Английская Википедия:Cubic honeycomb

Cubic honeycomb
Файл:Cubic honeycomb.pngФайл:Partial cubic honeycomb.png
Type	Regular honeycomb
Family	Hypercube honeycomb
Indexing^[1]	J_11,15, A₁ W₁, G₂₂
Schläfli symbol	{4,3,4}
Coxeter diagram	Шаблон:CDD
Cell type	{4,3} Файл:Uniform polyhedron-43-t0.png
Face type	square {4}
Vertex figure	Файл:Cubic honeycomb verf.svg octahedron
Space group Fibrifold notation	PmШаблон:Overlinem (221) 4⁻:2
Coxeter group	<math>{\tilde{C}}_3</math>, [4,3,4]
Dual	self-dual Cell: Файл:Cubic full domain.png
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.

Шаблон:Honeycomb

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	Файл:Monoclinic.svg	Файл:Orthorhombic.svg	Файл:Tetragonal.svg	Файл:Rhombohedral.svg	Файл:Cubic.svg
Space group Rotation subgroup	Pm (6) P1 (1)	Pmmm (47) P222 (16)	P4/mmm (123) P422 (89)	R3m (160) R3 (146)	PmШаблон:Overlinem (221) P432 (207)
Coxeter notation	-	[∞]_a×[∞]_b×[∞]_c	[4,4]_a×[∞]_c	-	[4,3,4]_a
Coxeter diagram	-	Шаблон:CDD	Шаблон:CDD	-	Шаблон:CDD

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] PmШаблон:Overlinem (221)	Шаблон:CDD Шаблон:CDD = Шаблон:CDD	{4,3,4}	Файл:Partial cubic honeycomb.png	1: aaaa/aaaa
[4,3^1,1] = [4,3,4,1⁺] FmШаблон:Overlinem (225)	Шаблон:CDD = Шаблон:CDD	{4,3^1,1}	Файл:Bicolor cubic honeycomb.png	2: abba/baab
[4,3,4] PmШаблон:Overlinem (221)	Шаблон:CDD	t_0,3{4,3,4}	Файл:Runcinated cubic honeycomb.png	4: abbc/bccd
[[4,3,4]] PmШаблон:Overlinem (229)	Шаблон:CDD	t_0,3{4,3,4}		4: abbb/bbba
[4,3,4,2,∞]	Шаблон:CDD or Шаблон:CDD	{4,4}×t{∞}	Файл:Square prismatic honeycomb.png	2: aaaa/bbbb
[4,3,4,2,∞]	Шаблон:CDD	t₁{4,4}×{∞}	Файл:Square prismatic 2-color honeycomb.png	2: abba/abba
[∞,2,∞,2,∞]	Шаблон:CDD	t{∞}×t{∞}×{∞}	Файл:Square 4-color prismatic honeycomb.png	4: abcd/abcd
[∞,2,∞,2,∞] = [4,(3,4)^*]	Шаблон:CDD = Шаблон:CDD	t{∞}×t{∞}×t{∞}	Файл:Cubic 8-color honeycomb.png	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	Файл:Cubic honeycomb-2.svg	Файл:Cubic honeycomb-1.svg	Файл:Cubic honeycomb-3.svg
Frame	Файл:Cubic honeycomb-2b.svg	Файл:Cubic honeycomb-1b.svg	Файл:Cubic honeycomb-3b.svg

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. Шаблон:Octahedral vertex figure tessellations

It in a sequence of regular polytopes and honeycombs with cubic cells. Шаблон:Cubic cell tessellations

Шаблон:Symmetric2 tessellations

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.

