 
[3^{[4]}]  (None)  
Fm3m (225) 
2^{−}:2  d2 
<[3^{[4]}]> ↔ [4,3^{1,1}] 
↔ 
×2_{1} ↔ 

Fd3m (227) 
2^{+}:2  g2 
[[3^{[4]}]] or [2^{+}[3^{[4]}]] 
↔ 
×2_{2}  
Pm3m (221) 
4^{−}:2  d4 
<2[3^{[4]}]> ↔ [4,3,4] 
↔ 
×4_{1} ↔ 

I3 (204) 
8^{−o}  r8 
[4[3^{[4]}]]^{+} ↔ [[4,3<sup>+</sup>,4]] 
↔ 
½×8 ↔ ½×2 

Im3m (229) 
8^{o}:2  [4[3^{[4]}]] ↔ [[4,3,4]] 
×8 ↔ ×2 
Rectified cubic honeycomb
Rectified cubic honeycomb  

Type  Uniform honeycomb 
Cells  Octahedron Cuboctahedron 
Schläfli symbol  r{4,3,4} or t_{1}{4,3,4} r{4,3^{1,1}} 2r{4,3^{1,1}} r{3^{[4]}} 
Coxeter diagrams  
Vertex figure  Cuboid 
Space group Fibrifold notation  Pm3m (221) 4^{−}:2 
Coxeter group  , [4,3,4] 
Dual  oblate octahedrille Cell: 
Properties  vertextransitive, edgetransitive 
The rectified cubic honeycomb or rectified cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of octahedra and cuboctahedra in a ratio of 1:1.
John Horton Conway calls this honeycomb a cuboctahedrille, and its dual an oblate octahedrille.
Projections
The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Symmetry
There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below.
Symmetry  [4,3,4] 
[1^{+},4,3,4] [4,3^{1,1}], 
[4,3,4,1^{+}] [4,3^{1,1}], 
[1^{+},4,3,4,1^{+}] [3^{[4]}], 

Space group  Pm3m (221)  Fm3m (225)  Fm3m (225)  F43m (216) 
Coloring  
Coxeter diagram 

Vertex figure  
Vertex figure symmetry 
D_{4h} [4,2] (*224) order 16 
D_{2h} [2,2] (*222) order 8 
C_{4v} [4] (*44) order 8 
C_{2v} [2] (*22) order 4 
This honeycomb can be divided on trihexagonal tiling planes, using the hexagon centers of the cuboctahedra, creating two triangular cupolae. This scaliform honeycomb is represented by Coxeter diagram
.
Related polytopes
A double symmetry construction can be made by placing octahedra on the cuboctahedra, resulting in a nonuniform honeycomb with two kinds of octahedra (regular octahedra and triangular antiprisms). The vertex figure is a square bifrustum. The dual is composed of elongated square bipyramids.
Truncated cubic honeycomb
Truncated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  t{4,3,4} or t_{0,1}{4,3,4} t{4,3^{1,1}} 
Coxeter diagrams  
Cell type  3.8.8, {3,4} 
Face type  {3}, {4}, {8} 
Vertex figure  Isosceles square pyramid 
Space group Fibrifold notation  Pm3m (221) 4^{−}:2 
Coxeter group  , [4,3,4] 
Dual  Pyramidille Cell: 
Properties  vertextransitive 
The truncated cubic honeycomb or truncated cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of truncated cubes and octahedra in a ratio of 1:1.
John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.
Projections
The truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Symmetry
There is a second uniform coloring by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells.
Construction  Bicantellated alternate cubic  Truncated cubic honeycomb 

Coxeter group  [4,3^{1,1}],  [4,3,4], =<[4,3^{1,1}]> 
Space group  Fm3m  Pm3m 
Coloring  
Coxeter diagram  
Vertex figure 
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 octahedra (regular octahedra and triangular antiprisms) and two kinds of tetrahedra (tetragonal disphenoids and digonal disphenoids). The vertex figure is an octakis square cupola.
Bitruncated cubic honeycomb
Bitruncated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  2t{4,3,4} t_{1,2}{4,3,4} 
CoxeterDynkin diagram  
Cell type  (4.6.6) 
Face types  square {4} hexagon {6} 
Edge figure  isosceles triangle {3} 
Vertex figure  (tetragonal disphenoid) 
Space group Fibrifold notation Coxeter notation  Im3m (229) 8^{o}:2 [[4,3,4]] 
Coxeter group  , [4,3,4] 
Dual  Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell: 
Properties  isogonal, isotoxal, isochoric 
The bitruncated cubic honeycomb is a spacefilling tessellation (or honeycomb) in Euclidean 3space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is celltransitive. It is also edgetransitive, with 2 hexagons and one square on each edge, and vertextransitive. 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.
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.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Symmetry
The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.
Space group  Im3m (229)  Pm3m (221)  Fm3m (225)  F43m (216)  Fd3m (227) 

