Cubic honeycomb

Cubic honeycomb

Type	Regular honeycomb
Family	Hypercube honeycomb
Indexing^[1]	J_11,15, 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	${\tilde {C}}_{3}$ , [4,3,4]
Dual	self-dual Cell:
Properties	Vertex-transitive, regular

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

Related honeycombs[edit]

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[edit]

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[edit]

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

Coxeter notation Space group	Coxeter diagram	Schläfli symbol	Colors by letters
[4,3,4] Pm3m (221)	=	{4,3,4}	1: aaaa/aaaa
[4,3^1,1] = [4,3,4,1⁺] Fm3m (225)	=	{4,3^1,1}	2: abba/baab
[4,3,4] Pm3m (221)		t_0,3{4,3,4}	4: abbc/bccd
[[4,3,4]] Pm3m (229)		t_0,3{4,3,4}	4: abbb/bbba
[4,3,4,2,∞]	or	{4,4}×t{∞}	2: aaaa/bbbb
[4,3,4,2,∞]		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[edit]

The cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a triangular tiling. A square symmetry projection forms a square tiling.

Orthogonal projections
Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

Related polytopes and honeycombs[edit]

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
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	S³	E³	H³
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	S³	Euclidean E³	H³
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,∞}

Related polytopes[edit]

The cubic honeycomb has a lower symmetry as a runcinated cubic honeycomb, with two sizes of cubes. A double symmetry construction can be constructed by placing a small cube into each large cube, resulting in a nonuniform honeycomb with cubes, square prisms, and rectangular trapezoprisms (a cube with D_2d symmetry). Its vertex figure is a triangular pyramid with its lateral faces augmented by tetrahedra.

Dual cell

The resulting honeycomb can be alternated to produce another nonuniform honeycomb with regular tetrahedra, two kinds of tetragonal disphenoids, triangular pyramids, and sphenoids. Its vertex figure has C_3v symmetry and has 26 triangular faces, 39 edges, and 15 vertices.

Related Euclidean tessellations[edit]

The [4,3,4], , Coxeter group generates 15 permutations of uniform tessellations, 9 with distinct geometry including the alternated cubic honeycomb. The expanded cubic honeycomb (also known as the runcinated cubic honeycomb) is geometrically identical to the cubic honeycomb.

C3 honeycombs
Space group	Fibrifold	Extended symmetry	Extended diagram	Order	Honeycombs
Pm3m (221)	4⁻:2	[4,3,4]		×1	₁, ₂, ₃, ₄, ₅, ₆
Fm3m (225)	2⁻:2	[1⁺,4,3,4] ↔ [4,3^1,1]	↔	Half	₇, ₁₁, ₁₂, ₁₃
I43m (217)	4^o:2	[[(4,3,4,2⁺)]]		Half × 2	₍₇₎,
Fd3m (227)	2⁺:2	[[1⁺,4,3,4,1⁺]] ↔ [[3^[4]]]	↔	Quarter × 2	₁₀,
Im3m (229)	8^o:2	[[4,3,4]]		×2	₍₁₎, ₈, ₉

The [4,3^1,1], , Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb.

B3 honeycombs
Space group	Fibrifold	Extended symmetry	Extended diagram	Order	Honeycombs
Fm3m (225)	2⁻:2	[4,3^1,1] ↔ [4,3,4,1⁺]	↔	×1	₁, ₂, ₃, ₄
Fm3m (225)	2⁻:2	<[1⁺,4,3^1,1]> ↔ <[3^[4]]>	↔	×2	₍₁₎, ₍₃₎
Pm3m (221)	4⁻:2	<[4,3^1,1]>		×2	₅, ₆, ₇, ₍₆₎, ₉, ₁₀, ₁₁

This honeycomb is one of five distinct uniform honeycombs^[2] constructed by the ${\tilde {A}}_{3}$ Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:

A3 honeycombs
Space group	Fibrifold	Square symmetry	Extended symmetry	Extended diagram	Extended group	Honeycomb diagrams
F43m (216)	1^o:2	a1	[3^[4]]		${\tilde {A}}_{3}$	(None)
Fm3m (225)	2⁻:2	d2	<[3^[4]]> ↔ [4,3^1,1]	↔	${\tilde {A}}_{3}$ ×2₁ ↔ ${\tilde {B}}_{3}$	₁, ₂
Fd3m (227)	2⁺:2	g2	[[3^[4]]] or [2⁺[3^[4]]]	↔	${\tilde {A}}_{3}$ ×2₂	₃
Pm3m (221)	4⁻:2	d4	<2[3^[4]]> ↔ [4,3,4]	↔	${\tilde {A}}_{3}$ ×4₁ ↔ ${\tilde {C}}_{3}$	₄
I3 (204)	8^−o	r8	[4[3^[4]]]⁺ ↔ [[4,3⁺,4]]	↔	½ ${\tilde {A}}_{3}$ ×8 ↔ ½ ${\tilde {C}}_{3}$ ×2	_(*)
Im3m (229)	8^o:2	r8	[4[3^[4]]] ↔ [[4,3,4]]	↔	${\tilde {A}}_{3}$ ×8 ↔ ${\tilde {C}}_{3}$ ×2	₅