Файл:Biruncinatocubic honeycomb dual cell.png
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], Шаблон:CDD, 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. Шаблон:C3 honeycombs

The [4,3^1,1], Шаблон:CDD, Coxeter group generates 9 permutations of uniform tessellations, 4 with distinct geometry including the alternated cubic honeycomb. Шаблон:B3 honeycombs

This honeycomb is one of five distinct uniform honeycombs^[2] constructed by the <math>{\tilde{A}}_3</math> Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams: Шаблон:A3 honeycombs

Rectified cubic honeycomb

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	Шаблон:CDD Шаблон:CDD = Шаблон:CDD Шаблон:CDD = Шаблон:CDD Шаблон:CDD = Шаблон:CDD = Шаблон:CDD = Шаблон:CDD
Cells	r{4,3} Файл:Uniform polyhedron-43-t1.png {3,4} Файл:Uniform polyhedron-43-t2.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Rectified cubic honeycomb verf.png square prism
Space group Fibrifold notation	PmШаблон:Overlinem (221) 4⁻:2
Coxeter group	<math>{\tilde{C}}_3</math>, [4,3,4]
Dual	oblate octahedrille Cell: Файл:Cubic square bipyramid.png
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.

Файл:Rectified cubic tiling.pngФайл:HC A3-P3.png

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	Файл:Rectified cubic honeycomb-2.png	Файл:Rectified cubic honeycomb-1.png	Файл:Rectified cubic honeycomb-3.png
Frame	Файл:Rectified cubic honeycomb-2b.png	Файл:Rectified cubic honeycomb-1b.png	Файл:Rectified cubic honeycomb-3b.png

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] <math>{\tilde{C}}_3</math>	[1⁺,4,3,4] [4,3^1,1], <math>{\tilde{B}}_3</math>	[4,3,4,1⁺] [4,3^1,1], <math>{\tilde{B}}_3</math>	[1⁺,4,3,4,1⁺] [3^[4]], <math>{\tilde{A}}_3</math>
Space group	PmШаблон:Overlinem (221)	FmШаблон:Overlinem (225)	FmШаблон:Overlinem (225)	FШаблон:Overline3m (216)
Coloring	Файл:Rectified cubic honeycomb.png	Файл:Rectified cubic honeycomb4.png	Файл:Rectified cubic honeycomb3.png	Файл:Rectified cubic honeycomb2.png
Coxeter diagram	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD
Coxeter diagram	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD
Vertex figure	Файл:Rectified cubic honeycomb verf.png	Файл:Rectified alternate cubic honeycomb verf.png	Файл:Cantellated alternate cubic honeycomb verf.png	Файл:T02 quarter cubic honeycomb verf.png
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 Шаблон:CDD, and symbol s₃{2,6,3}, with coxeter notation symmetry [2⁺,6,3].

Файл:Runcic snub 263 honeycomb.png.

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.

Файл:Biambocubic honeycomb dual cell.png
Dual cell

Truncated cubic honeycomb

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	Шаблон:CDD Шаблон:CDD = Шаблон:CDD
Cell type	t{4,3} Файл:Uniform polyhedron-43-t01.png {3,4} Файл:Uniform polyhedron-43-t2.png
Face type	triangle {3} octagon {8}
Vertex figure	Файл:Truncated cubic honeycomb verf.png isosceles square pyramid
Space group Fibrifold notation	PmШаблон:Overlinem (221) 4⁻:2
Coxeter group	<math>{\tilde{C}}_3</math>, [4,3,4]
Dual	Pyramidille Cell: Файл:Cubic square pyramid.png
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.

Файл:Truncated cubic tiling.pngФайл:HC A2-P3.png

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	Файл:Truncated cubic honeycomb-2.png	Файл:Truncated cubic honeycomb-1.png	Файл:Truncated cubic honeycomb-3.png
Frame	Файл:Truncated cubic honeycomb-2b.png	Файл:Truncated cubic honeycomb-1b.png	Файл:Truncated cubic honeycomb-3b.png

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	[4,3^1,1], <math>{\tilde{B}}_3</math>	[4,3,4], <math>{\tilde{C}}_3</math> =<[4,3^1,1]>
Space group	FmШаблон:Overlinem	PmШаблон:Overlinem
Coloring	Файл:Truncated cubic honeycomb2.png	Файл:Truncated cubic honeycomb.png
Coxeter diagram	Шаблон:CDD = Шаблон:CDD	Шаблон:CDD
Vertex figure	Файл:Bicantellated alternate cubic honeycomb verf.png	Файл:Truncated cubic honeycomb verf.png

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.

