Cubic honeycomb

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.

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

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

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 are 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

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

Related polytopes and honeycombs

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

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

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] ↔ [4,31,1]	↔	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 [4,3^1,1], , 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	[4,31,1] ↔ [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:2	a1	[3[4]]		${\displaystyle {\tilde {A}}_{3}}$	(None)
Fm3m (225)	2−:2	d2	<[3[4]]> ↔ [4,31,1]	↔	${\displaystyle {\tilde {A}}_{3}}$×21 ↔ ${\displaystyle {\tilde {B}}_{3}}$	1, 2
Fd3m (227)	2+:2	g2	[[3[4]]] or [2+[3[4]]]	↔	${\displaystyle {\tilde {A}}_{3}}$×22	3
Pm3m (221)	4−:2	d4	<2[3[4]]> ↔ [4,3,4]	↔	${\displaystyle {\tilde {A}}_{3}}$×41 ↔ ${\displaystyle {\tilde {C}}_{3}}$	4
I3 (204)	8−o	r8	[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

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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

Symmetry	p6m (*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].

.

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

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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} 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

Symmetry	p6m (*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.

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

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

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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]
Dual	Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell:
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated cubic honeycomb shown here in relation to a cubic honeycomb

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

Symmetry	p6m (*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: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

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.

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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,(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−	2o+	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

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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

Symmetry	p6m (*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

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

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.

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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

Symmetry	p6m (*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.

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

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.

Two views

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

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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 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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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]
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 T_h symmetry), and triangular prisms (as C_2v-symmetry wedges) filling the gaps.^[4]

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

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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

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

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.

Vertex figure

Dual cell

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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

Symmetry	p6m (*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−:2	8o:2
Coloring
Coxeter diagram
Vertex figure

Related polyhedra

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

Related polytopes

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.

Vertex figure

Dual cell

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).

Vertex figure

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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
Type	Dual 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.

Cell views

Net

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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]
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 D_2d symmetry), and triangular prisms (as C_2v-symmetry wedges) filling the gaps.

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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]]
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 D_2d symmetry).

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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.

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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.

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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.

See also

• Architectonic and catoptric tessellation
• Alternated cubic honeycomb
• List of regular polytopes
• Order-5 cubic honeycomb A hyperbolic cubic honeycomb with 5 cubes per edge
• Snub (geometry)
• Voxel

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.
4. cantic snub cubic honeycomb

• 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

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
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	0[3]	δ3	hδ3	qδ3	Hexagonal
E3	Uniform convex honeycomb	0[4]	δ4	hδ4	qδ4
E4	Uniform 4-honeycomb	0[5]	δ5	hδ5	qδ5	24-cell honeycomb
E5	Uniform 5-honeycomb	0[6]	δ6	hδ6	qδ6
E6	Uniform 6-honeycomb	0[7]	δ7	hδ7	qδ7	222
E7	Uniform 7-honeycomb	0[8]	δ8	hδ8	qδ8	133 • 331
E8	Uniform 8-honeycomb	0[9]	δ9	hδ9	qδ9	152 • 251 • 521
E9	Uniform 9-honeycomb	0[10]	δ10	hδ10	qδ10
E10	Uniform 10-honeycomb	0[11]	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	0[n]	δn	hδn	qδn	1k2 • 2k1 • k21