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Order-4 octahedral honeycomb

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Order-4 octahedral tiling honeycomb

Perspective projection view
within Poincaré disk model
Type Hyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols {3,4,4}
{3,41,1}
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]
[3,41,1]
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 {3,41,1}, 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

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}
t0,3{4,4,3}
tr{4,4,3}
t0,1,3{4,4,3}
t0,1,2,3{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}
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:

{p,4,4} honeycombs
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 t0,1{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, 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
Type Paracompact uniform honeycomb
Schläfli symbols rr{3,4,4} or t0,2{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, 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
Type Paracompact uniform honeycomb
Schläfli symbols tr{3,4,4} or t0,1,2{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, 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
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{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, 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
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+]
[41,1,3+]
[(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.

See also

References

  1. ^ 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