View source for Tetrahedral-octahedral honeycomb

{{Short description|Quasiregular space-filling tesselation}}
{| class="wikitable" align="right" style="margin-left:10px" width="360"
!bgcolor=#e7dcc3 colspan=2|Alternated cubic honeycomb 
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Alternated cubic tiling.png|210px]]&nbsp;[[File:HC P1-P3.png|130px]]
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Alternated hypercubic honeycomb]]<BR>[[Simplectic honeycomb]]
|-
|bgcolor=#e7dcc3|Indexing<ref>For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).</ref>
|J<sub>21,31,51</sub>, A<sub>2</sub><br>W<sub>9</sub>, G<sub>1</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||h{4,3,4}<br>{3<sup>[4]</sup>}<br>ht<sub>0,3</sub>{4,3,4}<BR>h{4,4}h{∞}<BR>ht<sub>0,2</sub>{4,4}h{∞}<BR>h{∞}h{∞}h{∞}<BR>s{∞}s{∞}s{∞}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|nodes_10ru|split2|node|4|node}} = {{CDD|node_h1|4|node|3|node|4|node}}<br>{{CDD|node_1|split1|nodes|split2|node}} = {{CDD|node_h1|4|node|split1|nodes}}<BR>{{CDD|node_h|4|node|3|node|4|node_h}}<br>{{CDD|node_h|4|node|4|node|2|node_h|infin|node}}<br>{{CDD|node_h|4|node|4|node_h|2|node_h|infin|node}}<br>{{CDD|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}} = {{CDD|node_h|4|node_g|3sg|node_g|4|node}}<br>{{CDD|node_h|infin|node_h|2|node_h|infin|node_h|2|node_h|infin|node_h}} = {{CDD|node_h|4|node_g|3sg|node_g|4g|node_g}}
|-
|bgcolor=#e7dcc3|Cells||[[Tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]<BR>[[Octahedron|{3,4}]] [[File:Uniform polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}
|-
|bgcolor=#e7dcc3|Edge figure||[{3,3}.{3,4}]<sup>2</sup><br>([[Rectangle|rectangle]])
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[Image:Alternated cubic honeycomb verf.svg|80px]]80px<br>80px80px<br>([[Cuboctahedron|cuboctahedron]])
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]||Fm{{overline|3}}m (225)
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_3</math>, [4,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|Dual||Dodecahedrille<br>[[Rhombic dodecahedral honeycomb|rhombic dodecahedral honeycomb]]<BR>Cell: [[File:Dodecahedrille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive, edge-transitive, quasiregular honeycomb
|}
The '''tetrahedral-octahedral honeycomb''', '''alternated cubic honeycomb''' is a quasiregular space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of alternating regular [[Octahedron|octahedra]] and [[Tetrahedron|tetrahedra]] in a ratio of 1:2.

Other names include '''half cubic honeycomb''', '''half cubic cellulation''', or '''tetragonal disphenoidal cellulation'''. [[Biography:John Horton Conway|John Horton Conway]] calls this honeycomb a '''tetroctahedrille''', and its dual a [[Rhombic dodecahedral honeycomb|dodecahedrille]].

R. Buckminster Fuller combines the two words <u>oct</u>ahedron and <u>tet</u>rahedron into '''octet''' truss, a rhombohedron consisting of one octahedron (or two square pyramids) and two opposite tetrahedra.

It is vertex-transitive with 8 tetrahedra and 6 octahedra around each [[Vertex (geometry)|vertex]]. It is edge-transitive with 2 tetrahedra and 2 octahedra alternating on each edge.

{{Honeycomb}}

It is part of an infinite family of [[Uniform honeycomb|uniform honeycomb]]s called [[Alternated hypercubic honeycomb|alternated hypercubic honeycomb]]s, formed as an [[Alternation (geometry)|alternation]] of a hypercubic honeycomb and being composed of [[Demihypercube|demihypercube]] and [[Cross-polytope|cross-polytope]] facets. It is also part of another infinite family of uniform honeycombs called [[Simplectic honeycomb|simplectic honeycomb]]s.

In this case of 3-space, the [[Cubic honeycomb|cubic honeycomb]] is alternated, reducing the cubic cells to tetrahedra, and the deleted vertices create octahedral voids. As such it can be represented by an extended [[Schläfli symbol]] h{4,3,4} as containing ''half'' the vertices of the {4,3,4} cubic honeycomb.

