# Cubic honeycomb

Cubic honeycomb
TypeRegular honeycomb
FamilyHypercube honeycomb
Indexing[1] J11,15, A1
W1, G22
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]
Dualself-dual
Propertiesvertex-transitive, quasiregular

The cubic honeycomb or cubic cellulation is the only 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 calls 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.

## Cartesian coordinates

The Cartesian coordinates of the vertices are:

(i, j, k)
for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1.

It is part of a multidimensional family of hypercube honeycombs, with Schläfli symbols of the form {4,3,...,3,4}, starting with the square tiling, {4,4} in the plane.

It is one of 28 uniform honeycombs using convex uniform polyhedral cells.

## Isometries of simple cubic lattices

Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:

Crystal system Monoclinic
Triclinic
Orthorhombic Tetragonal Rhombohedral Cubic
Unit cell Parallelepiped Rectangular cuboid Square cuboid Trigonal
trapezohedron
Cube
Point group
Order
Rotation subgroup
[ ], (*)
Order 2
[ ]+, (1)
[2,2], (*222)
Order 8
[2,2]+, (222)
[4,2], (*422)
Order 16
[4,2]+, (422)
[3], (*33)
Order 6
[3]+, (33)
[4,3], (*432)
Order 48
[4,3]+, (432)
Diagram
Space group
Rotation subgroup
Pm (6)
P1 (1)
Pmmm (47)
P222 (16)
P4/mmm (123)
P422 (89)
R3m (160)
R3 (146)
Pm3m (221)
P432 (207)
Coxeter notation - []a×[]b×[]c [4,4]a×[]c - [4,3,4]a
Coxeter diagram - -

## Uniform colorings

There is a large number of uniform colorings, derived from different symmetries. These include:

Coxeter notation
Space group
Coxeter diagram Schläfli symbol Partial
honeycomb
Colors by letters
[4,3,4]
Pm3m (221)

=
{4,3,4} 1: aaaa/aaaa
[4,31,1] = [4,3,4,1+]
Fm3m (225)
= {4,31,1} 2: abba/baab
[4,3,4]
Pm3m (221)
t0,3{4,3,4} 4: abbc/bccd
[[4,3,4]]
Pm3m (229)
t0,3{4,3,4} 4: abbb/bbba
[4,3,4,2,]
or
{4,4}×t{} 2: aaaa/bbbb
[4,3,4,2,] t1{4,4}×{} 2: abba/abba
[,2,,2,] t{}×t{}×{} 4: abcd/abcd
[,2,,2,] = [4,(3,4)*] = t{}×t{}×t{} 8: abcd/efgh

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 honeycomb with octahedral vertex figures.

It in a sequence of regular polytopes and honeycombs with cubic cells.

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.

The [4,31,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

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:

### Rectified cubic honeycomb

Rectified cubic honeycomb
TypeUniform honeycomb
CellsOctahedron
Cuboctahedron
Schläfli symbolr{4,3,4} or t1{4,3,4}
r{3[4]}
Coxeter diagrams
=
= =
Vertex figure
Cuboid
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group, [4,3,4]
Dualoblate octahedrille
(Square bipyramidal honeycomb)
Propertiesvertex-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.

John Horton Conway calls this honeycomb a cuboctahedrille, and its dual oblate octahedrille.

#### Symmetry

There are four uniform colorings for the cells of this honeycomb with reflective symmetry, listed by their Coxeter group, and Wythoff construction name, and the Coxeter diagram below.

Symmetry [4,3,4]
[1+,4,3,4]
[4,31,1],
[4,3,4,1+]
[4,31,1],
[1+,4,3,4,1+]
[3[4]],
Space groupPm3m
(221)
Fm3m
(225)
Fm3m
(225)
F43m
(216)
Coloring
Coxeter
diagram
Vertex figure
Vertex
figure
symmetry
D4h
[4,2]
(*224)
order 16
D2h
[2,2]
(*222)
order 8
C4v
[4]
(*44)
order 8
C2v
[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 s3{2,6,3}, with coxeter notation symmetry [2+,6,3].

