# Cubic honeycomb

Cubic honeycomb :
| Type | Regular honeycomb |
| Family | Hypercube honeycomb |
| Indexing | J$_{11,15}$, A$_1$ W$_1$, G$_{22}$ |
| Schläfli symbol | {4,3,4} |
| Coxeter diagram | |
| Cell type | {4,3} |
| Face type | square {4} |
| Vertex figure | octahedron |
| Space group Fibrifold notation | Pm3m (221) 4$^−$:2 |
| Coxeter group | ${\tilde {C}}_{3}$, [4,3,4] |
| Dual | self-dual Cell: |
| Properties | Vertex-transitive, regular |
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space 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 self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional 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 non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

## 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 are a large number of uniform colorings, derived from different symmetries. These include:
| Coxeter notation Space group | Coxeter diagram | Schläfli symbol | 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.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | pmm (*2222) |
| Solid | | | |
| Frame | | | |

## 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 octahedral vertex figures.
{p,3,4} regular honeycombs :
| Space | S$^3$ | E$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Finite | Affine | Compact | Paracompact | Noncompact | Noncompact | 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$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Finite | Affine | Compact | Paracompact | Noncompact | Noncompact | 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$^3$ | H$^3$ | H$^3$ | H$^3$ | H$^3$ |
| Form | Finite | Affine | Compact | Paracompact | Noncompact | Noncompact | 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.
Dual cell
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], , 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 cubic 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 | $_{(1)}$, $_8$, $_9$ |
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 | $_5$, $_6$, $_7$, $_{(6)}$, $_9$, $_{10}$, $_{11}$ |
This honeycomb is one of five distinct uniform honeycombs constructed by the ${\tilde {A}}_{3}$ 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$^{[4]}$] | | ${\tilde {A}}_{3}$ | (None) |
| Fm3m (225) | 2$^−$:2 | d2 | <[3$^{[4]}$]> ↔ [4,3$^{1,1}$] | ↔ | ${\tilde {A}}_{3}$×2$_1$ ↔ ${\tilde {B}}_{3}$ | $_1$, $_2$ |
| Fd3m (227) | 2$^+$:2 | g2 | [[3$^{[4]}$]] or [2$^+$[3$^{[4]}$]] | ↔ | ${\tilde {A}}_{3}$×2$_2$ | $_3$ |
| Pm3m (221) | 4$^−$:2 | d4 | <2[3$^{[4]}$]> ↔ [4,3,4] | ↔ | ${\tilde {A}}_{3}$×4$_1$ ↔ ${\tilde {C}}_{3}$ | $_4$ |
| I3 (204) | 8$^{−o}$ | r8 | [4[3$^{[4]}$]]$^+$ ↔ [[4,3$^+$,4]] | ↔ | ½${\tilde {A}}_{3}$×8 ↔ ½${\tilde {C}}_{3}$×2 | $_{(*)}$ |
| Im3m (229) | 8$^o$:2 | r8 | [4[3$^{[4]}$]] ↔ [[4,3,4]] | ↔ | ${\tilde {A}}_{3}$×8 ↔ ${\tilde {C}}_{3}$×2 | $_5$ |

### Rectified cubic honeycomb

Rectified cubic honeycomb :
| Type | Uniform honeycomb |
| 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 | = = = = = |
| Cells | r{4,3} {3,4} |
| Faces | triangle {3} square {4} |
| Vertex figure | square prism |
| Space group Fibrifold notation | Pm3m (221) 4$^−$:2 |
| Coxeter group | ${\tilde {C}}_{3}$, [4,3,4] |
| Dual | oblate octahedrille Cell: |
| Properties | Vertex-transitive, edge-transitive |
The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1, with a square prism vertex figure.
John Horton Conway calls this honeycomb a cuboctahedrille, and its dual an oblate octahedrille.

#### Projections

The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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] ${\tilde {C}}_{3}$ | [1$^+$,4,3,4] [4,3$^{1,1}$], ${\tilde {B}}_{3}$ | [4,3,4,1$^+$] [4,3$^{1,1}$], ${\tilde {B}}_{3}$ | [1$^+$,4,3,4,1$^+$] [3$^{[4]}$], ${\tilde {A}}_{3}$ |
| 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].