Fibrifold  8^{o}:2  4^{−}:2  2^{−}:2  1^{o}:2  2^{+}:2 
Coxeter group  ×2 [[4,3,4]] =[4[3^{[4]}]] 
[4,3,4] =[2[3^{[4]}]] 
[4,3^{1,1}] =<[3^{[4]}]> 
[3^{[4]}] 
×2 [[3^{[4]}]] =[[3^{[4]}]] 
Coxeter diagram  
truncated octahedra  1 
1:1 
2:1:1 
1:1:1:1 
1:1 
Vertex figure  
Vertex figure symmetry 
[2^{+},4] (order 8) 
[2] (order 4) 
[ ] (order 2) 
[ ]^{+} (order 1) 
[2]^{+} (order 2) 
Image Colored by cell 
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 octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C_{2v}symmetric triangular bipyramid.
This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra, octahedra (as triangular antiprisms), and tetrahedra (as sphenoids). Its vertex figure has C_{2v} symmetry and consists of 2 pentagons, 4 rectangles, 4 isosceles triangles (divided into two sets of 2), and 4 scalene triangles.
Alternated bitruncated cubic honeycomb
Alternated bitruncated cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  2s{4,3,4} 2s{4,3^{1,1}} sr{3^{[4]}} 
Coxeter diagrams  
Cells  tetrahedron icosahedron 
Vertex figure  
Coxeter group  [[4,3<sup>+</sup>,4]], 
Dual  Tenofdiamonds honeycomb Cell: 
Properties  vertextransitive 
The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb is nonuniform, with the highest symmetry construction reflecting an alternation of the uniform bitruncated cubic honeycomb. A lowersymmetry 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:
This honeycomb is represented in the boron atoms of the αrhombihedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[3]
Space group  I3 (204)  Pm3 (200)  Fm3 (202)  Fd3 (203)  F23 (196) 

Fibrifold  8^{−o}  4^{−}  2^{−}  2^{o+}  1^{o} 
Coxeter group  [[4,3^{+},4]]  [4,3^{+},4]  [4,(3^{1,1})^{+}]  [[3^{[4]}]]^{+}  [3^{[4]}]^{+} 
Coxeter diagram  
Order  double  full  half  quarter double 
quarter 
Cantellated cubic honeycomb
Cantellated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  rr{4,3,4} or t_{0,2}{4,3,4} rr{4,3^{1,1}} 
Coxeter diagram  
Cells  rr{4,3} r{4,3} {4,3} 
Vertex figure  (Wedge) 
Space group Fibrifold notation  Pm3m (221) 4^{−}:2 
Coxeter group  [4,3,4], 
Dual  quarter oblate octahedrille Cell: 
Properties  vertextransitive 
The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3.
John Horton Conway calls this honeycomb a 2RCOtrille, and its dual quarter oblate octahedrille.
Images
It is closely related to the 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.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Symmetry
There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells.
Construction  Truncated cubic honeycomb  Bicantellated alternate cubic 

Coxeter group  [4,3,4], =<[4,3^{1,1}]> 
[4,3^{1,1}], 
Space group  Pm3m  Fm3m 
Coxeter diagram  
Coloring  
Vertex figure  
Vertex figure symmetry 
[ ] order 2 
[ ]^{+} order 1 
Related polytopes
A double symmetry construction can be made by placing cuboctahedra on the rhombicuboctahedra, which results in the rectified cubic honeycomb, by taking the triangular antiprism gaps as regular octahedra, square antiprism pairs and zeroheight tetragonal disphenoids as components of the cuboctahedron. Other variants result in cuboctahedra, square antiprisms, octahedra (as triangular antipodiums), and tetrahedra (as tetragonal disphenoids), with a vertex figure topologically equivalent to a cube with a triangular prism attached to one of its square faces.
Quarter oblate octahedrille
The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram
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.
Cantitruncated cubic honeycomb
Cantitruncated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  tr{4,3,4} or t_{0,1,2}{4,3,4} tr{4,3^{1,1}} 
Coxeter diagram  
Vertex figure  (Sphenoid) 
Coxeter group  [4,3,4], 
Space group Fibrifold notation  Pm3m (221) 4^{−}:2 
Dual  triangular pyramidille Cells: 
Properties  vertextransitive 
The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3.
John Horton Conway calls this honeycomb a ntCOtrille, and its dual triangular pyramidille.
Images
Four cells exist around each vertex:
Projections
The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
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.
Construction  Cantitruncated cubic  Omnitruncated alternate cubic 

