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	↔ ↔
Cells	{4,3}
Faces	square {4}
Edge figure	pentagon {6}
Vertex figure	triangular tiling {3,6}
Coxeter group	BV3, [6,3,4] BP3, [4,3^[3]]
Dual	Order-4 hexagonal tiling honeycomb
Properties	Regular, quasiregular

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

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

It is similar to the 2D hyperbolic infinite-order square tiling, {4,∞} with square faces. All vertices are on the ideal surface.

Symmetry

A half symmetry construction exists as {4,3^[3]}, with alternating two types (colors) of cubic cells. ↔ . Another lower symmetry, [4,3^*,6], index 6 exists with a nonsimplex fundamental domain, .

This honeycomb contains that tile 2-hypercycle surfaces, similar to this paracompact tiling, :

Related polytopes and honeycombs

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

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 is related to the regular (order-4) cubic honeycomb of Euclidean 3-space, order-5 cubic honeycomb in hyperbolic space, which have 4 and 5 cubes per edge respectively.

It has a related alternation honeycomb, represented by ↔ , having hexagonal tiling and tetrahedron cells.

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form.

[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}

It in a sequence of regular polychora and honeycombs with cubic cells.

{4,3,p} regular honeycombs
Space	S^3	E^3	H^3
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 a 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 {}×{6}
Coxeter groups	BV3, [6,3,4] DV3, [6,3^1,1] [4,3^[3]] [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}

Space	H^3
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{6,3}	r{6,3}	r{∞,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	BV3, [6,3,4] [4,3^[3]]
Properties	Vertex-transitive

The truncated order-6 cubic honeycomb, t{4,3,6}, has truncated octahedron 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:

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}
Faces	Triangle {3} square {4} hexagon {6} octagon {8}
Vertex figure	triangular prism
Coxeter groups	BV3, [6,3,4] [4,3^[3]]
Properties	Vertex-transitive

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

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	DV3, [6,3^1,1]
Properties	Vertex-transitive, edge-transitive, quasiregular

In 3-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellations (or honeycombs). As an alternated order-6 cubic honeycomb and Schläfli symbol h{4,3,6}, with Coxeter diagram or . It can be considered a quasiregular honeycomb, alternating triangular tiling and tetrahedron around each vertex in an trihexagonal tiling vertex figure.

Symmetry

A half symmetry construction exists from {4,3^[3]}, with alternating two types (colors) of cubic cells. ↔ . Another lower symmetry, [4,3^*,6], index 6 exists with a nonsimplex fundamental domain, .

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

It has 3 related form cantic order-6 cubic honeycomb, h₂{4,3,6}, , runcic order-6 cubic honeycomb, h₃{4,3,6}, , 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} {6,3}
Faces	Triangle {3} hexagon {6}
Vertex figure
Coxeter group	DV3, [6,3^1,1]
Properties	Vertex-transitive

The cantic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs) with Schläfli symbol h₂{4,3,6}.

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} hexagon {6}
Vertex figure	triangular prism
Coxeter group	DV3, [6,3^1,1]
Properties	Vertex-transitive

The runcic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs). With Schläfli symbol h₃{4,3,6}, with a triangular prism 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	{6,3} tr{6,3} {3,3}
Faces	Triangle {3} square {4}
Vertex figure	tetrahedron
Coxeter group	DV3, [6,3^1,1]
Properties	Vertex-transitive

The runcicantic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs). With Schläfli symbol h_2,3{4,3,6}, with a tetrahedral vertex figure.

See also

• Convex uniform honeycombs in hyperbolic space

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, (2015) Chapter 13: Hyperbolic Coxeter groups