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 | square {4} |

Vertex figure | octahedron |

Space group Fibrifold notation | Pm3m (221) 4 ^{−}:2 |

Coxeter group | , [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**.

- Related honeycombs
- Isometries of simple cubic lattices
- Uniform colorings
- Projections
- Related polytopes and honeycombs
- Related polytopes
- Related Euclidean tessellations
- Rectified cubic honeycomb
- Truncated cubic honeycomb
- Bitruncated cubic honeycomb
- Alternated bitruncated cubic honeycomb
- Cantellated cubic honeycomb
- Cantitruncated cubic honeycomb
- Alternated cantitruncated cubic honeycomb
- Cantic snub cubic honeycomb
- Runcitruncated cubic honeycomb
- Omnitruncated cubic honeycomb
- Alternated omnitruncated cubic honeycomb
- Runcic cantitruncated cubic honeycomb
- Biorthosnub cubic honeycomb
- Truncated square prismatic honeycomb
- Snub square prismatic honeycomb
- Snub square antiprismatic honeycomb
- See also
- References

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.

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.

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 | - | - |

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 |

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 |

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} | ||||||||

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,∞} |

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.

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 |

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 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 | , [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.

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 |

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].

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 | |
---|---|

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} square {4} octagon {8} |

Vertex figure | isosceles square pyramid |

Space group Fibrifold notation | Pm3m (221) 4 ^{−}:2 |

Coxeter group | , [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.

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 |

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 |

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 | |
---|---|

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 | , [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.

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 |

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 |

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 | |
---|---|

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]], |

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.^{ [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 | |
---|---|

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], |

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.

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. |

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 |

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 |

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.

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 | |
---|---|

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], |

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.

Four cells exist around each vertex:

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 |

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 |

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.

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.

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 | |
---|---|

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.

| |

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

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 | |
---|---|

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], |

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.

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 |

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.

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 mid-edge point, two face centers, and the cube center.

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 | |
---|---|

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], |

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.

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 |

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 |

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 |
---|---|

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 | |
---|---|

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 | |
---|---|

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.

Net | |||

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

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

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 | |
---|---|

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 | |
---|---|

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.

Wikimedia Commons has media related to Cubic honeycomb .

- Architectonic and catoptric tessellation
- Alternated cubic honeycomb
- List of regular polytopes
- Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge
- voxel

In geometry, a **convex uniform honeycomb** is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In four-dimensional geometry, a **runcinated tesseract** is a convex uniform 4-polytope, being a runcination of the regular tesseract.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

The **bitruncated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. 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.

The **quarter cubic honeycomb**, **quarter cubic cellulation** or **bitruncated alternated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.

The **tetragonal disphenoid tetrahedral honeycomb** is a space-filling tessellation in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces. John Horton Conway calls it an *oblate tetrahedrille* or shortened to *obtetrahedrille*.

In hyperbolic geometry, the **order-5 cubic honeycomb** is one of four compact regular space-filling tessellations in hyperbolic 3-space. With Schläfli symbol {4,3,5}, it has five cubes {4,3} around each edge, and 20 cubes around each vertex. It is dual with the order-4 dodecahedral honeycomb.

In four-dimensional geometry, a **cantellated 5-cell** is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.

In four-dimensional geometry, a **cantellated 24-cell** is a convex uniform 4-polytope, being a cantellation of the regular 24-cell.

In four-dimensional geometry, a **runcinated 24-cell** is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

The **triangular prismatic honeycomb** or **triangular prismatic cellulation** is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.

In hyperbolic geometry, a **uniform honeycomb in hyperbolic space** is a uniform tessellation of uniform polyhedral cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by permutations of rings of the Coxeter diagrams for each family.

In four-dimensional Euclidean geometry, the **4-simplex honeycomb**, **5-cell honeycomb** or **pentachoric-dispentachoric honeycomb** is a space-filling tessellation honeycomb. It is composed of 5-cells and rectified 5-cells facets in a ratio of 1:1.

In the field of hyperbolic geometry, the **order-4 hexagonal tiling honeycomb** arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The **order-6 cubic honeycomb** is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is *paracompact* because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

In the geometry of hyperbolic 3-space, the **square tiling honeycomb** is one of 11 paracompact regular honeycombs. It is called *paracompact* because it has infinite cells, whose vertices exist on horospheres and converge to a single ideal point at infinity. Given by Schläfli symbol {4,4,3}, it has three square tilings, {4,4}, around each edge, and six square tilings around each vertex, in a cubic {4,3} vertex figure.

In the geometry of hyperbolic 3-space, the **cubic-octahedral honeycomb** is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.

- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008)
*The Symmetries of Things*, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms) - Coxeter, H.S.M.
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 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 3-space. Geombinatorics 4(1994), 49 - 56.
*Kaleidoscopes: Selected Writings of H.S.M. Coxeter*, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- (Paper 22) H.S.M. Coxeter,
*Regular and Semi Regular Polytopes I*, [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)

- (Paper 22) H.S.M. Coxeter,
- 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 3-Space: 01-Chon

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 4-honeycomb | {3^{[5]}} | δ_{5} | hδ_{5} | qδ_{5} | 24-cell honeycomb |

E^{5} | Uniform 5-honeycomb | {3^{[6]}} | δ_{6} | hδ_{6} | qδ_{6} | |

E^{6} | Uniform 6-honeycomb | {3^{[7]}} | δ_{7} | hδ_{7} | qδ_{7} | 2_{22} |

E^{7} | Uniform 7-honeycomb | {3^{[8]}} | δ_{8} | hδ_{8} | qδ_{8} | 1_{33} • 3_{31} |

E^{8} | Uniform 8-honeycomb | {3^{[9]}} | δ_{9} | hδ_{9} | qδ_{9} | 1_{52} • 2_{51} • 5_{21} |

E^{9} | Uniform 9-honeycomb | {3^{[10]}} | δ_{10} | hδ_{10} | qδ_{10} | |

E^{10} | Uniform 10-honeycomb | {3^{[11]}} | δ_{11} | hδ_{11} | qδ_{11} | |

E^{n-1} | Uniform (n-1)-honeycomb | {3^{[n]}} | δ_{n} | hδ_{n} | qδ_{n} | 1_{k2} • 2_{k1} • k_{21} |

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Images, videos and audio are available under their respective licenses.