Order-4 octahedral honeycomb

Order-4 octahedral honeycomb
File:H3 344 CC center.png Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{3,4,4} {3,41,1}
Coxeter diagrams	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	{3,4}
Faces	triangle {3}
Edge figure	square {4}
Vertex figure	square tiling, {4,4} Error creating thumbnail: File:Square tiling uniform coloring 7.png Error creating thumbnail: File:Square tiling uniform coloring 9.png
Dual	Square tiling honeycomb, {4,4,3}
Coxeter groups	${\bar{R}}_3$, [3,4,4] ${\bar{O}}_3$, [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.

Symmetry

A half symmetry construction, [3,4,4,1⁺], exists as {3,4^1,1}, with two alternating types (colors) of octahedral cells: Template:CDD ↔ Template:CDD.

A second half symmetry is [3,4,1⁺,4]: Template:CDD ↔ Template:CDD.

A higher index sub-symmetry, [3,4,4^*], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]: Template:CDD.

This honeycomb contains Template:CDD and Template:CDD that tile 2-hypercycle surfaces, which are similar to the paracompact infinite-order triangular tilings Template:CDD and Template:CDD, respectively:

File:H2chess 23ib.png

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. Template:Regular paracompact H3 honeycombs

There are fifteen uniform honeycombs in the [3,4,4] Coxeter group family, including this regular form. Template:443 family

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

It a part of a sequence of regular polychora and honeycombs with octahedral cells: Template:Octahedral cell tessellations

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	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	r{4,3} File:Uniform polyhedron-43-t1.svg {4,4}File:Uniform tiling 44-t0.svg
Faces	triangle {3} square {4}
Vertex figure	square prism
Coxeter groups	${\bar{R}}_3$, [3,4,4] ${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive, edge-transitive

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

Error creating thumbnail:

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	Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
Cells	t{3,4} File:Uniform polyhedron-43-t12.png {4,4}File:Uniform tiling 44-t0.svg
Faces	square {4} hexagon {6}
Vertex figure	File:Truncated order-4 octahedral honeycomb verf.png square pyramid
Coxeter groups	${\bar{R}}_3$, [3,4,4] ${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive

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

File:H3 443-0011.png

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	Template:CDD Template:CDD Template:CDD ↔ Template:CDD
Cells	rr{3,4} Error creating thumbnail: {}x4 File:Tetragonal prism.png r{4,4} File:Uniform tiling 44-t1.png
Faces	triangle {3} square {4}
Vertex figure	File:Cantellated order-4 octahedral honeycomb verf.png wedge
Coxeter groups	${\bar{R}}_3$, [3,4,4] ${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive

The cantellated order-4 octahedral honeycomb, t_0,2{3,4,4}, Template:CDD has rhombicuboctahedron, cube, and square tiling facets, with a wedge vertex figure.

File:H3 443-0101.png Template:Clear

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	tr{3,4} Error creating thumbnail: {}x{4} File:Tetragonal prism.png t{4,4} File:Uniform tiling 44-t01.png
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	mirrored sphenoid
Coxeter groups	${\bar{R}}_3$, [3,4,4] ${\bar{O}}_3$, [3,41,1]
Properties	Vertex-transitive

The cantitruncated order-4 octahedral honeycomb, t_0,1,2{3,4,4}, Template:CDD has truncated cuboctahedron, cube, and truncated square tiling facets, with a mirrored sphenoid vertex figure.

File:H3 443-0111.png

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	Template:CDD Template:CDD ↔ Template:CDD
Cells	t{3,4} File:Uniform polyhedron-43-t12.png {6}x{} File:Hexagonal prism.png rr{4,4} File:Uniform tiling 44-t02.svg
Faces	square {4} hexagon {6} octagon {8}
Vertex figure	File:Runcitruncated order-4 octahedral honeycomb verf.png square pyramid
Coxeter groups	${\bar{R}}_3$, [3,4,4]
Properties	Vertex-transitive

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

File:H3 443-1011.png

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	Template:CDD Template:CDD ↔ Template:CDD Template:CDD Template:CDD ↔ Template:CDD Template:CDD ↔ Template:CDD
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 Template:CDD. 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, Regular Polytopes, 3rd. ed., Dover Publications, 1973. Template:Isbn. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
• The Beauty of Geometry: Twelve Essays (1999), Dover Publications, Template:LCCN, Template:Isbn (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
• Jeffrey R. Weeks The Shape of Space, 2nd edition Template:Isbn (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