Файл:Bitruncatocubic honeycomb vertex figure.png
Vertex figure

Файл:Bitruncatocubic honeycomb dual cell.png
Dual cell

Bitruncated cubic honeycomb

Bitruncated cubic honeycomb
Файл:Bitruncated cubic tiling.png Файл:HC-A4.png
Type	Uniform honeycomb
Schläfli symbol	2t{4,3,4} t_1,2{4,3,4}
Coxeter-Dynkin diagram	Шаблон:CDD
Cells	t{3,4} Файл:Uniform polyhedron-43-t12.png
Faces	square {4} hexagon {6}
Edge figure	isosceles triangle {3}
Vertex figure	Файл:Bitruncated cubic honeycomb verf2.png tetragonal disphenoid
Symmetry group Fibrifold notation Coxeter notation	[[Cubic crystal system\|ImШаблон:Overlinem (229)]] 8^o:2 [[4,3,4]]
Coxeter group	<math>{\tilde{C}}_3</math>, [4,3,4]
Dual	Oblate tetrahedrille Disphenoid tetrahedral honeycomb Cell: Файл:Oblate tetrahedrille cell.png
Properties	Vertex-transitive, edge-transitive, cell-transitive

Файл:Cubes-A4 ani.gif

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	Файл:Bitruncated cubic honeycomb ortho2.png	Файл:Bitruncated cubic honeycomb ortho4.png	Файл:Bitruncated cubic honeycomb ortho1.png	Файл:Bitruncated cubic honeycomb ortho3.png	Файл:Bitruncated cubic honeycomb ortho5.png
Frame	Файл:Bitruncated cubic honeycomb orthoframe2.png	Файл:Bitruncated cubic honeycomb orthoframe4.png	Файл:Bitruncated cubic honeycomb orthoframe1.png	Файл:Bitruncated cubic honeycomb orthoframe3.png	Файл:Bitruncated cubic honeycomb orthoframe5.png

Symmetry

The vertex figure for this honeycomb is a disphenoid tetrahedron, and it is also the Goursat tetrahedron (fundamental domain) for the <math>{\tilde{A}}_3</math> 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	ImШаблон:Overlinem (229)	PmШаблон:Overlinem (221)	FmШаблон:Overlinem (225)	FШаблон:Overline3m (216)	FdШаблон:Overlinem (227)
Fibrifold	8^o:2	4⁻:2	2⁻:2	1^o:2	2⁺:2
Coxeter group	<math>{\tilde{C}}_3</math>×2 [[4,3,4]] =[4[3^[4]]] Шаблон:CDD = Шаблон:CDD	<math>{\tilde{C}}_3</math> [4,3,4] =[2[3^[4]]] Шаблон:CDD = Шаблон:CDD	<math>{\tilde{B}}_3</math> [4,3^1,1] =<[3^[4]]> Шаблон:CDD = Шаблон:CDD	<math>{\tilde{A}}_3</math> [3^[4]] Шаблон:CDD	<math>{\tilde{A}}_3</math>×2 [[3^[4]]] =[[3^[4]]] Шаблон:CDD
Coxeter diagram	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD
truncated octahedra	1 Файл:Uniform polyhedron-43-t12.svg	1:1 Файл:Uniform polyhedron-43-t12.svg:Файл:Uniform polyhedron-43-t12.svg	2:1:1 Файл:Uniform polyhedron-43-t12.svg:Файл:Uniform polyhedron-43-t12.svg:Файл:Uniform polyhedron-33-t012.png	1:1:1:1 Файл:Uniform polyhedron-33-t012.png:Файл:Uniform polyhedron-33-t012.png:Файл:Uniform polyhedron-33-t012.png:Файл:Uniform polyhedron-33-t012.png	1:1 Файл:Uniform polyhedron-33-t012.png:Файл:Uniform polyhedron-33-t012.png
Vertex figure	Файл:Bitruncated cubic honeycomb verf2.png	Файл:Bitruncated cubic honeycomb verf.png	Файл:Cantitruncated alternate cubic honeycomb verf.png	Файл:Omnitruncated 3-simplex honeycomb verf.png	Файл:Omnitruncated 3-simplex honeycomb verf2.png
Vertex figure symmetry	[2⁺,4] (order 8)	[2] (order 4)	[ ] (order 2)	[ ]⁺ (order 1)	[2]⁺ (order 2)
Image Colored by cell	Файл:Bitruncated Cubic Honeycomb1.svg	Файл:Bitruncated Cubic Honeycomb.svg	Файл:Bitruncated cubic honeycomb3.png	Файл:Bitruncated cubic honeycomb2.png	Файл:Bitruncated Cubic Honeycomb1.svg

