# Cubic honeycomb

Last updated
Cubic honeycomb
Type Regular honeycomb
Family Hypercube honeycomb
Indexing [1] J11,15, A1
W1, G22
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 ${\displaystyle {\tilde {C}}_{3}}$, [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.

## Contents

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.

## 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

### 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
Symmetryp6m (*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 S3 E3 H3
FormFiniteAffineCompactParacompactNoncompact
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 S3 E3 H3
FormFiniteAffineCompactParacompactNoncompact
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 S3 Euclidean E3 H3
FormFiniteAffineCompactParacompactNoncompact
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 D2d 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 C3v 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
OrderHoneycombs
Pm3m
(221)
4:2[4,3,4]×1 1, 2, 3, 4,
5, 6
Fm3m
(225)
2:2[1+,4,3,4]
↔ [4,31,1]

Half 7, 11, 12, 13
I43m
(217)
4o:2[[(4,3,4,2+)]]Half × 2 (7),
Fd3m
(227)
2+:2[[1+,4,3,4,1+]]
↔ [[3[4]]]

Quarter × 2 10,
Im3m
(229)
8o:2[[4,3,4]]×2

(1), 8, 9

The [4,31,1], , 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
OrderHoneycombs
Fm3m
(225)
2:2[4,31,1]
↔ [4,3,4,1+]

×1 1, 2, 3, 4
Fm3m
(225)
2:2<[1+,4,31,1]>
↔ <[3[4]]>

×2 (1), (3)
Pm3m
(221)
4:2<[4,31,1]>×2

5, 6, 7, (6), 9, 10, 11

This honeycomb is one of five distinct uniform honeycombs [2] constructed by the ${\displaystyle {\tilde {A}}_{3}}$ 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)
1o:2a1 [3[4]]${\displaystyle {\tilde {A}}_{3}}$(None)
Fm3m
(225)
2:2d2 <[3[4]]>
↔ [4,31,1]

${\displaystyle {\tilde {A}}_{3}}$×21
${\displaystyle {\tilde {B}}_{3}}$
1 ,  2
Fd3m
(227)
2+:2g2 [[3[4]]]
or [2+[3[4]]]

${\displaystyle {\tilde {A}}_{3}}$×22  3
Pm3m
(221)
4:2d4 <2[3[4]]>
↔ [4,3,4]

${\displaystyle {\tilde {A}}_{3}}$×41
${\displaystyle {\tilde {C}}_{3}}$
4
I3
(204)
8−or8 [4[3[4]]]+
↔ [[4,3+,4]]

½${\displaystyle {\tilde {A}}_{3}}$×8
↔ ½${\displaystyle {\tilde {C}}_{3}}$×2
(*)
Im3m
(229)
8o:2[4[3[4]]]
↔ [[4,3,4]]
${\displaystyle {\tilde {A}}_{3}}$×8
${\displaystyle {\tilde {C}}_{3}}$×2
5

### Rectified cubic honeycomb

Rectified cubic honeycomb
Type Uniform honeycomb
Schläfli symbol r{4,3,4} or t1{4,3,4}
r{4,31,1}
2r{4,31,1}
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 ${\displaystyle {\tilde {C}}_{3}}$, [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
Symmetryp6m (*632)p4m (*442)pmm (*2222)
Solid
Frame

#### Symmetry

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

Symmetry [4,3,4]
${\displaystyle {\tilde {C}}_{3}}$
[1+,4,3,4]
[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
[4,3,4,1+]
[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
[1+,4,3,4,1+]
[3[4]], ${\displaystyle {\tilde {A}}_{3}}$
Space group Pm3m
(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].

.

A double symmetry construction can be made by placing octahedra on the cuboctahedra, resulting in a nonuniform honeycomb with two kinds of octahedra (regular octahedra and triangular antiprisms). The vertex figure is a square bifrustum. The dual is composed of elongated square bipyramids.

### Truncated cubic honeycomb

Truncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t{4,3,4} or t0,1{4,3,4}
t{4,31,1}
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 ${\displaystyle {\tilde {C}}_{3}}$, [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
Symmetryp6m (*632)p4m (*442)pmm (*2222)
Solid
Frame

#### Symmetry

There is a second uniform coloring by reflectional symmetry of the Coxeter groups, the second seen with alternately colored truncated cubic cells.

ConstructionBicantellated alternate cubicTruncated cubic honeycomb
Coxeter group [4,31,1], ${\displaystyle {\tilde {B}}_{3}}$[4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
=<[4,31,1]>
Space group Fm3mPm3m
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

Bitruncated cubic honeycomb

Type Uniform honeycomb
Schläfli symbol 2t{4,3,4}
t1,2{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)
8o:2
[[4,3,4]]
Coxeter group ${\displaystyle {\tilde {C}}_{3}}$, [4,3,4]
DualOblate 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
Symmetryp6m (*632)p4m (*442)pmm (*2222)
Solid
Frame

#### Symmetry

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the ${\displaystyle {\tilde {A}}_{3}}$ 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 8o:24:22:21o:22+:2
Coxeter group ${\displaystyle {\tilde {C}}_{3}}$×2
[[4,3,4]]
=[4[3[4]]]
=
${\displaystyle {\tilde {C}}_{3}}$
[4,3,4]
=[2[3[4]]]
=
${\displaystyle {\tilde {B}}_{3}}$
[4,31,1]
=<[3[4]]>
=
${\displaystyle {\tilde {A}}_{3}}$
[3[4]]

${\displaystyle {\tilde {A}}_{3}}$×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

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 C2v-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 C2v 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{4,31,1}
sr{3[4]}
Coxeter diagrams
=
=
=
Cells {3,3}
s{3,3}
Faces triangle {3}
Vertex figure
Coxeter group [[4,3+,4]], ${\displaystyle {\tilde {C}}_{3}}$
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,(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-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice. [3]

