Order-4 octahedral honeycomb

Order-4 octahedral tiling honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols{3,4,4}
{3,41,1}
Coxeter diagrams
↔
↔
↔
Cellsoctahedron {3,4}
Facestriangle {3}
Edge figuresquare {4}
Vertex figuresquare tiling, {4,4}
DualSquare tiling honeycomb, {4,4,3}
Coxeter groups[4,4,3]
[3,41,1]
PropertiesRegular

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 {3,41,1}, 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.

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}
 t1{4,4,3}
 t0,1{4,4,3}
 t0,2{4,4,3}
 t0,3{4,4,3}
 t0,1,2{4,4,3}
 t0,1,3{4,4,3}
 t0,1,2,3{4,4,3}
{3,4,4}
 t1{3,4,4}
 t0,1{3,4,4}
 t0,2{3,4,4}
 t1,2{3,4,4}
 t0,1,2{3,4,4}
 t0,1,3{3,4,4}
 t0,1,2,3{3,4,4}

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

Rectified order-4 octahedral honeycomb

Rectified order-4 octahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{3,4,4} or t1{3,4,4}
Coxeter diagrams
↔
↔
↔
Cellsr{4,3}
{4,4}
Facestriangular {3}
square {4}
Vertex figure
Coxeter groups[4,4,3]
PropertiesVertex-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
TypeParacompact uniform honeycomb
Schläfli symbolst{3,4,4} or t0,1{3,4,4}
Coxeter diagrams
↔
↔
↔
Cellst{3,4}
{4,4}
Facessquare {4}
hexagon {6}
Vertex figure
Coxeter groups[4,4,3]
PropertiesVertex-transitive

The truncated order-4 octahedral honeycomb, t0,1{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
TypeParacompact uniform honeycomb
Schläfli symbolsrr{3,4,4} or t0,2{3,4,4}
s2{3,4,4}
Coxeter diagrams

↔
Cellsrr{3,4}
r{4,4}
Facestriangle {3}
square {4}
Vertex figure
triangular prism
Coxeter groups[4,4,3]
PropertiesVertex-transitive

The cantellated order-4 octahedral honeycomb, t0,2{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
TypeParacompact uniform honeycomb
Schläfli symbolstr{3,4,4} or t0,1,2{3,4,4}
Coxeter diagrams
↔
Cellstr{3,4}
r{4,4}
Facessquare {4}
hexagonal {6}
octagonal {8}
Vertex figure
tetrahedron
Coxeter groups[4,4,3]
PropertiesVertex-transitive

The cantitruncated order-4 octahedral honeycomb, t0,1,2{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
TypeParacompact uniform honeycomb
Schläfli symbolst0,1,3{3,4,4}
Coxeter diagrams
↔
Cellst{3,4}
rr{4,4}
Facestriangle {3}
square {4}
octagonal {8}
Vertex figure
square pyramid
Coxeter groups[4,4,3]
PropertiesVertex-transitive

The runcitruncated order-4 octahedral honeycomb, t0,1,3{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
TypeParacompact scaliform honeycomb
Schläfli symbolss{3,4,4}
Coxeter diagrams
↔

↔
↔
Cells
square tiling
icosahedra
square pyramid
Faces{3}
{4}
Vertex figure
Coxeter groups[4,4,3+]
[41,1,3+]
[(4,4,(3,3)+)]
PropertiesVertex-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.

See also

References

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