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.

Alternated bitruncated cubic honeycomb

Alternated bitruncated cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	2s{4,3,4} 2s{4,3^1,1} sr{3^[4]}
Coxeter diagrams	Шаблон:CDD Шаблон:CDD = Шаблон:CDD Шаблон:CDD = Шаблон:CDD Шаблон:CDD = Шаблон:CDD
Cells	{3,3} Файл:Uniform polyhedron-33-t0.png s{3,3} Файл:Uniform polyhedron-33-s012.png
Faces	triangle {3}
Vertex figure	Файл:Alternated bitruncated cubic honeycomb verf.png
Coxeter group	[[4,3⁺,4]], <math>{\tilde{C}}_3</math>
Dual	Ten-of-diamonds honeycomb Cell: Файл:Alternated bitruncated cubic honeycomb dual cell.png
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: Шаблон:CDD, Шаблон:CDD, and Шаблон:CDD. 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	IШаблон:Overline (204)	PmШаблон:Overline (200)	FmШаблон:Overline (202)	FdШаблон:Overline (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	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD	Шаблон:CDD
Order	double	full	half	quarter double	quarter

Cantellated cubic honeycomb

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	Шаблон:CDD Шаблон:CDD = Шаблон:CDD
Cells	rr{4,3} Файл:Uniform polyhedron-43-t02.png r{4,3} Файл:Uniform polyhedron-43-t1.png {}x{4} Файл:Tetragonal prism.png
Vertex figure	Файл:Cantellated cubic honeycomb verf.png wedge
Space group Fibrifold notation	PmШаблон:Overlinem (221) 4⁻:2
Coxeter group	[4,3,4], <math>{\tilde{C}}_3</math>
Dual	quarter oblate octahedrille Cell: Файл:Quarter oblate octahedrille cell.png
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.

Файл:Cantellated cubic tiling.png Файл:HC A5-A3-P2.png

Images

Файл:Cantellated cubic honeycomb.png	Файл:Perovskite.jpg 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	Файл:Cantellated cubic honeycomb-2.png	Файл:Cantellated cubic honeycomb-1.png	Файл:Cantellated cubic honeycomb-3.png
Frame	Файл:Cantellated cubic honeycomb-2b.png	Файл:Cantellated cubic honeycomb-1b.png	Файл:Cantellated cubic honeycomb-3b.png

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], <math>{\tilde{C}}_3</math> =<[4,3^1,1]>	[4,3^1,1], <math>{\tilde{B}}_3</math>
Space group	PmШаблон:Overlinem	FmШаблон:Overlinem
Coxeter diagram	Шаблон:CDD	Шаблон:CDD
Coloring	Файл:Cantellated cubic honeycomb.png	Файл:Cantellated cubic honeycomb2.png
Vertex figure	Файл:Cantellated cubic honeycomb verf.png	Файл:Runcicantellated alternate cubic honeycomb verf.png
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.