There is a similar honeycomb called gyrated tetrahedral-octahedral honeycomb which has layers rotated 60 degrees so half the edges have neighboring rather than alternating tetrahedra and octahedra.

The tetrahedral-octahedral honeycomb can have its symmetry doubled by placing tetrahedra on the octahedral cells, creating a nonuniform honeycomb consisting of [[Tetrahedron|tetrahedra]] and [[Octahedron|octahedra]] (as triangular antiprisms). Its vertex figure is an [[Truncated triakis tetrahedron#Full truncation|order-3 truncated triakis tetrahedron]]. This honeycomb is the dual of the [[Triakis truncated tetrahedral honeycomb|triakis truncated tetrahedral honeycomb]], with [[Triakis truncated tetrahedron|triakis truncated tetrahedral]] cells.

==Cartesian coordinates==

For an ''alternated cubic honeycomb'', with edges parallel to the axes and with an edge length of 1, the Cartesian coordinates of the vertices are: (For all integral values: ''i'',''j'',''k'' with ''i''+''j''+''k'' [[Even number|even]])
:(i, j, k)

[[Image:TetraOctaHoneycomb-VertexConfig.svg|180px|thumb|This diagram shows an exploded view of the cells surrounding each vertex.]]

==Symmetry==
There are two reflective constructions and many alternated [[Cubic honeycomb#Uniform colorings|cubic honeycomb]] ones; examples:
{| class=wikitable
!Symmetry
!<math>{\tilde{B}}_3</math>, [4,3<sup>1,1</sup>]<br>= ½<math>{\tilde{C}}_3</math>, [1<sup>+</sup>,4,3,4]
!<math>{\tilde{A}}_3</math>, [3<sup>[4]</sup>]<br>= ½<math>{\tilde{B}}_3</math>, [1<sup>+</sup>,4,3<sup>1,1</sup>]
!<nowiki></nowiki>(4,3,4,2<sup>+</sup>)
![(4,3,4,2<sup>+</sup>)]
|-
![[Space group]]
!Fm{{overline|3}}m (225)
!F{{overline|4}}3m (216)
!I{{overline|4}}3m (217)
!P{{overline|4}}3m (215)
|-
!Image
|[[Image:Tetrahedral-octahedral honeycomb.png|160px]]
|[[Image:Tetrahedral-octahedral honeycomb2.png|160px]]
|
|
|-
!Types of tetrahedra
!1
!2
!3
!4
|-
!Coxeter<br>diagram
!{{CDD|nodes_10ru|split2|node|4|node}} = {{CDD|node_h1|4|node|3|node|4|node}}
! {{CDD|node_1|split1|nodes|split2|node}} = {{CDD|nodes|split2|node|4|node_h1}} = {{CDD|node_h0|4|node|3|node|4|node_h1}}
!{{CDD|branch|4a4b|branch_hh|label2}}
!{{CDD|node_h|4|node|3|node|4|node_h}}
|}

=== Alternated cubic honeycomb slices ===
The ''alternated cubic honeycomb'' can be sliced into sections, where new square faces are created from inside of the octahedron. Each slice will contain up and downward facing [[Square pyramid|square pyramid]]s and tetrahedra sitting on their edges. A second slice direction needs no new faces and includes alternating tetrahedral and octahedral. This slab honeycomb is a scaliform honeycomb rather than uniform because it has nonuniform cells.
{| class=wikitable width=320
!{{CDD|node_h|2x|node_h|4|node|4|node}}
!{{CDD|node_h|2x|node_h|6|node|3|node}}
|-
|[[File:Alternated cubic slab honeycomb.png|240px]]
|[[File:Tetroctahedric semicheck.png|240px]]
|}

=== Projection by folding ===
The ''alternated cubic honeycomb'' can be orthogonally projected into the planar [[Square tiling|square tiling]] by a geometric folding operation that maps one pairs of mirrors into each other. The projection of the ''alternated cubic honeycomb'' creates two offset copies of the square tiling [[Vertex arrangement|vertex arrangement]] of the plane:

{|class=wikitable
|-
![[Coxeter group|Coxeter<br>group]]
!<math>{\tilde{A}}_3</math>
!<math>{\tilde{C}}_2</math>
|- align=center
!Coxeter<br>diagram
|{{CDD|node_1|split1|nodes|split2|node}}
|{{CDD|node_1|4|node|4|node}}
|- align=center
!Image
|[[File:Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg|100px]]
|[[File:Uniform tiling 44-t0.svg|100px]]
|- align=center
!Name
|alternated cubic honeycomb
|[[Square tiling|square tiling]]
|}

==A3/D3 lattice==
Its [[Vertex arrangement|vertex arrangement]] represents an [[Root system#Explicit construction of the irreducible root systems|A<sub>3</sub> lattice]] or ''D<sub>3</sub> lattice''.<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html|title = The Lattice D3}}</ref><ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A3.html|title = The Lattice A3}}</ref> This lattice is known as the [[Cubic crystal system|face-centered cubic lattice]] in crystallography and is also referred to as the [[Close-packing of equal spheres#FCC and HCP Lattices|cubic close packed lattice]] as its vertices are the centers of a close-packing with equal spheres that achieves the highest possible average density.  The tetrahedral-octahedral honeycomb is the 3-dimensional case of a [[Simplectic honeycomb|simplectic honeycomb]]. Its Voronoi cell is a [[Rhombic dodecahedron|rhombic dodecahedron]], the dual of the [[Cuboctahedron|cuboctahedron]] vertex figure for the tet-oct honeycomb.

The D{{sup sub|+|3}} packing can be constructed by the union of two D<sub>3</sub> (or A<sub>3</sub>) lattices. The D{{sup sub|+|n}} packing is only a lattice for even dimensions. The kissing number is 2<sup>2</sup>=4, (2<sup>n-1</sup> for n<8, 240 for n=8, and 2n(n-1) for n>8).<ref>Conway (1998), p. 119</ref>
:{{CDD|node_1|split1|nodes|split2|node}} ∪ {{CDD|node|split1|nodes|split2|node_1}}

The A{{sup sub|*|3}} or D{{sup sub|*|3}} lattice (also called A{{sup sub|4|3}} or D{{sup sub|4|3}}) can be constructed by the union of all four A<sub>3</sub> lattices, and is identical to the [[Vertex arrangement|vertex arrangement]] of the disphenoid tetrahedral honeycomb, dual honeycomb of the uniform [[Bitruncated cubic honeycomb|bitruncated cubic honeycomb]]:<ref>{{Cite web|url=http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds3.html|title=The Lattice D3}}</ref>  It is also the body centered cubic, the union of two [[Cubic honeycomb|cubic honeycomb]]s in dual positions.
:{{CDD|node_1|split1|nodes|split2|node}} ∪ {{CDD|node|split1|nodes_10luru|split2|node}} ∪ {{CDD|node|split1|nodes_01lr|split2|node}} ∪ {{CDD|node|split1|nodes|split2|node_1}} = dual of {{CDD|node_1|split1|nodes_11|split2|node_1}} = {{CDD|node_1|4|node|3|node|4|node}} ∪ {{CDD|node|4|node|3|node|4|node_1}}.

The [[Kissing number|kissing number]] of the D{{sup sub|*|3}} lattice is 8<ref>Conway (1998), p. 120</ref> and its [[Voronoi tessellation]] is a [[Bitruncated cubic honeycomb|bitruncated cubic honeycomb]], {{CDD|branch_11|4a4b|nodes}}, containing all [[Truncated octahedron|truncated octahedral]] Voronoi cells, {{CDD|node|4|node_1|3|node_1}}.<ref>Conway (1998), p. 466</ref>

== Related honeycombs==

=== C3 honeycombs===
The [4,3,4], {{CDD|node|4|node|3|node|4|node}}, [[Coxeter group]] generates 15 permutations of uniform honeycombs, 9 with distinct geometry including the alternated cubic honeycomb. The [[Expansion (geometry)|expanded]] cubic honeycomb (also known as the runcinated tesseractic honeycomb) is geometrically identical to the cubic honeycomb.

=== B3 honeycombs===
The [4,3<sup>1,1</sup>], {{CDD|node|4|node|split1|nodes}}, [[Coxeter group]] generates 9 permutations of uniform honeycombs, 4 with distinct geometry including the alternated cubic honeycomb.

