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} {3,41,1} |
| Coxeter diagrams |       ↔    ↔   ↔  
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| Cells | octahedron {3,4} |
| Faces | triangle {3} |
| Edge figure | square {4} |
| Vertex figure | square tiling, {4,4}   
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| 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 
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 | |||||||||||
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 {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}
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r{4,4,3}
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t{4,4,3}
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rr{4,4,3}
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t0,3{4,4,3}
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tr{4,4,3}
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t0,1,3{4,4,3}
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t0,1,2,3{4,4,3}
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| {3,4,4}
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r{3,4,4}
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t{3,4,4}
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rr{3,4,4}
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2t{3,4,4}
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tr{3,4,4}
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t0,1,3{3,4,4}
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t0,1,2,3{3,4,4}
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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
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| Cells |  {2,4}
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 {3,4}
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{4,4}
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 {5,4}
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 {6,4}
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 {∞,4}
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It a part of a sequence of regular polychora and honeycombs with octahedral cells.
| {3,4,p} polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Space | S3 | H3 | |||||||||
| Form | Finite | Paracompact | Noncompact | ||||||||
| Name | {3,4,3} 
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{3,4,4}
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{3,4,5}
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{3,4,6}
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{3,4,7}
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{3,4,8} 
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... {3,4,∞}
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| Vertex figure |
 {4,3}
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{4,4} 
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 {4,5}
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 {4,6}
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 {4,7}
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 {4,8}
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 {4,∞}
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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 | ↔ ↔ ↔
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| Cells | r{4,3}  {4,4} 
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| Faces | triangular {3} square {4} |
| Vertex figure | 
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| 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 | ↔ ↔ ↔
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| Cells | t{3,4}  {4,4}
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| Faces | square {4} hexagon {6} |
| Vertex figure | 
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| 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 |  ↔
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| Cells | rr{3,4}  r{4,4} 
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| 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 | ↔
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| Cells | tr{3,4}  r{4,4}
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| 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 | ↔
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| Cells | t{3,4}  rr{4,4} 
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| 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 | ↔  ↔  ↔
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| 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
- ^ 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 Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336