Quarter oblate octahedrille

The dual of the cantellated cubic honeycomb is called a quarter oblate octahedrille, a catoptric tessellation with Coxeter diagram Шаблон:CDD, 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.

Файл:Quarter oblate octahedrille cell.png

Cantitruncated cubic honeycomb

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	Шаблон:CDD Шаблон:CDD = Шаблон:CDD
Cells	tr{4,3} Файл:Uniform polyhedron-43-t012.png t{3,4} Файл:Uniform polyhedron-43-t12.png {}x{4} Файл:Tetragonal prism.png
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	Файл:Cantitruncated cubic honeycomb verf.pngФайл:Omnitruncated alternated cubic honeycomb verf.png mirrored sphenoid
Coxeter group	[4,3,4], <math>{\tilde{C}}_3</math>
Symmetry group Fibrifold notation	PmШаблон:Overlinem (221) 4⁻:2
Dual	triangular pyramidille Cells: Файл:Triangular pyramidille cell1.png
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.

Файл:Cantitruncated cubic tiling.png Файл:HC A6-A4-P2.png

Images

Four cells exist around each vertex:

Файл:2-Kuboktaederstumpf 1-Oktaederstumpf 1-Hexaeder.png

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	Файл:Cantitruncated cubic honeycomb-2.png	Файл:Cantitruncated cubic honeycomb-1.png	Файл:Cantitruncated cubic honeycomb-3.png
Frame	Файл:Cantitruncated cubic honeycomb-2b.png	Файл:Cantitruncated cubic honeycomb-1b.png	Файл:Cantitruncated cubic honeycomb-3b.png

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], <math>{\tilde{C}}_3</math> =<[4,3^1,1]>	[4,3^1,1], <math>{\tilde{B}}_3</math>
Space group	PmШаблон:Overlinem (221)	FmШаблон:Overlinem (225)
Fibrifold	4⁻:2	2⁻:2
Coloring	Файл:Cantitruncated Cubic Honeycomb.svg	Файл:Cantitruncated Cubic Honeycomb2.svg
Coxeter diagram	Шаблон:CDD	Шаблон:CDD
Vertex figure	Файл:Cantitruncated cubic honeycomb verf.png	Файл:Omnitruncated alternated cubic honeycomb verf.png
Vertex figure symmetry	[ ] order 2	[ ]⁺ order 1

Triangular pyramidille

The dual of the cantitruncated cubic honeycomb is called a triangular pyramidille, with Coxeter diagram, Шаблон:CDD. This honeycomb cells represents the fundamental domains of <math>{\tilde{B}}_3</math> 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.

Файл:Triangular pyramidille cell1.png

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
Файл:Cantitruncated cubic honeycomb apeirohedron 4466.png	Файл:Omnitruncated cubic honeycomb apeirohedron 4466.png

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.

Файл:Bicantitruncatocubic honeycomb vertex figure.png
Vertex figure

Файл:Bicantitruncatocubic honeycomb dual cell.png
Dual cell

Alternated cantitruncated cubic honeycomb

Alternated cantitruncated cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	sr{4,3,4} sr{4,3^1,1}
Coxeter diagrams	Шаблон:CDD Шаблон:CDD = Шаблон:CDD
Cells	s{4,3} Файл:Uniform polyhedron-43-s012.png s{3,3} Файл:Uniform polyhedron-33-s012.png {3,3} Файл:Uniform polyhedron-33-t0.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Alternated cantitruncated cubic honeycomb vertex figure.pngФайл:Alternated cantitruncated cubic honeycomb verf.png
Coxeter group	[(4,3)⁺,4]
Dual	Шаблон:CDD Cell: Файл:Alternated cantitruncated cubic honeycomb dual cell.png
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 Шаблон:CDD or Шаблон:CDD.

