Order-4 octahedral honeycomb
| Order-4 octahedral 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 | {3,4} 
|
| Faces | triangle {3} |
| Edge figure | square {4} |
| Vertex figure | square tiling, {4,4} 40px 40px 
|
| Dual | Square tiling honeycomb, {4,4,3} |
| Coxeter groups | [math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4] [math]\displaystyle{ \overline{O}_3 }[/math], [3,41,1] |
| Properties | Regular |
The order-4 octahedral honeycomb is a regular paracompact honeycomb in hyperbolic 3-space. It is 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 ideal octahedra around each edge, and infinite octahedra around each vertex in a square tiling vertex figure.[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 two alternating types (colors) of octahedral cells: ↔ .
A second half symmetry is [3,4,1+,4]: ↔ .
A higher index sub-symmetry, [3,4,4*], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: .
This honeycomb contains and 
that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings 
and 
, respectively:
Related polytopes and honeycombs
The order-4 octahedral honeycomb is a regular hyperbolic honeycomb in 3-space, and is one of eleven regular paracompact honeycombs.
There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form.
It is a part of a sequence of honeycombs with a square tiling vertex figure:
It a part of a sequence of regular polychora and honeycombs with octahedral cells:
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 | triangle {3} square {4} |
| Vertex figure |  square prism |
| Coxeter groups | [math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4] [math]\displaystyle{ \overline{O}_3 }[/math], [3,41,1] |
| Properties | Vertex-transitive, edge-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 |  square pyramid |
| Coxeter groups | [math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4] [math]\displaystyle{ \overline{O}_3 }[/math], [3,41,1] |
| 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.
Bitruncated order-4 octahedral honeycomb
The bitruncated order-4 octahedral honeycomb is the same as the bitruncated square tiling honeycomb.
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}  {}x4 40px r{4,4} 
|
| Faces | triangle {3} square {4} |
| Vertex figure |  wedge |
| Coxeter groups | [math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4] [math]\displaystyle{ \overline{O}_3 }[/math], [3,41,1] |
| Properties | Vertex-transitive |
The cantellated order-4 octahedral honeycomb, t0,2{3,4,4}, has rhombicuboctahedron, cube, and square tiling facets, with a wedge 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}  {}x{4} 40px t{4,4} 
|
| Faces | square {4} hexagon {6} octagon {8} |
| Vertex figure |  mirrored sphenoid |
| Coxeter groups | [math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4] [math]\displaystyle{ \overline{O}_3 }[/math], [3,41,1] |
| Properties | Vertex-transitive |
The cantitruncated order-4 octahedral honeycomb, t0,1,2{3,4,4}, has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.
Runcinated order-4 octahedral honeycomb
The runcinated order-4 octahedral honeycomb is the same as the runcinated square tiling honeycomb.
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} {6}x{} 40px rr{4,4} 
|
| Faces | square {4} hexagon {6} octagon {8} |
| Vertex figure |  square pyramid |
| Coxeter groups | [math]\displaystyle{ \overline{R}_3 }[/math], [3,4,4] |
| Properties | Vertex-transitive |
The runcitruncated order-4 octahedral honeycomb, t0,1,3{3,4,4}, has truncated octahedron, hexagonal prism, and square tiling facets, with a square pyramid vertex figure.
Runcicantellated order-4 octahedral honeycomb
The runcicantellated order-4 octahedral honeycomb is the same as the runcitruncated square tiling honeycomb.
Omnitruncated order-4 octahedral honeycomb
The omnitruncated order-4 octahedral honeycomb is the same as the omnitruncated square tiling honeycomb.
Snub order-4 octahedral honeycomb
| Snub order-4 octahedral honeycomb | |
|---|---|
| Type | Paracompact scaliform honeycomb |
| Schläfli symbols | s{3,4,4} |
| Coxeter diagrams | ↔  ↔  ↔
|
| Cells | square tiling icosahedron square pyramid |
| Faces | triangle {3} square {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 tiling, and icosahedron facets.
See also
- Convex uniform honeycombs in hyperbolic space
- Regular tessellations of hyperbolic 3-space
- Paracompact uniform honeycombs
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
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