Cubic honeycomb
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 
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.
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.
Contents
Cartesian coordinatesEdit
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 honeycombsEdit
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 latticesEdit
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 coloringsEdit
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 
ProjectionsEdit
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 honeycombsEdit
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 Euclidean tessellationsEdit
The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.
C3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Pm3m (221) 
4^{−}:2  [4,3,4]  ×1  _{1}, _{2}, _{3}, _{4}, _{5}, _{6}  
Fm3m (225) 
2^{−}:2  [1^{+},4,3,4] ↔ [4,3^{1,1}] 
↔ 
Half  _{7}, _{11}, _{12}, _{13} 
I43m (217) 
4^{o}:2  [[(4,3,4,2^{+})]]  Half × 2  _{(7)},  
Fd3m (227) 
2^{+}:2  [[1^{+},4,3,4,1^{+}]] ↔ [[3^{[4]}]] 
↔ 
Quarter × 2  _{10}, 
Im3m (229) 
8^{o}:2  [[4,3,4]]  ×2 
The [4,3^{1,1}], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.
B3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Fm3m (225) 
2^{−}:2  [4,3^{1,1}] ↔ [4,3,4,1^{+}] 
↔ 
×1  _{1}, _{2}, _{3}, _{4} 
Fm3m (225) 
2^{−}:2  <[1^{+},4,3^{1,1}]> ↔ <[3^{[4]}]> 
↔ 
×2  _{(1)}, _{(3)} 
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} ↔ 
_{1}, _{2} 
Fd3m (227) 
2^{+}:2  g2  [[3^{[4]}]] or [2^{+}[3^{[4]}]] 
↔ 
×2_{2}  _{3} 
Pm3m (221) 
4^{−}:2  d4  <2[3^{[4]}]> ↔ [4,3,4] 
↔ 
×4_{1} ↔ 
_{4} 
I3 (204) 
8^{−o}  r8  [4[3^{[4]}]]^{+} ↔ [[4,3^{+},4]] 
↔ 
½ ×8 ↔ ½ ×2 
_{(*)} 
Im3m (229) 
8^{o}:2  [4[3^{[4]}]] ↔ [[4,3,4]] 
×8 ↔ ×2 
_{5} 
Rectified cubic honeycombEdit
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.
ProjectionsEdit
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 
SymmetryEdit
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 , and symbol s_{3}{2,6,3}, with coxeter notation symmetry [2^{+},6,3].
Truncated cubic honeycombEdit
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.
ProjectionsEdit
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 
SymmetryEdit
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 
Alternated bitruncated cubic honeycombEdit
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  [4,3,4], 
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: , , and . These have symmetry [4,3^{+},4], [4,(3^{1,1})^{+}] and [3^{[4]}]^{+} respectively. The first and last symmetry can be doubled as [[4,3^{+},4]] and [[3^{[4]}]]^{+}.
This honeycomb is represented in the boron atoms of the 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 honeycombEdit
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.
ImagesEdit
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. 
ProjectionsEdit
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 
SymmetryEdit
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 
Quarter oblate octahedrilleEdit
The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram , 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.
Cantitruncated cubic honeycombEdit
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  (Irreg. tetrahedron) 
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.
ImagesEdit
Four cells exist around each vertex:
ProjectionsEdit
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 
SymmetryEdit
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 pyramidilleEdit
The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, . This honeycomb cells represents the fundamental domains of 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.
Related polyhedra and honeycombsEdit
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.
Alternated cantitruncated cubic honeycombEdit
Alternated cantitruncated cubic honeycomb  

Type  Convex honeycomb 
Schläfli symbol  sr{4,3,4} sr{4,3^{1,1}} 
Coxeter diagrams  = 
Cells  tetrahedron pseudoicosahedron snub cube 
Vertex figure  
Coxeter group  [4,3^{1,1}], 
Dual  Cell: 
Properties  vertextransitive 
The alternated cantitruncated cubic honeycomb or snub rectified cubic honeycomb contains three types of cells: snub cubes, icosahedra (snub tetrahedron), and tetrahedra. In addition the gaps created at the alternated vertices form tetrahedral cells.
Although it is not uniform, constructionally it can be given as Coxeter diagrams or .
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.


Runcitruncated cubic honeycombEdit
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, with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb.
John Horton Conway calls this honeycomb a 1RCOtrille, and its dual square quarter pyramidille.
ProjectionsEdit
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 apeirohedronEdit
A related uniform skew apeirohedron exists with the same vertex arrangement, but some of the square and all of the octagons removed. It can be seen as truncated tetrahedra and truncated cubes augmented together.
Square quarter pyramidilleEdit
The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram . Faces exist in 3 of 4 hyperplanes of the [4,3,4], Coxeter group.
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.
Omnitruncated cubic honeycombEdit
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.
ProjectionsEdit
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 
SymmetryEdit
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 polyhedraEdit
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.
4.4.4.6 
4.8.4.8 

Alternated omnitruncated cubic honeycombEdit
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  Phyllic disphenoidal honeycomb 
Properties  vertextransitive 
An alternated omnitruncated cubic honeycomb or full snub 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: and has symmetry [[4,3,4]]^{+}. It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms and with new tetrahedral cells created in the gaps.
Dual alternated omnitruncated cubic honeycombEdit
Dual alternated omnitruncated cubic honeycomb  

Type  Dual alternated uniform honeycomb 
Schläfli symbol  dht_{0,1,2,3}{4,3,4} 
Coxeter diagram  
Cells  
Symmetry  [[4,3,4]]^{+} 
Properties  Celltransitive 
A dual alternated omnitruncated cubic honeycomb is a spacefilling honeycomb constructed as the dual of the alternated omnitruncated cubic honeycomb.
Cells can be seen from tetrahedra as 1/48 of a cube, augmented by new center point of adjacent tetrahedra.
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 honeycombEdit
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  
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 . It has rhombicuboctahedra (with T_{h} symmetry), snub cubes, rectangular trapezoprisms (topologically equivalent to a cube but with D_{2d} symmetry), and triangular prisms (as C_{2v}symmetry wedges) filling the gaps.
Truncated square prismatic honeycombEdit
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 honeycombEdit
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.
See alsoEdit
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
ReferencesEdit
 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 [2]
 (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
 

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{3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal  
{3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
{3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb  
{3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
{3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22}  
{3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31}  
{3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21}  
{3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
{3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 