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.

Файл:Alternated cantitruncated cubic honeycomb.png Шаблон:CDD	Файл:Althalfcell-honeycomb-cube3x3x3.png Шаблон:CDD

Cantic snub cubic honeycomb

Orthosnub cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	2s₀{4,3,4}
Coxeter diagrams	Шаблон:CDD
Cells	s₂{3,4} Файл:Uniform polyhedron-43-t02.png s{3,3} Файл:Uniform polyhedron-33-s012.png {}x{3} Файл:Triangular prism.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Orthosnub cubic honeycomb vertex figure.png
Coxeter group	[4⁺,3,4]
Dual	Cell: Файл:Orthosnub cubic honeycomb dual cell.png
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 Шаблон:CDD. 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.

Файл:Biorthopyritohedral honeycomb vertex figure.png
Vertex figure

Файл:Biorthopyritohedral honeycomb dual cell.png
Dual cell

Runcitruncated cubic honeycomb

Runcitruncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t_0,1,3{4,3,4}
Coxeter diagrams	Шаблон:CDD
Cells	rr{4,3} Файл:Uniform polyhedron-43-t02.png t{4,3} Файл:Uniform polyhedron-43-t01.png {}x{8} Файл:Octagonal prism.png {}x{4} Файл:Tetragonal prism.png
Faces	triangle {3} square {4} octagon {8}
Vertex figure	Файл:Runcitruncated cubic honeycomb verf.png isosceles-trapezoidal pyramid
Coxeter group	[4,3,4], <math>{\tilde{C}}_3</math>
Space group Fibrifold notation	PmШаблон:Overlinem (221) 4⁻:2
Dual	square quarter pyramidille Cell Файл:Square quarter pyramidille cell.png
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, Шаблон:CDD 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.

Файл:Runcitruncated cubic tiling.png

Файл:HC A5-A2-P2-Pr8.png

Файл:Runcitruncated cubic honeycomb.jpg

Файл:Runcitruncated cubic honeycomb (Schoute 1911).jpg

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	Файл:Runcitruncated cubic honeycomb-2.png	Файл:Runcitruncated cubic honeycomb-1.png	Файл:Runcitruncated cubic honeycomb-3.png
Frame	Файл:Runcitruncated cubic honeycomb-2b.png	Файл:Runcitruncated cubic honeycomb-1b.png	Файл:Runcitruncated cubic honeycomb-3b.png

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.

Файл:Skew polyhedron 34444.pngФайл:Skew polyhedron 3448.png

Square quarter pyramidille

The dual to the runcitruncated cubic honeycomb is called a square quarter pyramidille, with Coxeter diagram Шаблон:CDD. Faces exist in 3 of 4 hyperplanes of the [4,3,4], <math>{\tilde{C}}_3</math> 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.

Файл:Square quarter pyramidille cell.png

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.

Файл:Biruncitruncatocubic honeycomb vertex figure.png
Vertex figure

Файл:Biruncitruncatocubic honeycomb dual cell.png
Dual cell

Omnitruncated cubic honeycomb

Omnitruncated cubic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t_0,1,2,3{4,3,4}
Coxeter diagram	Шаблон:CDD
Cells	tr{4,3} Файл:Uniform polyhedron-43-t012.png {}x{8} Файл:Octagonal prism.png
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	Файл:Omnitruncated cubic honeycomb verf.png phyllic disphenoid
Symmetry group Fibrifold notation Coxeter notation	[[Cubic crystal system\|ImШаблон:Overlinem (229)]] 8^o:2 [[4,3,4]]
Coxeter group	[4,3,4], <math>{\tilde{C}}_3</math>
Dual	eighth pyramidille Cell Файл:Fundamental tetrahedron1.png
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.