#### 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.
Dual cell

### 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 | t{4,3} {3,4} |
| Face type | triangle {3} octagon {8} |
| Vertex figure | isosceles square pyramid |
| Space group Fibrifold notation | Pm3m (221) 4$^−$:2 |
| Coxeter group | ${\tilde {C}}_{3}$, [4,3,4] |
| Dual | Pyramidille Cell: |
| Properties | Vertex-transitive |
The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1, with an isosceles square pyramid vertex figure.
John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.

#### Projections

The truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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$^{1,1}$], ${\tilde {B}}_{3}$ | [4,3,4], ${\tilde {C}}_{3}$ =<[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.
Vertex figure
Dual cell

### Bitruncated cubic honeycomb

Bitruncated cubic honeycomb :
| Type | Uniform honeycomb |
| Schläfli symbol | 2t{4,3,4} t$_{1,2}${4,3,4} |
| Coxeter-Dynkin diagram | |
| Cells | t{3,4} |
| Faces | square {4} hexagon {6} |
| Edge figure | isosceles triangle {3} |
| Vertex figure | tetragonal disphenoid |
| Symmetry group Fibrifold notation Coxeter notation | Im3m (229) 8$^o$:2 [[4,3,4]] |
| Coxeter group | ${\tilde {C}}_{3}$, [4,3,4] |
| Dual | Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell: |
| Properties | Vertex-transitive, edge-transitive, cell-transitive |
The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has four truncated octahedra around each vertex, in a tetragonal disphenoid vertex figure. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.
John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list, with its dual called an oblate tetrahedrille, also called a disphenoid tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.

#### 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.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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 ${\tilde {A}}_{3}$ 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.
Five uniform colorings by cell :
| 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 | ${\tilde {C}}_{3}$×2 [[4,3,4]] =[4[3$^{[4]}$]] = | ${\tilde {C}}_{3}$ [4,3,4] =[2[3$^{[4]}$]] = | ${\tilde {B}}_{3}$ [4,3$^{1,1}$] =<[3$^{[4]}$]> = | ${\tilde {A}}_{3}$ [3$^{[4]}$] | ${\tilde {A}}_{3}$×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 | {3,3} s{3,3} |
| Faces | triangle {3} |
| Vertex figure | |
| Coxeter group | [[4,3$^+$,4]], ${\tilde {C}}_{3}$ |
| Dual | Ten-of-diamonds honeycomb Cell: |
| 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 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 α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.
Five uniform colorings :
| 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}x{4} |
| Vertex figure | wedge |
| Space group Fibrifold notation | Pm3m (221) 4$^−$:2 |
| Coxeter group | [4,3,4], ${\tilde {C}}_{3}$ |
| Dual | quarter oblate octahedrille Cell: |
| Properties | Vertex-transitive |
The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3, with a wedge vertex figure.
John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.

#### 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.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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.
Vertex uniform colorings by cell :
| Construction | Truncated cubic honeycomb | Bicantellated alternate cubic |
| Coxeter group | [4,3,4], ${\tilde {C}}_{3}$ =<[4,3$^{1,1}$]> | [4,3$^{1,1}$], ${\tilde {B}}_{3}$ |
| 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 zero-height 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 , 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 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 | = |
| Cells | tr{4,3} t{3,4}x{4} |
| Faces | square {4} hexagon {6} octagon {8} |
| Vertex figure | mirrored sphenoid |
| Coxeter group | [4,3,4], ${\tilde {C}}_{3}$ |
| Symmetry group Fibrifold notation | Pm3m (221) 4$^−$:2 |
| Dual | triangular pyramidille Cells: |
| Properties | Vertex-transitive |
The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3, with a mirrored sphenoid vertex figure.
John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille.

