Order-6 cubic honeycomb

Order-6 cubic honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol	{4,3,6} {4,3^[3]}
Coxeter diagram	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{4,3}
Faces	square {4}
Edge figure	hexagon {6}
Vertex figure	triangular tiling
Coxeter group	${\overline{B V}}_{3}$ , [4,3,6] ${\overline{B P}}_{3}$ , [4,3^[3]]
Dual	Order-4 hexagonal tiling honeycomb
Properties	Regular, quasiregular

The order-6 cubic honeycomb is a paracompact regular space-filling tessellation (or honeycomb) in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

Template:Honeycomb

Images

File:Order-6 cubic honeycomb cell.png One cell viewed outside of the Poincaré sphere model	File:H2 tiling 24i-4.png The order-6 cubic honeycomb is analogous to the 2D hyperbolic infinite-order square tiling, {4,∞} with square faces. All vertices are on the ideal surface.

Symmetry

A half-symmetry construction of the order-6 cubic honeycomb exists as {4,3^[3]}, with two alternating types (colors) of cubic cells. This construction has Coxeter-Dynkin diagram Template:CDD ↔ Template:CDD.

Another lower-symmetry construction, [4,3^*,6], of index 6, exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram Template:CDD.

This honeycomb contains Template:CDD that tile 2-hypercycle surfaces, similar to the paracompact order-3 apeirogonal tiling, Template:CDD:

File:H2-I-3-dual.svg

Related polytopes and honeycombs

The order-6 cubic honeycomb is a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact. Template:Regular paracompact H3 honeycombs

It has a related alternation honeycomb, represented by Template:CDD ↔ Template:CDD. This alternated form has hexagonal tiling and tetrahedron cells.

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including the order-6 cubic honeycomb itself. Template:634 family

The order-6 cubic honeycomb is part of a sequence of regular polychora and honeycombs with cubic cells. Template:Cubic cell tessellations

It is also part of a sequence of honeycombs with triangular tiling vertex figures. Template:Triangular tiling vertex figure tessellations small

Rectified order-6 cubic honeycomb

Rectified order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{4,3,6} or t₁{4,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	r{3,4} File:Uniform polyhedron-43-t1.svg {3,6} Error creating thumbnail:
Faces	triangle {3} square {4}
Vertex figure	File:Rectified order-6 cubic honeycomb verf.png hexagonal prism
Coxeter groups	${\overline{B V}}_{3}$ , [4,3,6] ${\overline{D V}}_{3}$ , [6,3^1,1] ${\overline{B P}}_{3}$ , [4,3^[3]] ${\overline{D P}}_{3}$ , [3^[]×[]]
Properties	Vertex-transitive, edge-transitive

The rectified order-6 cubic honeycomb, r{4,3,6}, Template:CDD has cuboctahedral and triangular tiling facets, with a hexagonal prism vertex figure.

File:H3 436 CC center 0100.png

It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{4,∞}, Template:CDD alternating apeirogonal and square faces:

File:H2 tiling 24i-2.png

Template:Hexagonal tiling vertex figure tessellations Template:Clear

Truncated order-6 cubic honeycomb

Truncated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{4,3,6} or t_0,1{4,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{4,3} File:Uniform polyhedron-43-t01.png {3,6} Error creating thumbnail:
Faces	triangle {3} octagon {8}
Vertex figure	Error creating thumbnail: hexagonal pyramid
Coxeter groups	${\overline{B V}}_{3}$ , [4,3,6] ${\overline{B P}}_{3}$ , [4,3^[3]]
Properties	Vertex-transitive

The truncated order-6 cubic honeycomb, t{4,3,6}, Template:CDD has truncated cube and triangular tiling facets, with a hexagonal pyramid vertex figure.

File:H3 634-0011.png

It is similar to the 2D hyperbolic truncated infinite-order square tiling, t{4,∞}, Template:CDD with apeirogonal and octagonal (truncated square) faces:

File:H2 tiling 24i-6.png

Template:Clear

Bitruncated order-6 cubic honeycomb

The bitruncated order-6 cubic honeycomb is the same as the bitruncated order-4 hexagonal tiling honeycomb.

