Order-4 octahedral honeycomb

Order-4 octahedral tiling honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{3,4,4} {3,4^1,1}
Coxeter diagrams	↔ ↔ ↔
Cells	octahedron {3,4}
Faces	triangle {3}
Edge figure	square {4}
Vertex figure	square tiling, {4,4}
Dual	Square tiling honeycomb, {4,4,3}
Coxeter groups	[4,4,3] [3,4^1,1]
Properties	Regular

In the geometry of hyperbolic 3-space, the order-4 octahedral honeycomb is a regular paracompact honeycomb. It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. Given by Schläfli symbol {3,4,4}, it has four octahedra, {3,4} around each edge, and infinite octahedra 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

A half symmetry construction, [3,4,4,1⁺], exists as {3,4^1,1}, with alternating two types (colors) of octahedral cells. ↔ . A second half symmetry, [3,4,1⁺,4]: ↔ . A higher index subsymmetry, [3,4,4^*], index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: .

This honeycomb contains , that tile 2-hypercycle surfaces, similar to the paracompact tiling or

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

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


{3,4,4}	r{3,4,4}	t{3,4,4}	rr{3,4,4}	2t{3,4,4}	tr{3,4,4}	t_0,1,3{3,4,4}	t_0,1,2,3{3,4,4}

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}

Rectified order-4 octahedral honeycomb

Rectified order-4 octahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	r{3,4,4} or t₁{3,4,4}
Coxeter diagrams	↔ ↔ ↔
Cells	r{4,3} {4,4}
Faces	triangular {3} square {4}
Vertex figure
Coxeter groups	[4,4,3]
Properties	Vertex-transitive

The rectified order-4 octahedral honeycomb, t₁{3,4,4}, has cuboctahedron and square tiling facets, with a square prism vertex figure.

Truncated order-4 octahedral honeycomb

Truncated order-4 octahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	t{3,4,4} or t_0,1{3,4,4}
Coxeter diagrams	↔ ↔ ↔
Cells	t{3,4} {4,4}
Faces	square {4} hexagon {6}
Vertex figure
Coxeter groups	[4,4,3]
Properties	Vertex-transitive

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

Cantellated order-4 octahedral honeycomb

Cantellated order-4 octahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	rr{3,4,4} or t_0,2{3,4,4} s₂{3,4,4}
Coxeter diagrams	↔
Cells	rr{3,4} r{4,4}
Faces	triangle {3} square {4}
Vertex figure	triangular prism
Coxeter groups	[4,4,3]
Properties	Vertex-transitive

The cantellated order-4 octahedral honeycomb, t_0,2{3,4,4}, has rhombicuboctahedron and square tiling facets, with a triangular prism vertex figure.

Cantitruncated order-4 octahedral honeycomb

Cantitruncated order-4 octahedral honeycomb
Type	Paracompact uniform honeycomb
Schläfli symbols	tr{3,4,4} or t_0,1,2{3,4,4}
Coxeter diagrams	↔
Cells	tr{3,4} r{4,4}
Faces	square {4} hexagonal {6} octagonal {8}
Vertex figure	tetrahedron
Coxeter groups	[4,4,3]
Properties	Vertex-transitive

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

Runcitruncated order-4 octahedral honeycomb

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

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

Snub order-4 octahedral honeycomb

Truncated order-4 octahedral honeycomb
Type	Paracompact scaliform honeycomb
Schläfli symbols	s{3,4,4}
Coxeter diagrams	↔ ↔ ↔
Cells	square tiling icosahedra square pyramid
Faces	{3} {4}
Vertex figure
Coxeter groups	[4,4,3⁺] [4^1,1,3⁺] [(4,4,(3,3)⁺)]
Properties	Vertex-transitive

The snub order-4 octahedral honeycomb, s{3,4,4}, has Coxeter diagram . It is a scaliform honeycomb, with square pyramid, square tilings, and icosahedra.

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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