#### Images

Four cells exist around each vertex:

#### Projections

The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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], ${\tilde {C}}_{3}$ =<[4,3$^{1,1}$]> | [4,3$^{1,1}$], ${\tilde {B}}_{3}$ |
| 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, . This honeycomb cells represents the fundamental domains of ${\tilde {B}}_{3}$ 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 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.
Vertex figure
Dual cell

### 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 | s{4,3} s{3,3} {3,3} |
| Faces | triangle {3} square {4} |
| Vertex figure | |
| Coxeter group | [(4,3)$^+$,4] |
| Dual | Cell: |
| Properties | Vertex-transitive, non-uniform |
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 or .
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.

### Cantic snub cubic honeycomb

Orthosnub cubic honeycomb :
| Type | Convex honeycomb |
| Schläfli symbol | 2s$_0${4,3,4} |
| Coxeter diagrams | |
| Cells | s$_2${3,4} s{3,3}x{3} |
| Faces | triangle {3} square {4} |
| Vertex figure | |
| Coxeter group | [4$^+$,3,4] |
| Dual | Cell: |
| Properties | Vertex-transitive, non-uniform |
The cantic snub cubic honeycomb is constructed by snubbing the truncated octahedra in a way that leaves only rectangles from the cubes (square prisms). It is not uniform but it can be represented as Coxeter diagram . It has rhombicuboctahedra (with T$_h$ symmetry), icosahedra (with T$_h$ symmetry), and triangular prisms (as C$_{2v}$-symmetry wedges) filling the gaps.

#### 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.
Vertex figure
Dual cell

### Runcitruncated cubic honeycomb

Runcitruncated cubic honeycomb :
| Type | Uniform honeycomb |
| Schläfli symbol | t$_{0,1,3}${4,3,4} |
| Coxeter diagrams | |
| Cells | rr{4,3} t{4,3}x{8}x{4} |
| Faces | triangle {3} square {4} octagon {8} |
| Vertex figure | isosceles-trapezoidal pyramid |
| Coxeter group | [4,3,4], ${\tilde {C}}_{3}$ |
| Space group Fibrifold notation | Pm3m (221) 4$^−$:2 |
| Dual | square quarter pyramidille Cell |
| Properties | Vertex-transitive |
The runcitruncated cubic honeycomb or runcitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3, with an isosceles-trapezoidal pyramid vertex figure.
Its name is derived from its Coxeter diagram, with three ringed nodes representing 3 active mirrors in the Wythoff construction from its relation to the regular cubic honeycomb.
John Horton Conway calls this honeycomb a 1-RCO-trille, and its dual square quarter pyramidille.

#### Projections

The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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 . Faces exist in 3 of 4 hyperplanes of the [4,3,4], ${\tilde {C}}_{3}$ Coxeter group.
Cells are irregular pyramids and can be seen as 1/24 of a cube, using one corner, one mid-edge point, two face centers, and the cube center.

#### 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.
Vertex figure
Dual cell

### Omnitruncated cubic honeycomb

Omnitruncated cubic honeycomb :
| Type | Uniform honeycomb |
| Schläfli symbol | t$_{0,1,2,3}${4,3,4} |
| Coxeter diagram | |
| Cells | tr{4,3}x{8} |
| Faces | square {4} hexagon {6} octagon {8} |
| Vertex figure | phyllic disphenoid |
| Symmetry group Fibrifold notation Coxeter notation | Im3m (229) 8$^o$:2 [[4,3,4]] |
| Coxeter group | [4,3,4], ${\tilde {C}}_{3}$ |
| Dual | eighth pyramidille Cell |
| Properties | Vertex-transitive |
The omnitruncated cubic honeycomb or omnitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cuboctahedra and octagonal prisms 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.