Cantellated order-6 cubic honeycomb

Cantellated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	rr{4,3,6} or t_0,2{4,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	rr{4,3} r{3,6} File:Uniform tiling 63-t1.png {}x{6} File:Hexagonal prism.png
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	wedge
Coxeter groups	${\overline{B V}}_{3}$ , [4,3,6] ${\overline{B P}}_{3}$ , [4,3^[3]]
Properties	Vertex-transitive

The cantellated order-6 cubic honeycomb, rr{4,3,6}, Template:CDD has rhombicuboctahedron, trihexagonal tiling, and hexagonal prism facets, with a wedge vertex figure.

File:H3 634-0101.png

Template:Clear

Cantitruncated order-6 cubic honeycomb

Cantitruncated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{4,3,6} or t_0,1,2{4,3,6}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD
Cells	tr{4,3} File:Uniform polyhedron-43-t012.png t{3,6} File:Uniform tiling 63-t12.svg {}x{6} File:Hexagonal prism.png
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter groups	${\overline{B V}}_{3}$ , [4,3,6] ${\overline{B P}}_{3}$ , [4,3^[3]]
Properties	Vertex-transitive

The cantitruncated order-6 cubic honeycomb, tr{4,3,6}, Template:CDD has truncated cuboctahedron, hexagonal tiling, and hexagonal prism facets, with a mirrored sphenoid vertex figure.

Template:Clear

Runcinated order-6 cubic honeycomb

The runcinated order-6 cubic honeycomb is the same as the runcinated order-4 hexagonal tiling honeycomb.

Runcitruncated order-6 cubic honeycomb

Cantellated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t_0,1,3{4,3,6}
Coxeter diagrams	Template:CDD
Cells	t{4,3} File:Uniform polyhedron-43-t01.png rr{3,6} {}x{6} File:Hexagonal prism.png {}x{8} File:Octagonal prism.png
Faces	triangle {3} square {4} hexagon {6} octagon {8}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter groups	${\overline{B V}}_{3}$ , [4,3,6]
Properties	Vertex-transitive

The runcitruncated order-6 cubic honeycomb, rr{4,3,6}, Template:CDD has truncated cube, rhombitrihexagonal tiling, hexagonal prism, and octagonal prism facets, with an isosceles-trapezoidal pyramid vertex figure.

File:H3 634-1011.png

Template:Clear

Runcicantellated order-6 cubic honeycomb

The runcicantellated order-6 cubic honeycomb is the same as the runcitruncated order-4 hexagonal tiling honeycomb.

Omnitruncated order-6 cubic honeycomb

The omnitruncated order-6 cubic honeycomb is the same as the omnitruncated order-4 hexagonal tiling honeycomb.

Alternated order-6 cubic honeycomb

Alternated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbol	h{4,3,6}
Coxeter diagram	Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{3,3} Error creating thumbnail: {3,6} Error creating thumbnail:
Faces	triangle {3}
Vertex figure	File:Uniform tiling 63-t1.png trihexagonal tiling
Coxeter group	${\overline{D V}}_{3}$ , [6,3^1,1] ${\overline{D P}}_{3}$ , [3^[]x[]]
Properties	Vertex-transitive, edge-transitive, quasiregular

In three-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellation (or honeycomb). As an alternation, with Schläfli symbol h{4,3,6} and Coxeter-Dynkin diagram Template:CDD or Template:CDD, it can be considered a quasiregular honeycomb, alternating triangular tilings and tetrahedra around each vertex in a trihexagonal tiling vertex figure.

Symmetry

A half-symmetry construction from the form {4,3^[3]} exists, with two alternating types (colors) of triangular tiling cells. This form has Coxeter-Dynkin diagram Template:CDD ↔ Template:CDD. Another lower-symmetry form of index 6, [4,3^*,6], exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram Template:CDD.

Related honeycombs

The alternated order-6 cubic honeycomb is part of a series of quasiregular polychora and honeycombs.

