Order-4 square tiling honeycomb

Order-4 square tiling honeycomb

Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{4,4,4} h{4,4,4} ↔ {4,4^1,1} {4^[4]}
Coxeter diagrams	↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔
Cells	{4,4}
Faces	Square {4}
Edge figure	Square {4}
Vertex figure	Square tiling, {4,4}
Dual	Self-dual
Coxeter groups	[4,4,4] [4^1,1,1] [4^[4]]
Properties	Regular, quasiregular

In the geometry of hyperbolic 3-space, the order-4 square tiling honeycomb, is one of 11 paracompact regular honeycombs. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {4,4,4}, has four square tilings, {4,4} around each edge, and infinite square tilings around each vertex in an square tiling {4,4} vertex arrangement.^[1]

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.

Symmetry

It has many reflective symmetry constructions, as a regular honeycomb, ↔ with alternate types (colors) of square tilings, and with 3 types (colors) of square tilings, with a ratio of 2:1:1. Two more half symmetry construction with pyramidal domains have [4,4,1⁺,4] symmetry: ↔ , and ↔ .

There are two high index subgroups, both index 8: [4,4,4^*] ↔ [(4,4,4,4,1⁺)] exists with a pyramidal fundamental domain, [((4,∞,4)),((4,∞,4))] or , and secondly [4,4^*,4], with 4 orthogonal sets of ultraparallel mirrors in an octahedral fundamental domain: .

Images

It is analogous to the 2D hyperbolic infinite-order apeirogonal tiling, {∞,∞} with infinite apeirogonal faces and all vertices are on the ideal surface.

This honeycomb contains , that tile 2-hypercycle surfaces, similar to these paracompact tilings:

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}

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

[4,4,4] family honeycombs
{4,4,4}	r{4,4,4}	t{4,4,4}	rr{4,4,4}	t_0,3{4,4,4}	2t{4,4,4}	tr{4,4,4}	t_0,1,3{4,4,4}	t_0,1,2,3{4,4,4}

It is a part of a sequence of honeycombs with a square tiling vertex figure:

{p,4,4} honeycombs

{p,4,4}
Space	E³	H³
Form	Affine	Paracompact		Noncompact
Name	{2,4,4}	{3,4,4}	{4,4,4}	{5,4,4}	{6,4,4}...	{∞,4,4}
Coxeter
Image
Cells	{2,4}	{3,4}	{4,4}	{5,4}	{6,4}	{∞,4}

It is a part of a sequence of honeycombs with square tiling cells:

{4,4,p} honeycombs
Space	E³	H³
Form	Affine	Paracompact		Noncompact
Name	4,4,2}	{4,4,3}	{4,4,4}	{4,4,5}	{4,4,6}	...{4,4,∞}
Coxeter
Image
Vertex figure	{4,2}	{4,3}	{4,4}	{4,5}	{4,6}	{4,∞}

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

Rectified order-4 square tiling honeycomb

The rectified order-4 hexagonal tiling honeycomb, t₁{4,4,4}, has square tiling facets, with a square prism, , vertex figure. It is the same as the regular square tiling honeycomb, {4,4,3}, , with a cube vertex figure, .

Truncated order-4 square tiling honeycomb

Truncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{4,4,4} or t_0,1{4,4,4}
Coxeter diagrams	↔ ↔ ↔
Cells	{4,4} t{4,4}
Faces	square {4}
Vertex figure	square pyramid
Coxeter groups	[4,4,4] [4^1,1,1]
Properties	Vertex-transitive

The truncated order-4 square tiling honeycomb, t_0,1{4,4,4}, has square tiling and truncated square tiling facets, with a square pyramid vertex figure.

Bitruncated order-4 square tiling honeycomb

Bitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	2t{4,4,4} or t_1,2{4,4,4}
Coxeter diagrams	↔ ↔ ↔
Cells	t{4,4}
Faces	square {4}, octagon {8}
Vertex figure	tetragonal disphenoid
Coxeter groups	[[4,4,4]] [4^1,1,1] [4^[4]]
Properties	Vertex-transitive, edge-transitive, cell-transitive

The bitruncated order-4 square tiling honeycomb, t_1,2{4,4,4}, has truncated square tiling facets, with a tetragonal disphenoid vertex figure.

Cantellated order-4 square tiling honeycomb

The cantellated order-4 square tiling honeycomb, is the same thing as the rectified square tiling honeycomb, .

Cantitruncated order-4 square tiling honeycomb

The cantitruncated order-4 square tiling honeycomb, is the same thing as the truncated square tiling honeycomb, .

Runcinated order-4 square tiling honeycomb

Runcinated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t_0,3{4,4,4}
Coxeter diagrams	↔ ↔
Cells	{4,4} {4,3}
Faces	square {4}
Vertex figure	square antiprism
Coxeter groups	[[4,4,4]]
Properties	Vertex-transitive, edge-transitive

The runcinated order-4 square tiling honeycomb, t_0,3{4,4,4}, has square tiling and cube facets, with a square antiprism vertex figure.

Runcitruncated order-4 square tiling honeycomb

Runcitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t_0,1,3{4,4,4}
Coxeter diagrams	↔
Cells	t{4,4} rr{4,4} {4,3} {8}x{}
Faces	square {4} Octagon {8}
Vertex figure	square pyramid
Coxeter groups	[4,4,4]
Properties	Vertex-transitive

The runcitruncated order-4 square tiling honeycomb, t_0,1,3{4,4,4}, has square tiling, truncated square tiling and cube facets, with a square pyramid vertex figure.

Omnitruncated order-4 square tiling honeycomb

Omnitruncated order-4 square tiling honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t_0,1,2,3{4,4,4}
Coxeter diagrams
Cells	{4,4} {8}x{}
Faces	square {4} Octagon {8}
Vertex figure	Phyllic disphenoid
Coxeter groups	[[4,4,4]]
Properties	Vertex-transitive

The omnitruncated order-4 square tiling honeycomb, t_0,1,2,3{4,4,4}, has truncated square tiling and octagonal prism facets, with a tetrahedron vertex figure.

Quarter order-4 square tiling honeycomb

The quarter order-4 square tiling honeycomb, q{4,4,4}, has truncated square tiling and octagonal prism facets, with a tetrahedron vertex figure.

Honeycomb name Coxeter diagram	Cells by location (and count around each vertex)				Vertex figure
Honeycomb name Coxeter diagram	0	1	2	3	Vertex figure
quarter order-4 square ↔	(4.8.8)	(4.4.4.4)	(4.4.4.4)	(4.8.8)

References

↑ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III

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
- Norman W. Johnson and Asia Ivic Weiss Quadratic Integers and Coxeter Groups PDF Canad. J. Math. Vol. 51 (6), 1999 pp. 1307–1336

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