Projections

The rectified cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

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.

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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

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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].

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Related polytopes

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

Dual cell

Projections

The truncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Symmetry

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

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Construction	Bicantellated alternate cubic	Truncated cubic honeycomb
Coxeter group	[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$	[4,3,4], ${\displaystyle {\tilde {C}}_{3}}$ =<[4,31,1]>
Space group	Fm3m	Pm3m
Coloring
Coxeter diagram	=
Vertex figure

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Related polytopes

A double symmetry construction can be made by placing octahedra on the truncated cubes, resulting in a nonuniform honeycomb with two kinds of octahedra (regular octahedra and triangular antiprisms) and two kinds of tetrahedra (tetragonal disphenoids and digonal disphenoids). The vertex figure is an octakis square cupola.

Vertex figure

Dual cell

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.

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.

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Five uniform colorings by cell

Space group	Im3m (229)	Pm3m (221)	Fm3m (225)	F43m (216)	Fd3m (227)
Fibrifold	8o:2	4−:2	2−:2	1o:2	2+: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

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Related polytopes

Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with truncated octahedra and hexagonal prisms (as ditrigonal trapezoprisms). Its vertex figure is a C_2v-symmetric triangular bipyramid.

This honeycomb can then be alternated to produce another nonuniform honeycomb with pyritohedral icosahedra, octahedra (as triangular antiprisms), and tetrahedra (as sphenoids). Its vertex figure has C_2v symmetry and consists of 2 pentagons, 4 rectangles, 4 isosceles triangles (divided into two sets of 2), and 4 scalene triangles.

Images

Projections

The cantellated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Symmetry

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

More information [4,3,4], ...

Vertex uniform colorings by cell

Construction	Truncated cubic honeycomb	Bicantellated alternate cubic
Coxeter group	[4,3,4], ${\displaystyle {\tilde {C}}_{3}}$ =<[4,31,1]>	[4,31,1], ${\displaystyle {\tilde {B}}_{3}}$
Space group	Pm3m	Fm3m
Coxeter diagram
Coloring
Vertex figure
Vertex figure symmetry	[ ] order 2	[ ]+ order 1

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Related polytopes

A double symmetry construction can be made by placing cuboctahedra on the rhombicuboctahedra, which results in the rectified cubic honeycomb, by taking the triangular antiprism gaps as regular octahedra, square antiprism pairs and zero-height tetragonal disphenoids as components of the cuboctahedron. Other variants result in cuboctahedra, square antiprisms, octahedra (as triangular antipodiums), and tetrahedra (as tetragonal disphenoids), with a vertex figure topologically equivalent to a cube with a triangular prism attached to one of its square faces.

Quarter oblate octahedrille

The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram , containing faces from two of four hyperplanes of the cubic [4,3,4] fundamental domain.

It has irregular triangle bipyramid cells which can be seen as 1/12 of a cube, made from the cube center, 2 face centers, and 2 vertices.

Images

Four cells exist around each vertex:

Projections

The cantitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

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.

More information [4,3,4], ...

Construction	Cantitruncated cubic	Omnitruncated 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−:2	2−:2
Coloring
Coxeter diagram
Vertex figure
Vertex figure symmetry	[ ] order 2	[ ]+ order 1

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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.

Related polyhedra and honeycombs

It is related to a skew apeirohedron with vertex configuration 4.4.6.6, with the octagons and some of the squares removed. It can be seen as constructed by augmenting truncated cuboctahedral cells, or by augmenting alternated truncated octahedra and cubes.

Related polytopes

A double symmetry construction can be made by placing truncated octahedra on the truncated cuboctahedra, resulting in a nonuniform honeycomb with truncated octahedra, hexagonal prisms (as ditrigonal trapezoprisms), cubes (as square prisms), triangular prisms (as C_2v-symmetric wedges), and tetrahedra (as tetragonal disphenoids). Its vertex figure is topologically equivalent to the octahedron.

Vertex figure

Dual cell

Related polytopes

A double symmetry construction can be made by placing icosahedra on the rhombicuboctahedra, resulting in a nonuniform honeycomb with icosahedra, octahedra (as triangular antiprisms), triangular prisms (as C_2v-symmetric wedges), and square pyramids.

Vertex figure

Dual cell

Projections

The runcitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements.

Related skew apeirohedron

Two related uniform skew apeirohedrons exists with the same vertex arrangement, seen as boundary cells from a subset of cells. One has triangles and squares, and the other triangles, squares, and octagons.

Square quarter pyramidille

The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram . Faces exist in 3 of 4 hyperplanes of the [4,3,4], ${\displaystyle {\tilde {C}}_{3}}$ 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.

Related polytopes

A double symmetry construction can be made by placing rhombicuboctahedra on the truncated cubes, resulting in a nonuniform honeycomb with rhombicuboctahedra, octahedra (as triangular antiprisms), cubes (as square prisms), two kinds of triangular prisms (both C_2v-symmetric wedges), and tetrahedra (as digonal disphenoids). Its vertex figure is topologically equivalent to the augmented triangular prism.