View source for Order-4 octahedral honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Order-4 octahedral honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[File:H3 344 CC center.png|320px]]<BR>[[Perspective projection]] view<BR>within [[Poincaré disk model]]
|-
|bgcolor=#e7dcc3|Type||[[List of regular polytopes#Tessellations of hyperbolic 3-space|Hyperbolic regular honeycomb]]<BR>[[Paracompact uniform honeycomb#.3B4.2C4.2C4.5D family|Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||{3,4,4}<BR>{3,4<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|4|node|4|node}}<BR>{{CDD|node_1|3|node|split1-44|nodes}} ↔ {{CDD|node_1|3|node|4|node|4|node_h0}}<BR>{{CDD|node_1|split1|nodes|2a2b-cross|nodes}} ↔ {{CDD|node_1|3|node|4|node_h0|4|node}}<BR>{{CDD|branchu|split2|node_1|split1|branchu}} ↔ {{CDD|node_1|3|node|4|node_g|4sg|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[octahedron|{3,4}]] [[File:Uniform polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}
|-
|bgcolor=#e7dcc3|[[Edge figure]]||[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[square tiling]], {4,4}<BR>[[File:Square tiling uniform coloring 1.svg|40px]] [[File:Square tiling uniform coloring 7.png|40px]] [[File:Square tiling uniform coloring 8.png|40px]] [[File:Square tiling uniform coloring 9.png|40px]]
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[Square tiling honeycomb]], {4,4,3}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>\overline{R}_3</math>, [3,4,4]<BR><math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Regular
|}
The '''order-4 octahedral honeycomb''' is a regular paracompact honeycomb in [[Hyperbolic space|hyperbolic 3-space]]. It is ''paracompact'' because it has infinite [[vertex figure]]s, with all vertices as [[ideal point]]s at infinity. Given by [[Schläfli symbol]] {3,4,4}, it has four [[ideal polyhedron|ideal]] [[octahedra]] around each edge, and infinite octahedra around each vertex in a [[square tiling]] [[vertex figure]].<ref>Coxeter ''The Beauty of Geometry'', 1999, Chapter 10, Table III</ref>

{{Honeycomb}}

== Symmetry ==

A half symmetry construction, [3,4,4,1<sup>+</sup>], exists as {3,4<sup>1,1</sup>}, with two alternating types (colors) of octahedral cells: {{CDD|node_1|3|node|4|node|4|node_h0}} ↔ {{CDD|node_1|3|node|split1-44|nodes}}.

A second half symmetry is [3,4,1<sup>+</sup>,4]: {{CDD|node_1|3|node|4|node_h0|4|node}} ↔ {{CDD|node_1|split1|nodes|2a2b-cross|nodes}}.

A higher index sub-symmetry, [3,4,4<sup>*</sup>], which is index 8, exists with a pyramidal fundamental domain, [((3,∞,3)),((3,∞,3))]:  {{CDD|branchu|split2|node_1|split1|branchu}}.

This honeycomb contains {{CDD|node_1|split1|branchu}} and {{CDD|node_1|3|node|ultra|node}} that tile 2-[[hypercycle (geometry)|hypercycle]] surfaces, which are similar to the paracompact [[infinite-order triangular tiling]]s {{CDD|node_1|split1|branch|labelinfin}} and {{CDD|node_1|3|node|infin|node}}, respectively:<BR>
: [[File:H2chess 23ib.png|120px]]

== Related polytopes and honeycombs ==
The order-4 octahedral honeycomb is a [[List of regular polytopes#Tessellations of hyperbolic 3-space|regular hyperbolic honeycomb]] in 3-space, and is one of eleven regular paracompact honeycombs. 
{{Regular_paracompact_H3_honeycombs}}

There are [[Paracompact uniform honeycomb#.5B4.2C4.2C3.5D family|fifteen uniform honeycombs]] in the [3,4,4] [[Coxeter group]] family, including this regular form.
{{443_family}}

It is a part of a sequence of honeycombs with a [[square tiling]] vertex figure:
{{Square tiling vertex figure tessellations}}

It a part of a sequence of [[regular polychora]] and honeycombs with [[octahedron|octahedral]] [[cell (geometry)|cells]]:
{{Octahedral cell tessellations}}