Файл:Omnitruncated cubic tiling.png Файл:HC A6-Pr8.png

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	Файл:Omnitruncated cubic honeycomb-2.png	Файл:Omnitruncated cubic honeycomb-1.png	Файл:Omnitruncated cubic honeycomb-3.png
Frame	Файл:Omnitruncated cubic honeycomb-2b.png	Файл:Omnitruncated cubic honeycomb-1b.png	Файл:Omnitruncated cubic honeycomb-3b.png

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	<math>{\tilde{C}}_3</math>, [4,3,4]	<math>{\tilde{C}}_3</math>×2, Шаблон:Brackets
Space group	PmШаблон:Overlinem (221)	ImШаблон:Overlinem (229)
Fibrifold	4⁻:2	8^o:2
Coloring	Файл:Omnitruncated cubic honeycomb1.png	Файл:Omnitruncated cubic honeycomb2.png
Coxeter diagram	Шаблон:CDD	Шаблон:CDD
Vertex figure	Файл:Omnitruncated cubic honeycomb verf.png	Файл:Omnitruncated cubic honeycomb verf2.png

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 Файл:Omnitruncated cubic honeycomb skew1 verf.png	4.8.4.8 Файл:Omnitruncated cubic honeycomb skew2 verf.png
Файл:Omnitruncated cubic honeycomb apeirohedron 4446.png	Файл:Skew polyhedron 4848.png

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.

Файл:Biomnitruncatocubic honeycomb vertex figure.png
Vertex figure

Файл:Biomnitruncatocubic honeycomb dual cell.png
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).

Файл:Alternated biomnitruncatocubic honeycomb vertex figure.png
Vertex figure

Alternated omnitruncated cubic honeycomb

Alternated omnitruncated cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	ht_0,1,2,3{4,3,4}
Coxeter diagram	Шаблон:CDD
Cells	s{4,3} Файл:Uniform polyhedron-43-s012.png s{2,4} Файл:Square antiprism.png {3,3} Файл:Uniform polyhedron-33-t0.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Omnisnub cubic honeycomb vertex figure.pngФайл:Snub cubic honeycomb verf.png
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: Шаблон:CDD 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	dht_0,1,2,3{4,3,4}
Coxeter diagram	Шаблон:CDD
Cell	Файл:Omnisnub cubic honeycomb dual cell.png
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:

Файл:Altbasetet-24-in-cube.png

Individual cells have 2-fold rotational symmetry. In 2D orthogonal projection, this looks like a mirror symmetry.

Cell views
Файл:Altbasetet net.png Net	Файл:Altbasetet.png	Файл:Altbasetet-frame1.png	Файл:Altbasetet-frame2.png
Файл:Altbasetet-frame3.png	Файл:Altbasetet-frame4.png	Файл:Altbasetet-frame5.png	Файл:Altbasetet-frame7.png

Runcic cantitruncated cubic honeycomb

Runcic cantitruncated cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	sr₃{4,3,4}
Coxeter diagrams	Шаблон:CDD
Cells	s₂{3,4} Файл:Uniform polyhedron-43-t02.png s{4,3} Файл:Uniform polyhedron-43-s012.png {}x{4} Файл:Tetragonal prism.png {}x{3} Файл:Triangular prism.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Bialternatosnub cubic honeycomb vertex figure.png
Coxeter group	[4,3⁺,4]
Dual	Cell: Файл:Bialternatosnub cubic honeycomb dual cell.png
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 Шаблон:CDD. 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.

Biorthosnub cubic honeycomb

Biorthosnub cubic honeycomb
Type	Convex honeycomb
Schläfli symbol	2s_0,3{4,3,4}
Coxeter diagrams	Шаблон:CDD
Cells	s₂{3,4} Файл:Uniform polyhedron-43-t02.png {}x{4} Файл:Tetragonal prism.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Biorthosnub cubic honeycomb vertex figure.png (Tetragonal antiwedge)
Coxeter group	[[4,3⁺,4]]
Dual	Cell: Файл:Biorthosnub cubic honeycomb dual cell.png
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 Шаблон:CDD. 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).