Quasiregular polychora and honeycombs: h{4,p,q}
Space	Finite	Affine	Compact	Paracompact
Schläfli symbol	h{4,3,3}	h{4,3,4}	h{4,3,5}	h{4,3,6}	h{4,4,3}	h{4,4,4}
Schläfli symbol	${3, \binom{3}{3}}$	${3, \binom{4}{3}}$	${3, \binom{5}{3}}$	${3, \binom{6}{3}}$	${4, \binom{4}{3}}$	${4, \binom{4}{4}}$
Coxeter diagram	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD	Template:CDD ↔ Template:CDD
Coxeter diagram	Template:CDD ↔ Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD	Template:CDD ↔ Template:CDD
Image	Error creating thumbnail:	File:Tetrahedral-octahedral honeycomb.png	File:Alternated order 5 cubic honeycomb.png			File:H3 444 FC boundary.png
Vertex figure r{p,3}	File:Uniform polyhedron-33-t1.svg Template:CDD	File:Uniform polyhedron-43-t1.svg Template:CDD	File:Uniform polyhedron-53-t1.svg Template:CDD	Error creating thumbnail: Template:CDD	File:Uniform polyhedron-43-t1.svg Template:CDD	File:Uniform tiling 44-t1.svg Template:CDD

It also has 3 related forms: the cantic order-6 cubic honeycomb, h₂{4,3,6}, Template:CDD; the runcic order-6 cubic honeycomb, h₃{4,3,6}, Template:CDD; and the runcicantic order-6 cubic honeycomb, h_2,3{4,3,6}, Template:CDD.

Template:Clear

Cantic order-6 cubic honeycomb

Cantic order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h₂{4,3,6}
Coxeter diagram	Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD ↔ Template:CDD
Cells	t{3,3} File:Truncated tetrahedron.png r{6,3} File:Uniform tiling 63-t1.png t{3,6} File:Uniform tiling 63-t12.svg
Faces	triangle {3} hexagon {6}
Vertex figure	File:Cantic order-6 cubic honeycomb verf.png rectangular pyramid
Coxeter group	${\overline{D V}}_{3}$ , [6,3^1,1] ${\overline{D P}}_{3}$ , [3^[]x[]]
Properties	Vertex-transitive

The cantic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb) with Schläfli symbol h₂{4,3,6}. It is composed of truncated tetrahedron, trihexagonal tiling, and hexagonal tiling facets, with a rectangular pyramid vertex figure. Template:Clear

Runcic order-6 cubic honeycomb

Runcic order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h₃{4,3,6}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	{3,3} Error creating thumbnail: {6,3} Error creating thumbnail: rr{6,3} Error creating thumbnail:
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	Error creating thumbnail: triangular cupola
Coxeter group	${\overline{D V}}_{3}$ , [6,3^1,1]
Properties	Vertex-transitive

The runcic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb) with Schläfli symbol h₃{4,3,6}. It is composed of tetrahedron, hexagonal tiling, and rhombitrihexagonal tiling facets, with a triangular cupola vertex figure. Template:Clear

Runcicantic order-6 cubic honeycomb

Runcicantic order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h_2,3{4,3,6}
Coxeter diagram	Template:CDD ↔ Template:CDD
Cells	t{6,3} Error creating thumbnail: tr{6,3} Error creating thumbnail: t{3,3} File:Uniform polyhedron-33-t01.png
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	File:Runcicantic order-6 cubic honeycomb verf.png mirrored sphenoid
Coxeter group	${\overline{D V}}_{3}$ , [6,3^1,1]
Properties	Vertex-transitive

The runcicantic order-6 cubic honeycomb is a uniform compact space-filling tessellation (or honeycomb), with Schläfli symbol h_2,3{4,3,6}. It is composed of truncated hexagonal tiling, truncated trihexagonal tiling, and truncated tetrahedron facets, with a mirrored sphenoid vertex figure. Template:Clear

References

Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:Isbn. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, Template:Isbn (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
Jeffrey R. Weeks The Shape of Space, 2nd edition Template:Isbn (Chapter 16-17: Geometries on Three-manifolds I, II)
Norman Johnson Uniform Polytopes, Manuscript
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups

Order-6 cubic honeycomb

Contents

Images

Symmetry