Rectified cubic honeycomb[edit]

Rectified cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	r{4,3,4} or t₁{4,3,4} r{4,3^1,1} 2r{4,3^1,1} r{3^[4]}
Coxeter diagrams	= = = = =
Cells	r{4,3} {3,4}
Faces	triangle {3} square {4}
Vertex figure	square prism
Space group Fibrifold notation	Pm3m (221) 4⁻:2
Coxeter group	${\tilde {C}}_{3}$ , [4,3,4]
Dual	oblate octahedrille Cell:
Properties	Vertex-transitive, edge-transitive

The rectified cubic honeycomb or rectified cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1, with a square prism vertex figure.

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

Projections[edit]

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

Orthogonal projections
Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

Symmetry[edit]

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

Symmetry	[4,3,4] ${\tilde {C}}_{3}$	[1⁺,4,3,4] [4,3^1,1], ${\tilde {B}}_{3}$	[4,3,4,1⁺] [4,3^1,1], ${\tilde {B}}_{3}$	[1⁺,4,3,4,1⁺] [3^[4]], ${\tilde {A}}_{3}$
Space group	Pm3m (221)	Fm3m (225)	Fm3m (225)	F43m (216)
Coloring
Coxeter diagram
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 s₃{2,6,3}, with coxeter notation symmetry [2⁺,6,3].

.

Related polytopes[edit]

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

Dual cell

Truncated cubic honeycomb[edit]

Truncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t{4,3,4} or t_0,1{4,3,4} t{4,3^1,1}
Coxeter diagrams	=
Cell type	t{4,3} {3,4}
Face type	triangle {3} octagon {8}
Vertex figure	isosceles square pyramid
Space group Fibrifold notation	Pm3m (221) 4⁻:2
Coxeter group	${\tilde {C}}_{3}$ , [4,3,4]
Dual	Pyramidille Cell:
Properties	Vertex-transitive

The truncated cubic honeycomb or truncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1, with an isosceles square pyramid vertex figure.

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

Projections[edit]

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

Orthogonal projections
Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

Symmetry[edit]

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

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

Related polytopes[edit]

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

Vertex figure

Dual cell

Bitruncated cubic honeycomb[edit]

Bitruncated cubic honeycomb

Type	Uniform honeycomb
Schläfli symbol	2t{4,3,4} t_1,2{4,3,4}
Coxeter-Dynkin diagram
Cells	t{3,4}
Faces	square {4} hexagon {6}
Edge figure	isosceles triangle {3}
Vertex figure	tetragonal disphenoid
Symmetry group Fibrifold notation Coxeter notation	Im3m (229) 8^o:2 [[4,3,4]]
Coxeter group	${\tilde {C}}_{3}$ , [4,3,4]
Dual	Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell:
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra (or, equivalently, bitruncated cubes). It has four truncated octahedra around each vertex, in a tetragonal disphenoid vertex figure. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

John Horton Conway calls this honeycomb a truncated octahedrille in his Architectonic and catoptric tessellation list, with its dual called an oblate tetrahedrille, also called a disphenoid tetrahedral honeycomb. Although a regular tetrahedron can not tessellate space alone, this dual has identical disphenoid tetrahedron cells with isosceles triangle faces.

Projections[edit]

The bitruncated cubic honeycomb can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform rhombitrihexagonal tiling. A square symmetry projection forms two overlapping truncated square tiling, which combine together as a chamfered square tiling.

Orthogonal projections
Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

Symmetry[edit]

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the ${\tilde {A}}_{3}$ Coxeter group. This honeycomb has four uniform constructions, with the truncated octahedral cells having different Coxeter groups and Wythoff constructions. These uniform symmetries can be represented by coloring differently the cells in each construction.