Truncated square prismatic honeycomb

Truncated square prismatic honeycomb
Type	Uniform honeycomb
Schläfli symbol	t{4,4}×{∞} or t_0,1,3{4,4,2,∞} tr{4,4}×{∞} or t_0,1,2,3{4,4,∞}
Coxeter-Dynkin diagram	Шаблон:CDD Шаблон:CDD
Cells	{}x{8} Файл:Octagonal prism.png {}x{4} Файл:Tetragonal prism.png
Faces	square {4} octagon {8}
Coxeter group	[4,4,2,∞]
Dual	Tetrakis square prismatic tiling Cell: Файл:Cubic half domain.png
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.

Файл:Truncated square prismatic honeycomb.png

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	Шаблон:CDD Шаблон:CDD
Cells	{}x{4} Файл:Tetragonal prism.png {}x{3} Файл:Triangular prism.png
Faces	triangle {3} square {4}
Coxeter group	[4⁺,4,2,∞] [(4,4)⁺,2,∞]
Dual	Cairo pentagonal prismatic honeycomb Cell: Файл:Snub square prismatic honeycomb dual cell.png
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.

Файл:Snub square prismatic honeycomb.png

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	ht_1,2,3{4,4,2,∞} ht_0,1,2,3{4,4,∞}
Coxeter-Dynkin diagram	Шаблон:CDD Шаблон:CDD
Cells	s{2,4} Файл:Square antiprism.png {3,3} Файл:Uniform polyhedron-33-t0.png
Faces	triangle {3} square {4}
Vertex figure	Файл:Snub square antiprismatic honeycomb vertex figure.png
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: Шаблон:CDD 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.

References

Шаблон:Reflist

John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, Шаблон:ISBN (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, Шаблон:ISBN 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, Шаблон:ISBN [1]
- (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.
Шаблон:KlitzingPolytopes
Uniform Honeycombs in 3-Space: 01-Chon

Шаблон:Honeycombs

↑ 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
↑ Williams, 1979, p 199, Figure 5-38.
↑ cantic snub cubic honeycomb

[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] [2], A000029 6-1 cases, skipping one with zero marks

[3] Williams, 1979, p 199, Figure 5-38.

[4] tic snub cubic honeycomb

[1]

[2]

[3]

[4]

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Английская Википедия:Cubic honeycomb

Related honeycombs

Isometries of simple cubic lattices

Uniform colorings

Projections

Related polytopes and honeycombs

Related polytopes

Related Euclidean tessellations

Rectified cubic honeycomb

Projections

Symmetry

Related polytopes

Truncated cubic honeycomb

Projections

Symmetry

Related polytopes

Bitruncated cubic honeycomb

Projections

Symmetry

Related polytopes

Alternated bitruncated cubic honeycomb

Cantellated cubic honeycomb

Images

Projections

Symmetry

Related polytopes

Quarter oblate octahedrille

Cantitruncated cubic honeycomb

Images

Projections

Symmetry

Triangular pyramidille

Related polyhedra and honeycombs

Related polytopes

Alternated cantitruncated cubic honeycomb

Cantic snub cubic honeycomb

Related polytopes

Runcitruncated cubic honeycomb

Projections

Related skew apeirohedron

Square quarter pyramidille

Related polytopes

Omnitruncated cubic honeycomb

Projections

Symmetry

Related polyhedra

Related polytopes

Alternated omnitruncated cubic honeycomb

Dual alternated omnitruncated cubic honeycomb

Runcic cantitruncated cubic honeycomb

Biorthosnub cubic honeycomb

Truncated square prismatic honeycomb

Snub square prismatic honeycomb

Snub square antiprismatic honeycomb

See also

References

Навигация

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