.

### Truncated cubic honeycomb

Truncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolt{4,3,4} or t0,1{4,3,4}
Coxeter diagrams
=
Cell type3.8.8, {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]
DualPyramidille
(Hexakis cubic honeycomb)
Propertiesvertex-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.

John Horton Conway calls this honeycomb a truncated cubille, and its dual pyramidille.

#### 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 [4,31,1], [4,3,4],
=<[4,31,1]>
Space groupFm3mPm3m
Coloring
Coxeter diagram =
Vertex figure

### Bitruncated cubic honeycomb

Bitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbol2t{4,3,4} or t1,2{4,3,4}
Coxeter diagram or

=
=

Cell type(4.6.6)
Face typessquare {4}
hexagon {6}
Edge figureisosceles triangle {3}
Vertex figure
(disphenoid tetrahedron)
Space group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group, [4,3,4]
DualOblate tetrahedrille
Disphenoid tetrahedral honeycomb
Propertiesisogonal, isotoxal, isochoric

The bitruncated cubic honeycomb or bitruncated cubic cellulation is a space-filling tessellation (or honeycomb) 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.

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.

It can be realized as the Voronoi tessellation of the body-centred cubic lattice. Lord Kelvin conjectured that a variant of the bitruncated cubic honeycomb (with curved faces and edges, but the same combinatorial structure) is the optimal soap bubble foam. However, the Weaire–Phelan structure is a less symmetrical, but more efficient, foam of soap bubbles.

#### Symmetry

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.

Five uniform colorings by cell
Space groupIm3m (229)Pm3m (221)Fm3m (225)F43m (216)Fd3m (227)
Fibrifold8o:24:22:21o:22+:2
Coxeter group ×2
[[4,3,4]]
=[4[3[4]]]
=

[4,3,4]
=[2[3[4]]]
=

[4,31,1]
=<[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

#### Projection by folding

The bitruncated cubic honeycomb can be orthogonally projected into the planar truncated square tiling by a geometric folding operation that maps two pairs of mirrors into each other. The projection of the bitruncated cubic honeycomb creating two offset copies of the truncated square tiling vertex arrangement of the plane:

Coxeter
group
Coxeter
diagram
Graph
Bitruncated cubic honeycomb

Truncated square tiling

### Alternated bitruncated cubic honeycomb

Alternated bitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbol2s{4,3,4}
Coxeter diagrams
=
=
Cellstetrahedron
icosahedron
Vertex figure
Coxeter group[4,3,4],
Propertiesvertex-transitive

The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb can be creating regular icosahedron from the truncated octahedra with irregular tetrahedral cells created in the gaps. There are three constructions from three related Coxeter diagrams: , , and . These have symmetry [4,3+,4], [4,(31,1)+] 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-rhombihedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.[3]

Five uniform colorings
Space groupI3 (204) Pm3 (200) Fm3 (202)Fd3 (203) F23 (196)
Fibrifold8−o422o+ 1o
Coxeter group[[4,3+,4]] [4,3+,4] [4,(31,1)+] [[3[4]]]+[3[4]]+
Coxeter diagram
Order double full half quarter
double
quarter

### Cantellated cubic honeycomb

Cantellated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolrr{4,3,4} or t0,2{4,3,4}
Coxeter diagram
=
Cellsrr{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
Propertiesvertex-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.

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.

#### 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],
=<[4,31,1]>
[4,31,1],
Space groupPm3mFm3m
Coxeter diagram
Coloring
Vertex figure
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1

### Cantitruncated cubic honeycomb

Cantitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symboltr{4,3,4} or t0,1,2{4,3,4}
Coxeter diagram
=
Vertex figure
(Irreg. tetrahedron)
Coxeter group[4,3,4],
Space group
Fibrifold notation
Pm3m (221)
4:2
Dualtriangular pyramidille
Propertiesvertex-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.

John Horton Conway calls this honeycomb a n-tCO-trille, and its dual triangular pyramidille.