=== Rectified order-4 octahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified order-4 octahedral honeycomb 
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||r{3,4,4} or t<sub>1</sub>{3,4,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|3|node_1|4|node|4|node}}<BR>{{CDD|node|3|node_1|split1-44|nodes}} ↔ {{CDD|node|3|node_1|4|node|4|node_h0}}<BR>{{CDD|node|split1|nodes_11|2a2b-cross|nodes}} ↔ {{CDD|node|3|node_1|4|node_h0|4|node}}<BR>{{CDD|branchu_11|split2|node|split1|branchu_11}} ↔ {{CDD|node|3|node_1|4|node_g|4sg|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[cuboctahedron|r{4,3}]] [[File:Uniform polyhedron-43-t1.svg|40px]]<BR>[[square tiling|{4,4}]][[File:Uniform tiling 44-t0.svg|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Rectified order-4 octahedral honeycomb verf.png|80px]]<BR>[[square prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>\overline{R}_3</math>, [3,4,4]<BR><math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive, edge-transitive
|}
The '''rectified order-4 octahedral honeycomb''', t<sub>1</sub>{3,4,4}, {{CDD|node|3|node_1|4|node|4|node}} has [[cuboctahedron]] and [[square tiling]] facets, with a [[square prism]] [[vertex figure]].

[[File:H3 344 CC center 0100.png|480px]]

{{Clear}}

=== Truncated order-4 octahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Truncated order-4 octahedral honeycomb 
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||t{3,4,4} or t<sub>0,1</sub>{3,4,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node_1|4|node|4|node}}<BR>{{CDD|node_1|3|node_1|split1-44|nodes}} ↔ {{CDD|node_1|3|node_1|4|node|4|node_h0}}<BR>{{CDD|node_1|split1|nodes_11|2a2b-cross|nodes}} ↔ {{CDD|node_1|3|node_1|4|node_h0|4|node}}<BR>{{CDD|branchu_11|split2|node_1|split1|branchu_11}} ↔ {{CDD|node_1|3|node_1|4|node_g|4sg|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated octahedron|t{3,4}]] [[File:Uniform polyhedron-43-t12.png|40px]]<BR>[[square tiling|{4,4}]][[File:Uniform tiling 44-t0.svg|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:truncated order-4 octahedral honeycomb verf.png|80px]]<BR>[[square pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>\overline{R}_3</math>, [3,4,4]<BR><math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''truncated order-4 octahedral honeycomb''', t<sub>0,1</sub>{3,4,4}, {{CDD|node_1|3|node_1|4|node|4|node}} has [[truncated octahedron]] and [[square tiling]] facets, with a [[square pyramid]] [[vertex figure]].

[[Image:H3 443-0011.png|480px]]

{{Clear}}

=== 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 ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantellated order-4 octahedral honeycomb 
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||rr{3,4,4} or t<sub>0,2</sub>{3,4,4}<BR>s<sub>2</sub>{3,4,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node|4|node_1|4|node}}<BR>{{CDD|node_h|3|node_h|4|node_1|4|node}}<BR>{{CDD|node_1|3|node|split1-44|nodes_11}} ↔ {{CDD|node_1|3|node|4|node_1|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|rr{3,4}]] [[File:Uniform polyhedron-43-t02.png|40px]]<BR>[[cube|{}x4]] [[File:Tetragonal prism.png|40px]]<BR>[[square tiling|r{4,4}]] [[File:Uniform tiling 44-t1.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Cantellated order-4 octahedral honeycomb verf.png|80px]]<BR>[[wedge (geometry)|wedge]]
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>\overline{R}_3</math>, [3,4,4]<BR><math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantellated order-4 octahedral honeycomb''', t<sub>0,2</sub>{3,4,4}, {{CDD|node_1|3|node|4|node_1|4|node}} has [[rhombicuboctahedron]], [[cube]], and [[square tiling]] facets, with a [[wedge (geometry)|wedge]] [[vertex figure]].

[[Image:H3_443-0101.png|480px]]
{{Clear}}

=== Cantitruncated order-4 octahedral honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Cantitruncated order-4 octahedral honeycomb 
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||tr{3,4,4} or t<sub>0,1,2</sub>{3,4,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node_1|4|node_1|4|node}}<BR>{{CDD|node_1|3|node_1|split1-44|nodes_11}} ↔ {{CDD|node_1|3|node_1|4|node_1|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated cuboctahedron|tr{3,4}]] [[File:Uniform polyhedron-43-t012.png|40px]]<BR>[[cube|{}x{4}]] [[File:Tetragonal prism.png|40px]]<BR>[[truncated square tiling|t{4,4}]] [[File:Uniform tiling 44-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Cantitruncated order-4 octahedral honeycomb verf.png|80px]]<BR>[[mirrored sphenoid]]
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>\overline{R}_3</math>, [3,4,4]<BR><math>\overline{O}_3</math>, [3,4<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''cantitruncated order-4 octahedral honeycomb''', t<sub>0,1,2</sub>{3,4,4}, {{CDD|node_1|3|node_1|4|node_1|4|node}} has [[truncated cuboctahedron]], [[cube]], and [[truncated square tiling]] facets, with a [[mirrored sphenoid]] [[vertex figure]].