Coxeter group  [4,3,4], =<[4,3^{1,1}]> 
[4,3^{1,1}], 
Space group  Pm3m (221)  Fm3m (225) 
Fibrifold  4^{−}:2  2^{−}:2 
Coloring  
Coxeter diagram  
Vertex figure  
Vertex figure symmetry 
[ ] order 2 
[ ]^{+} order 1 
Triangular pyramidille
The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram,
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.
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.
Related polytopes
A double symmetry construction can be made by placing truncated octahedra on the truncated cuboctahedra, resulting in a nonuniform honeycomb with truncated octahedra, hexagonal prisms (as ditrigonal trapezoprisms), cubes (as square prisms), triangular prisms (as C_{2v}symmetric wedges), and tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.
Alternated cantitruncated cubic honeycomb
Alternated cantitruncated cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  sr{4,3,4} sr{4,3^{1,1}} 
Coxeter diagrams  
Cells  snub cube icosahedron tetrahedron 
Vertex figure  
Coxeter group  [(4,3)^{+},4] 
Dual  Cell: 
Properties  vertextransitive 
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
Despite being nonuniform, there is a nearmiss 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.
Orthosnub cubic honeycomb
Orthosnub cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  2s_{0}{4,3,4} 
Coxeter diagrams  
Cells  rhombicuboctahedron icosahedron triangular prism 
Vertex figure  
Coxeter group  [4^{+},3,4] 
Dual  Cell: 
Properties  vertextransitive 
The orthosnub 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
Related polytopes
A double symmetry construction can be made by placing icosahedra on the rhombicuboctahedra, resulting in a nonuniform honeycomb with icosahedra, octahedra (as triangular antiprisms), triangular prisms (as C_{2v}symmetric wedges), and square pyramids.
Runcitruncated cubic honeycomb
Runcitruncated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  t_{0,1,3}{4,3,4} 
Coxeter diagrams  
Cells  rhombicuboctahedron truncated cube octagonal prism cube 
Vertex figure  (Trapezoidal pyramid) 
Coxeter group  [4,3,4], 
Space group Fibrifold notation  Pm3m (221) 4^{−}:2 
Dual  square quarter pyramidille Cell 
Properties  vertextransitive 
The runcitruncated cubic honeycomb or runcitruncated cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3.
Its name is derived from its Coxeter diagram,
John Horton Conway calls this honeycomb a 1RCOtrille, and its dual square quarter pyramidille.
Projections
The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Related skew apeirohedron
Two related uniform skew apeirohedrons 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.
Square quarter pyramidille
The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram
Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one midedge point, two face centers, and the cube center.
Related polytopes
A double symmetry construction can be made by placing rhombicuboctahedra on the truncated cubes, resulting in a nonuniform honeycomb with rhombicuboctahedra, octahedra (as triangular antiprisms), cubes (as square prisms), two kinds of triangular prisms (both C_{2v}symmetric wedges), and tetrahedra (as digonal disphenoids). Its vertex figure is topologically equivalent to the augmented triangular prism.
Omnitruncated cubic honeycomb
Omnitruncated cubic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  t_{0,1,2,3}{4,3,4} 
Coxeter diagram  
Vertex figure  Phyllic disphenoid 
Space group Fibrifold notation Coxeter notation  Im3m (229) 8^{o}:2 [[4,3,4]] 
Coxeter group  [4,3,4], 
Dual  eighth pyramidille Cell 
Properties  vertextransitive 
The omnitruncated cubic honeycomb or omnitruncated cubic cellulation is a uniform spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3.
John Horton Conway calls this honeycomb a btCOtrille, and its dual eighth pyramidille.
Projections
The omnitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Symmetry  p6m (*632)  p4m (*442)  pmm (*2222)  

Solid  
Frame 
Symmetry
Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octagonal prisms. 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.
Symmetry  , [4,3,4]  ×2, [[4,3,4]] 

Space group  Pm3m (221)  Im3m (229) 
Fibrifold  4^{−}:2  8^{o}:2 
Coloring  
Coxeter diagram  
Vertex figure 
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.
4.4.4.6 
4.8.4.8 