#### Images

Four cells exist around each vertex:

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.

#### Symmetry

Cells can be shown in two different symmetries. The linear Coxeter diagram form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of truncated cuboctahedron cells alternating.

Construction Cantitruncated cubic Omnitruncated alternate cubic
Coxeter group [4,3,4],
=<[4,31,1]>
[4,31,1],
Space groupPm3m (221)Fm3m (225)
Fibrifold4:22:2
Coloring
Coxeter diagram
Vertex figure
Vertex
figure
symmetry
[ ]
order 2
[ ]+
order 1

### Alternated cantitruncated cubic honeycomb

Alternated cantitruncated cubic honeycomb
TypeConvex honeycomb
Schläfli symbolsr{4,3,4}
sr{4,31,1}
Coxeter diagrams
=
Cellstetrahedron
pseudoicosahedron
snub cube
Vertex figure
Coxeter group[4,31,1],
Dual square quarter pyramidille
Propertiesvertex-transitive

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 .

### Runcic cantitruncated cubic honeycomb

Runcic cantitruncated cubic honeycomb
TypeConvex honeycomb
Schläfli symbolsr3{4,3,4}
Coxeter diagrams
Cellsrhombicuboctahedron
snub cube
cube
Vertex figure
Coxeter group[4,3,4],
Dual
Propertiesvertex-transitive

The runcic cantitruncated cubic honeycomb or runcic cubic cellulation contains cells: snub cubes, rhombicuboctahedrons, and cubes. In addition the gaps created at the alternated vertices form an irregular cell.
Although it is not uniform, constructionally it can be given as Coxeter diagram .

### Runcitruncated cubic honeycomb

Runcitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolt0,1,3{4,3,4}
Coxeter diagrams
Cellsrhombicuboctahedron
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
Propertiesvertex-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.

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.

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.

### Omnitruncated cubic honeycomb

Omnitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolt0,1,2,3{4,3,4}
Coxeter diagram
Vertex figure
Phyllic disphenoid
Space group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group[4,3,4],
Dualeighth pyramidille
Propertiesvertex-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.

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

#### Symmetry

Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octahedral 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 octahedral prism cells.

Two uniform colorings
Symmetry , [4,3,4] ×2, 4,3,4
Space groupPm3m (221)Im3m (229)
Fibrifold4:28o: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.

4.4.4.6
4.8.4.8

### Alternated omnitruncated cubic honeycomb

Alternated omnitruncated cubic honeycomb
TypeUniform honeycomb
Schläfli symbolht0,1,2,3{4,3,4}
Coxeter diagram
Cells snub cube
square antiprism
tetrahedron
Vertex figure
Symmetry[[4,3,4]]+
Propertiesvertex-transitive

A 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.

### Truncated square prismatic honeycomb

Truncated square prismatic honeycomb
TypeUniform honeycomb
Schläfli symbolt{4,4}×{∞} or t0,1,3{4,4,2,∞}
tr{4,4}×{∞} or t0,1,2,3{4,4,∞}
Coxeter-Dynkin diagram
Coxeter group[4,4,2,∞]
DualTetrakis square prismatic tiling
Propertiesvertex-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
TypeUniform honeycomb
Schläfli symbols{4,4}×{∞}
sr{4,4}×{∞}
Coxeter-Dynkin diagram
Coxeter group[4+,4,2,∞]
[(4,4)+,2,∞]
DualCairo pentagonal prismatic honeycomb
Propertiesvertex-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.

 Wikimedia Commons has media related to Cubic honeycomb.

## References

1. For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).
2. , A000029 6-1 cases, skipping one with zero marks
3. Williams, 1979, p 199, Figure 5-38.
• 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)
• 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
Fundamental convex regular and uniform honeycombs in dimensions 3–10 (or 2-9)
Family / /
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
Uniform 10-honeycomb {3[10]} δ10 hδ10 qδ10
Uniform n-honeycomb {3[n]} δn hδn qδn 1k22k1k21