Five uniform colorings
Space group I3 (204)Pm3 (200)Fm3 (202)Fd3 (203)F23 (196)
Fibrifold 8−o422o+1o
Coxeter group [[4,3+,4]][4,3+,4][4,(31,1)+][[3[4]]]+[3[4]]+
Coxeter diagram
Orderdoublefullhalfquarter
double
quarter

### Cantellated cubic honeycomb

Cantellated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol rr{4,3,4} or t0,2{4,3,4}
rr{4,31,1}
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], ${\displaystyle {\tilde {C}}_{3}}$
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
Symmetryp6m (*632)p4m (*442)pmm (*2222)
Solid
Frame

#### Symmetry

There is a second uniform colorings by reflectional symmetry of the Coxeter groups, the second seen with alternately colored rhombicuboctahedral cells.

Vertex uniform colorings by cell
ConstructionTruncated cubic honeycombBicantellated alternate cubic
Coxeter group [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
=<[4,31,1]>
[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
Space group Pm3mFm3m
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.

#### 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 t0,1,2{4,3,4}
tr{4,31,1}
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], ${\displaystyle {\tilde {C}}_{3}}$
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
Symmetryp6m (*632)p4m (*442)pmm (*2222)
Solid
Frame

#### Symmetry

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

ConstructionCantitruncated cubicOmnitruncated alternate cubic
Coxeter group [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
=<[4,31,1]>
[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
Space group Pm3m (221)Fm3m (225)
Fibrifold 4:22:2
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 ${\displaystyle {\tilde {B}}_{3}}$ 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.

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 C2v-symmetric wedges), and tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.

### Alternated cantitruncated cubic honeycomb

Alternated cantitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol sr{4,3,4}
sr{4,31,1}
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 Th 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 2s0{4,3,4}
Coxeter diagrams
Cells s2{3,4}
s{3,3}
{}x{3}
Faces triangle {3}
square {4}
Vertex figure
Coxeter group [4+,3,4]
DualCell:
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 Th symmetry), icosahedra (with Th symmetry), and triangular prisms (as C2v-symmetry wedges) filling the gaps. [4]

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 C2v-symmetric wedges), and square pyramids.

### Runcitruncated cubic honeycomb

Runcitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t0,1,3{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], ${\displaystyle {\tilde {C}}_{3}}$
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
Symmetryp6m (*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.

#### 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], ${\displaystyle {\tilde {C}}_{3}}$ 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.

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 C2v-symmetric wedges), and tetrahedra (as digonal disphenoids). Its vertex figure is topologically equivalent to the augmented triangular prism.

### Omnitruncated cubic honeycomb

Omnitruncated cubic honeycomb
Type Uniform honeycomb
Schläfli symbol t0,1,2,3{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)
8o:2
[[4,3,4]]
Coxeter group [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$
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
Symmetryp6m (*632)p4m (*442)pmm (*2222)
Solid
Frame

#### Symmetry

Cells can be shown in two different symmetries. The Coxeter diagram form has two colors of truncated cuboctahedra and octagonal prisms. The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octagonal prism cells.

Two uniform colorings
Symmetry ${\displaystyle {\tilde {C}}_{3}}$, [4,3,4]${\displaystyle {\tilde {C}}_{3}}$×2, [[4,3,4]]
Space group Pm3m (221)Im3m (229)
Fibrifold 4:28o: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, 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 C2v-symmetric variants). Its vertex figure is an irregular triangular bipyramid.

This honeycomb can then be alternated to produce another nonuniform honeycomb with snub cubes, square antiprisms, octahedra (as triangular antiprisms), and three kinds of tetrahedra (as tetragonal disphenoids, phyllic disphenoids, and irregular tetrahedra).

### Alternated omnitruncated cubic honeycomb

Alternated omnitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol ht0,1,2,3{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
TypeDual alternated uniform honeycomb
Schläfli symbol dht0,1,2,3{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

Runcic cantitruncated cubic honeycomb
Type Convex honeycomb
Schläfli symbol sr3{4,3,4}
Coxeter diagrams
Cells s2{3,4}
s{4,3}
{}x{4}
{}x{3}
Faces triangle {3}
square {4}
Vertex figure
Coxeter group [4,3+,4]
DualCell:
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 Th symmetry), snub cubes, two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry), and triangular prisms (as C2v-symmetry wedges) filling the gaps.

### Biorthosnub cubic honeycomb

Biorthosnub cubic honeycomb
Type Convex honeycomb
Schläfli symbol 2s0,3{4,3,4}
Coxeter diagrams
Cells s2{3,4}
{}x{4}
Faces triangle {3}
square {4}
Vertex figure
(Tetragonal antiwedge)
Coxeter group [[4,3+,4]]
DualCell:
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 Th symmetry) and two kinds of cubes: square prisms and rectangular trapezoprisms (topologically equivalent to a cube but with D2d symmetry).

### Truncated square prismatic honeycomb

Truncated square prismatic honeycomb
Type Uniform honeycomb
Schläfli symbol t{4,4}×{∞} or t0,1,3{4,4,2,∞}
tr{4,4}×{∞} or t0,1,2,3{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 ht1,2,3{4,4,2,∞}
ht0,1,2,3{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.

## Related Research Articles

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.

## 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
Space Family ${\displaystyle {\tilde {A}}_{n-1}}$${\displaystyle {\tilde {C}}_{n-1}}$${\displaystyle {\tilde {B}}_{n-1}}$${\displaystyle {\tilde {D}}_{n-1}}$${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
E10Uniform 10-honeycomb{3[11]}δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21