#### Projections

The omnitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.
Orthogonal projections :
| Symmetry | p6m (*632) | p4m (*442) | pmm (*2222) | pmm (*2222) | 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.
Two uniform colorings :
| Symmetry | ${\tilde {C}}_{3}$, [4,3,4] | ${\tilde {C}}_{3}$×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.
Vertex figure
Dual cell
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).
Vertex figure

### Alternated omnitruncated cubic honeycomb

Alternated omnitruncated cubic honeycomb :
| Type | Convex honeycomb |
| Schläfli symbol | ht$_{0,1,2,3}${4,3,4} |
| Coxeter diagram | |
| Cells | s{4,3} s{2,4} {3,3} |
| Faces | triangle {3} square {4} |
| Vertex figure | |
| Symmetry | [[4,3,4]]$^+$ |
| Dual | Dual alternated omnitruncated cubic honeycomb |
| Properties | Vertex-transitive, non-uniform |
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: and has symmetry [[4,3,4]]$^+$. It makes snub cubes from the truncated cuboctahedra, square antiprisms from the octagonal prisms, and creates new tetrahedral cells from the gaps.

#### 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 | Cell-transitive |
A dual alternated omnitruncated cubic honeycomb is a space-filling 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 3-dimensions:
Individual cells have 2-fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.
Cell views :
* Net

### Runcic cantitruncated cubic honeycomb

Runcic cantitruncated cubic honeycomb :
| Type | Convex honeycomb |
| Schläfli symbol | sr$_3${4,3,4} |
| Coxeter diagrams | |
| Cells | s$_2${3,4} s{4,3}x{4}x{3} |
| Faces | triangle {3} square {4} |
| Vertex figure | |
| Coxeter group | [4,3$^+$,4] |
| Dual | Cell: |
| 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 . It has rhombicuboctahedra (with T$_h$ symmetry), snub cubes, two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D$_{2d}$ symmetry), and triangular prisms (as C$_{2v}$-symmetry wedges) filling the gaps.

### Biorthosnub cubic honeycomb

Biorthosnub cubic honeycomb :
| Type | Convex honeycomb |
| Schläfli symbol | 2s$_{0,3}${4,3,4} |
| Coxeter diagrams | |
| Cells | s$_2${3,4}x{4} |
| Faces | triangle {3} square {4} |
| Vertex figure | (Tetragonal antiwedge) |
| Coxeter group | [[4,3$^+$,4]] |
| Dual | Cell: |
| 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 . It has rhombicuboctahedra (with T$_h$ symmetry) and two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D$_{2d}$ symmetry).

### 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,∞} |
| Coxeter-Dynkin diagram | |
| Cells |x{8}x{4} |
| Faces | square {4} octagon {8} |
| Coxeter group | [4,4,2,∞] |
| Dual | Tetrakis square prismatic tiling Cell: |
| Properties | Vertex-transitive |
The truncated square prismatic honeycomb or tomo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octagonal prisms and cubes in a ratio of 1:1.
It is constructed from a truncated square tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.

### Snub square prismatic honeycomb

Snub square prismatic honeycomb :
| Type | Uniform honeycomb |
| Schläfli symbol | s{4,4}×{∞} sr{4,4}×{∞} |
| Coxeter-Dynkin diagram | |
| Cells |x{4}x{3} |
| Faces | triangle {3} square {4} |
| Coxeter group | [4$^+$,4,2,∞] [(4,4)$^+$,2,∞] |
| Dual | Cairo pentagonal prismatic honeycomb Cell: |
| Properties | Vertex-transitive |
The snub square prismatic honeycomb or simo-square prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.
It is constructed from a snub square tiling extruded into prisms.
It is one of 28 convex uniform honeycombs.

### Snub square antiprismatic honeycomb

Snub square antiprismatic honeycomb :
| Type | Convex honeycomb |
| Schläfli symbol | ht$_{1,2,3}${4,4,2,∞} ht$_{0,1,2,3}${4,4,∞} |
| Coxeter-Dynkin diagram | |
| Cells | s{2,4} {3,3} |
| Faces | triangle {3} square {4} |
| Vertex figure | |
| Symmetry | [4,4,2,∞]$^+$ |
| Properties | Vertex-transitive, non-uniform |
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: and has symmetry [4,4,2,∞]$^+$. It makes square antiprisms from the octagonal prisms, tetrahedra (as tetragonal disphenoids) from the cubes, and two tetrahedra from the triangular bipyramids.