Order-6 cubic honeycomb

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.

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	↔ ↔
Cells	{4,3}
Faces	square {4}
Edge figure	hexagon {6}
Vertex figure	triangular tiling
Coxeter group	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]]
Dual	Order-4 hexagonal tiling honeycomb
Properties	Regular, quasiregular

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.

Images

One cell viewed outside of the Poincaré sphere model

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 ↔ .

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

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

Related polytopes and honeycombs

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

11 paracompact regular honeycombs
{6,3,3}	{6,3,4}	{6,3,5}	{6,3,6}	{4,4,3}	{4,4,4}
{3,3,6}	{4,3,6}	{5,3,6}	{3,6,3}	{3,4,4}

It has a related alternation honeycomb, represented by ↔ . 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.

[6,3,4] family honeycombs
{6,3,4}	r{6,3,4}	t{6,3,4}	rr{6,3,4}	t0,3{6,3,4}	tr{6,3,4}	t0,1,3{6,3,4}	t0,1,2,3{6,3,4}
{4,3,6}	r{4,3,6}	t{4,3,6}	rr{4,3,6}	2t{4,3,6}	tr{4,3,6}	t0,1,3{4,3,6}	t0,1,2,3{4,3,6}

The order-6 cubic honeycomb is part of a sequence of regular polychora 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,∞}

It is also part of a sequence of honeycombs with triangular tiling vertex figures.

Hyperbolic uniform honeycombs: {p,3,6}

Form	Paracompact				Noncompact
Name	{3,3,6}	{4,3,6}	{5,3,6}	{6,3,6}	{7,3,6}	{8,3,6}	... {∞,3,6}
Image
Cells	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}

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	↔ ↔ ↔
Cells	r{3,4} {3,6}
Faces	triangle {3} square {4}
Vertex figure	hexagonal prism
Coxeter groups	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {DV}}_{3}}$, [6,31,1] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]] ${\displaystyle {\overline {DP}}_{3}}$, [3[]×[]]
Properties	Vertex-transitive, edge-transitive

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

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

r{p,3,6}

• v
• t
• e

Space	H3
Form	Paracompact				Noncompact
Name	r{3,3,6}	r{4,3,6}	r{5,3,6}	r{6,3,6}	r{7,3,6}	... r{∞,3,6}
Image
Cells {3,6}	r{3,3}	r{4,3}	r{5,3}	r{6,3}	r{7,3}	r{∞,3}

Truncated order-6 cubic honeycomb

Truncated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{4,3,6} or t0,1{4,3,6}
Coxeter diagrams	↔
Cells	t{4,3} {3,6}
Faces	triangle {3} octagon {8}
Vertex figure	hexagonal pyramid
Coxeter groups	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]]
Properties	Vertex-transitive

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

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

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 t0,2{4,3,6}
Coxeter diagrams	↔
Cells	rr{4,3} r{3,6} {}x{6}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	wedge
Coxeter groups	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]]
Properties	Vertex-transitive

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

Cantitruncated order-6 cubic honeycomb

Cantitruncated order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{4,3,6} or t0,1,2{4,3,6}
Coxeter diagrams	↔
Cells	tr{4,3} t{3,6} {}x{6}
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter groups	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6] ${\displaystyle {\overline {BP}}_{3}}$, [4,3[3]]
Properties	Vertex-transitive

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

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	t0,1,3{4,3,6}
Coxeter diagrams
Cells	t{4,3} rr{3,6} {}x{6} {}x{8}
Faces	triangle {3} square {4} hexagon {6} octagon {8}
Vertex figure	isosceles-trapezoidal pyramid
Coxeter groups	${\displaystyle {\overline {BV}}_{3}}$, [4,3,6]
Properties	Vertex-transitive

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

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	↔ ↔ ↔ ↔
Cells	{3,3} {3,6}
Faces	triangle {3}
Vertex figure	trihexagonal tiling
Coxeter group	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1] ${\displaystyle {\overline {DP}}_{3}}$, [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 or , 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 ↔ . Another lower-symmetry form of index 6, [4,3^*,6], exists with a non-simplex fundamental domain, with Coxeter-Dynkin diagram .

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}
	${\displaystyle \left\{3,{3 \atop 3}\right\}}$	${\displaystyle \left\{3,{4 \atop 3}\right\}}$	${\displaystyle \left\{3,{5 \atop 3}\right\}}$	${\displaystyle \left\{3,{6 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 3}\right\}}$	${\displaystyle \left\{4,{4 \atop 4}\right\}}$
Coxeter diagram	↔	↔	↔	↔	↔	↔
	↔					↔
Image
Vertex figure r{p,3}

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

Cantic order-6 cubic honeycomb

Cantic order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2{4,3,6}
Coxeter diagram	↔ ↔ ↔
Cells	t{3,3} r{6,3} t{3,6}
Faces	triangle {3} hexagon {6}
Vertex figure	rectangular pyramid
Coxeter group	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1] ${\displaystyle {\overline {DP}}_{3}}$, [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.

Runcic order-6 cubic honeycomb

Runcic order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h3{4,3,6}
Coxeter diagram	↔
Cells	{3,3} {6,3} rr{6,3}
Faces	triangle {3} square {4} hexagon {6}
Vertex figure	triangular cupola
Coxeter group	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1]
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.

Runcicantic order-6 cubic honeycomb

Runcicantic order-6 cubic honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbol	h2,3{4,3,6}
Coxeter diagram	↔
Cells	t{6,3} tr{6,3} t{3,3}
Faces	triangle {3} square {4} hexagon {6} dodecagon {12}
Vertex figure	mirrored sphenoid
Coxeter group	${\displaystyle {\overline {DV}}_{3}}$, [6,31,1]
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.

See also

• Convex uniform honeycombs in hyperbolic space
• Regular tessellations of hyperbolic 3-space
• Paracompact uniform honeycombs

References

• Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (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