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 cuboctahedra, octagonal prisms, hexagonal prisms (as ditrigonal trapezoprisms), and two kinds of cubes (as rectangular trapezoprisms and their C_{2v}symmetric variants). Its vertex figure is an irregular triangular bipyramid.
This honeycomb can then be alternated to produce another nonuniform honeycomb with snub cubes, square antiprisms, octahedra (as triangular antiprisms), and three kinds of tetrahedra (as tetragonal disphenoids, phyllic disphenoids, and irregular tetrahedra).
Alternated omnitruncated cubic honeycomb
Alternated omnitruncated cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  ht_{0,1,2,3}{4,3,4} 
Coxeter diagram  
Cells  snub cube square antiprism tetrahedron 
Vertex figure  
Symmetry  [[4,3,4]]^{+} 
Dual  Dual alternated omnitruncated cubic honeycomb 
Properties  vertextransitive 
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:
Dual alternated omnitruncated cubic honeycomb
Dual alternated omnitruncated cubic honeycomb  

Type  Dual alternated uniform honeycomb 
Schläfli symbol  dht_{0,1,2,3}{4,3,4} 
Coxeter diagram  
Cell  
Vertex figures  pentagonal icositetrahedron tetragonal trapezohedron tetrahedron 
Symmetry  [[4,3,4]]^{+} 
Dual  Alternated omnitruncated cubic honeycomb 
Properties  Celltransitive 
A dual alternated omnitruncated cubic honeycomb is a spacefilling honeycomb constructed as the dual of the alternated omnitruncated cubic honeycomb.
24 cells fit around a vertex, making a chiral octahedral symmetry that can be stacked in all 3dimensions:
Individual cells have 2fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.
Net 

Bialternatosnub cubic honeycomb
Bialternatosnub cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  sr_{3}{4,3,4} 
Coxeter diagrams  
Cells  rhombicuboctahedron snub cube cube triangular prism 
Vertex figure  
Coxeter group  [4,3^{+},4] 
Dual  Cell: 
Properties  vertextransitive 
The bialternatosnub cubic honeycomb or 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
Biorthosnub cubic honeycomb
Biorthosnub cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  2s_{0,3}{4,3,4} 
Coxeter diagrams  
Cells  rhombicuboctahedron cube 
Vertex figure  (Tetragonal antiwedge) 
Coxeter group  [[4,3<sup>+</sup>,4]] 
Dual  Cell: 
Properties  vertextransitive 
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
Truncated square prismatic honeycomb
Truncated square prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  t{4,4}×{∞} or t_{0,1,3}{4,4,2,∞} tr{4,4}×{∞} or t_{0,1,2,3}{4,4,∞} 
CoxeterDynkin diagram  
Coxeter group  [4,4,2,∞] 
Dual  Tetrakis square prismatic tiling Cell: 
Properties  vertextransitive 
The truncated square prismatic honeycomb or tomosquare prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. 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.
Snub square prismatic honeycomb
Snub square prismatic honeycomb  

Type  Uniform honeycomb 
Schläfli symbol  s{4,4}×{∞} sr{4,4}×{∞} 
CoxeterDynkin diagram  
Coxeter group  [4^{+},4,2,∞] [(4,4)^{+},2,∞] 
Dual  Cairo pentagonal prismatic honeycomb Cell: 
Properties  vertextransitive 
The snub square prismatic honeycomb or simosquare prismatic cellulation is a spacefilling tessellation (or honeycomb) in Euclidean 3space. 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.
Snub square antiprismatic honeycomb
Snub square antiprismatic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  ht_{0,1,3}{4,4,2,∞} ht_{0,1,2,3}{4,4,∞} 
CoxeterDynkin diagram  
Cells  square antiprism tetrahedron 
Vertex figure  
Symmetry  [4,4,2,∞]^{+} 
Properties  vertextransitive 
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:
See also
Wikimedia Commons has media related to Cubic honeycomb. 
 Architectonic and catoptric tessellation
 Alternated cubic honeycomb
 List of regular polytopes
 Order5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge
 voxel
References
 For crossreferencing, they are given with list indices from Andreini (122), Williams(12,919), Johnson (1119, 2125, 3134, 4149, 5152, 6165), and Grünbaum(128).
 , A000029 61 cases, skipping one with zero marks
 Williams, 1979, p 199, Figure 538.
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292298, includes all the nonprismatic forms)
 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808 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 3space. 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, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 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.
 Klitzing, Richard. "3D Euclidean Honeycombs x4o3o4o  chon  O1".
 Uniform Honeycombs in 3Space: 01Chon
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 