[[Image:H3_443-0111.png|480px]]

{{Clear}}

=== 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 ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Runcitruncated order-4 octahedral honeycomb 
|-
|bgcolor=#e7dcc3|Type||[[Paracompact uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||t<sub>0,1,3</sub>{3,4,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_1|3|node_1|4|node|4|node_1}}<BR>{{CDD|node_1|split1|nodes_11|2a2b-cross|nodes_11}} ↔ {{CDD|node_1|3|node_1|4|node_h0|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[truncated octahedron|t{3,4}]] [[File:Uniform polyhedron-43-t12.png|40px]]<BR>[[hexagonal prism|{6}x{}]] [[File:Hexagonal prism.png|40px]]<BR>[[square tiling|rr{4,4}]] [[File:Uniform tiling 44-t02.svg|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[square]] {4}<BR>[[hexagon]] {6}<BR>[[octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Runcitruncated order-4 octahedral honeycomb verf.png|80px]]<BR>[[square pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||<math>\overline{R}_3</math>, [3,4,4]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''runcitruncated order-4 octahedral honeycomb''', t<sub>0,1,3</sub>{3,4,4}, {{CDD|node_1|3|node_1|4|node|4|node_1}} has [[truncated octahedron]], [[hexagonal prism]], and [[square tiling]] facets, with a [[square pyramid]] [[vertex figure]].

[[Image:H3 443-1011.png|480px]]

{{Clear}}

=== 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 ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Snub order-4 octahedral honeycomb 
|-
|bgcolor=#e7dcc3|Type||Paracompact scaliform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||s{3,4,4}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node_h|3|node_h|4|node|4|node}}<BR>{{CDD|node_h|3|node_h|split1-44|nodes}} ↔ {{CDD|node_h|3|node_h|4|node|4|node_h0}}<BR>{{CDD|node|split1-44|nodes_hh|split2|node_h}}<BR>{{CDD|node_h|split1|nodes_hh|2a2b-cross|nodes}} ↔ {{CDD|node_h|3|node_h|4|node_h0|4|node}}<BR>{{CDD|branchu_hh|split2|node_h|split1|branchu_hh}} ↔ {{CDD|node_h|3|node_h|4|node_g|4sg|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[square tiling]]<BR>[[icosahedron]]<BR>[[square pyramid]]
|-
|bgcolor=#e7dcc3|Faces||[[triangle]] {3}<BR>[[square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram#Ranks 4.E2.80.9310|Coxeter groups]]||[4,4,3<sup>+</sup>]<BR>[4<sup>1,1</sup>,3<sup>+</sup>]<BR>[(4,4,(3,3)<sup>+</sup>)]
|-
|bgcolor=#e7dcc3|Properties||Vertex-transitive
|}
The '''snub order-4 octahedral honeycomb''', s{3,4,4}, has Coxeter diagram {{CDD|node_h|3|node_h|4|node|4|node}}. It is a [[scaliform honeycomb]], with [[square pyramid]], [[square tiling]], and [[icosahedron]] facets.

{{Clear}}

== See also ==
* [[Convex uniform honeycombs in hyperbolic space]]
* [[List of regular polytopes#Tessellations of hyperbolic 3-space|Regular tessellations of hyperbolic 3-space]]
* [[Paracompact uniform honeycomb]]s

== References ==
{{reflist}}
*[[H.S.M. Coxeter|Coxeter]], ''[[Regular Polytopes (book)|Regular Polytopes]]'', 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294–296)
* ''The Beauty of Geometry: Twelve Essays'' (1999), Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}} (Chapter 10, [http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf Regular Honeycombs in Hyperbolic Space]) Table III
* [[Jeffrey Weeks (mathematician)|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 (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript
** [[Norman Johnson (mathematician)|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 [https://cms.math.ca/cjm/v51/weisscox8.pdf Quadratic Integers and Coxeter Groups] [[PDF]] Can. J. Math. Vol. 51 (6), 1999 pp.&nbsp;1307–1336

[[Category:Regular 3-honeycombs]]