Five uniform colorings by cell
Space group	Im3m (229)	Pm3m (221)	Fm3m (225)	F43m (216)	Fd3m (227)
Fibrifold	8^o:2	4⁻:2	2⁻:2	1^o:2	2⁺:2
Coxeter group	${\tilde {C}}_{3}$ ×2 [[4,3,4]] =[4[3^[4]]] =	${\tilde {C}}_{3}$ [4,3,4] =[2[3^[4]]] =	${\tilde {B}}_{3}$ [4,3^1,1] =<[3^[4]]> =	${\tilde {A}}_{3}$ [3^[4]]	${\tilde {A}}_{3}$ ×2 [[3^[4]]] =[[3^[4]]]
Coxeter diagram
truncated octahedra	1	1:1 :	2:1:1 ::	1:1:1:1 :::	1:1 :
Vertex figure
Vertex figure symmetry	[2⁺,4] (order 8)	[2] (order 4)	[ ] (order 2)	[ ]⁺ (order 1)	[2]⁺ (order 2)
Image Colored by cell

Related polytopes[edit]

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[edit]

Alternated bitruncated cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	2s{4,3,4} 2s{4,3^1,1} sr{3^[4]}
Coxeter diagrams	= = =
Cells	{3,3} s{3,3}
Faces	triangle {3}
Vertex figure
Coxeter group	[[4,3⁺,4]], ${\tilde {C}}_{3}$
Dual	Ten-of-diamonds honeycomb Cell:
Properties	Vertex-transitive, non-uniform

The alternated bitruncated cubic honeycomb or bisnub cubic honeycomb is non-uniform, with the highest symmetry construction reflecting an alternation of the uniform bitruncated cubic honeycomb. A lower-symmetry construction involves regular icosahedra paired with golden icosahedra (with 8 equilateral triangles paired with 12 golden triangles). There are three constructions from three related Coxeter diagrams: , , and . These have symmetry [4,3⁺,4], [4,(3^1,1)⁺] and [3^[4]]⁺ respectively. The first and last symmetry can be doubled as [[4,3⁺,4]] and [[3^[4]]]⁺.

This honeycomb is represented in the boron atoms of the α-rhombohedral crystal. The centers of the icosahedra are located at the fcc positions of the lattice.^[3]

Five uniform colorings
Space group	I3 (204)	Pm3 (200)	Fm3 (202)	Fd3 (203)	F23 (196)
Fibrifold	8^−o	4⁻	2⁻	2^o+	1^o
Coxeter group	[[4,3⁺,4]]	[4,3⁺,4]	[4,(3^1,1)⁺]	[[3^[4]]]⁺	[3^[4]]⁺
Coxeter diagram
Order	double	full	half	quarter double	quarter

Cantellated cubic honeycomb[edit]

Cantellated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	rr{4,3,4} or t_0,2{4,3,4} rr{4,3^1,1}
Coxeter diagram	=
Cells	rr{4,3} r{4,3} {}x{4}
Vertex figure	wedge
Space group Fibrifold notation	Pm3m (221) 4⁻:2
Coxeter group	[4,3,4], ${\tilde {C}}_{3}$
Dual	quarter oblate octahedrille Cell:
Properties	Vertex-transitive

The cantellated cubic honeycomb or cantellated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3, with a wedge vertex figure.

John Horton Conway calls this honeycomb a 2-RCO-trille, and its dual quarter oblate octahedrille.

Images[edit]

	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[edit]

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

Orthogonal projections
Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

Symmetry[edit]

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

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

Related polytopes[edit]

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[edit]

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[edit]

Cantitruncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	tr{4,3,4} or t_0,1,2{4,3,4} tr{4,3^1,1}
Coxeter diagram	=
Cells	tr{4,3} t{3,4} {}x{4}
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter group	[4,3,4], ${\tilde {C}}_{3}$
Symmetry group Fibrifold notation	Pm3m (221) 4⁻:2
Dual	triangular pyramidille Cells:
Properties	Vertex-transitive

The cantitruncated cubic honeycomb or cantitruncated cubic cellulation is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3, with a mirrored sphenoid vertex figure.

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

Images[edit]

Four cells exist around each vertex:

Projections[edit]

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

Orthogonal projections
Symmetry	p6m (*632)	p4m (*442)	pmm (*2222)
Solid
Frame

Symmetry[edit]

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

Construction	Cantitruncated cubic	Omnitruncated alternate cubic
Coxeter group	[4,3,4], ${\tilde {C}}_{3}$ =<[4,3^1,1]>	[4,3^1,1], ${\tilde {B}}_{3}$
Space group	Pm3m (221)	Fm3m (225)
Fibrifold	4⁻:2	2⁻:2
Coloring
Coxeter diagram
Vertex figure
Vertex figure symmetry	[ ] order 2	[ ]⁺ order 1

Triangular pyramidille[edit]

The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, . This honeycomb cells represents the fundamental domains of ${\tilde {B}}_{3}$ symmetry.

A cell can be as 1/24 of a translational cube with vertices positioned: taking two corner, ne face center, and the cube center. The edge colors and labels specify how many cells exist around the edge.

Related polyhedra and honeycombs[edit]

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.

Two views

Related polytopes[edit]

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

Vertex figure

Dual cell

Alternated cantitruncated cubic honeycomb[edit]

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.

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