=== A3 honeycombs ===
This honeycomb is one of [[Uniform 5-polytope#Regular and uniform honeycombs|five distinct uniform honeycombs]]<ref>[http://mathworld.wolfram.com/Necklace.html], {{OEIS el|A000029}} 6-1 cases, skipping one with zero marks</ref> constructed by the <math>{\tilde{A}}_3</math> [[Coxeter group]]. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams: 
{{clear}}
=== Quasiregular honeycombs===

=== Cantic cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Cantic cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|nodes_10ru|split2|node_1|4|node}} = {{CDD|node_h1|4|node|3|node_1|4|node}}<br>{{CDD|node_1|split1|nodes_11|split2|node}} = {{CDD|node_h1|4|node|split1|nodes_11}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated octahedron|t{3,4}]] [[File:Uniform polyhedron-43-t12.png|40px]]<br>[[Cuboctahedron|r{4,3}]] 40px<br>[[Truncated tetrahedron|t{3,3}]] [[File:Uniform polyhedron-33-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}<BR>[[Hexagon|hexagon]] {6}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Truncated alternated cubic honeycomb verf.png|80px]]<BR>rectangular [[Pyramid (geometry)|pyramid]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||[4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math><br>[3<sup>[4]</sup>], <math>{\tilde{A}}_3</math>
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]||Fm{{overline|3}}m (225)
|-
|bgcolor=#e7dcc3|Dual|| half oblate octahedrille<BR>Cell: [[File:Half oblate octahedrille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''cantic cubic honeycomb''', '''cantic cubic cellulation''' or '''truncated half cubic honeycomb''' is a uniform space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[Truncated octahedron|truncated octahedra]], [[Cuboctahedron|cuboctahedra]] and [[Truncated tetrahedron|truncated tetrahedra]] in a ratio of 1:1:2. Its [[Vertex figure|vertex figure]] is a rectangular [[Pyramid (geometry)|pyramid]].

[[Biography:John Horton Conway|John Horton Conway]] calls this honeycomb a '''truncated tetraoctahedrille''', and its dual half oblate octahedrille.
:[[File:Truncated alternated cubic tiling.png|210px]]&nbsp;[[File:HC A1-A3-A4.png|130px]]

==== Symmetry ====
It has two different uniform constructions. The <math>{\tilde{A}}_3</math> construction can be seen with alternately colored [[Truncated tetrahedron|truncated tetrahedra]].
{| class="wikitable" width=320
!Symmetry
![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math><br>=<[3<sup>[4]</sup>]>
![3<sup>[4]</sup>], <math>{\tilde{A}}_3</math>
|-
![[Space group]]
!Fm{{overline|3}}m (225)
!F{{overline|4}}3m (216)
|- align=center
|Coloring
|[[Image:Truncated Alternated Cubic Honeycomb.svg|150px]]
|[[File:Truncated Alternated Cubic Honeycomb2.png|150px]]
|- align=center
!Coxeter
|{{CDD|nodes_10ru|split2|node_1|4|node}} = {{CDD|node_h1|4|node|3|node_1|4|node}}
|{{CDD|node_1|split1|nodes_11|split2|node}} = {{CDD|node_h1|4|node|split1|nodes_11}}
|- align=center
![[Vertex figure]]
||[[File:Truncated alternated cubic honeycomb verf.png|80px]]
||[[File:T012 quarter cubic honeycomb verf.png|80px]]
|}

==== Related honeycombs====
It is related to the cantellated cubic honeycomb. Rhombicuboctahedra are reduced to truncated octahedra, and cubes are reduced to truncated tetrahedra.
{| class=wikitable
|- align=center
|[[File:Cantellated cubic honeycomb.png|160px]]<br>cantellated cubic<br>{{CDD|node_1|4|node|3|node_1|4|node}}
|[[File:Truncated Alternated Cubic Honeycomb.svg|160px]]<br>Cantic cubic<br>{{CDD|node_h1|4|node|3|node_1|4|node}}<br>{{CDD|nodes_10ru|split2|node_1|4|node}}
|- align=center
|{{CDD|node_1|4|node|3|node_1}}, {{CDD|node|3|node_1|4|node}}, {{CDD|node_1|2|node_1|4|node}}<br>[[Rhombicuboctahedron|rr{4,3}]], [[Cuboctahedron|r{4,3}]], [[Cube|{4,3}]]
|{{CDD|node_h1|4|node|3|node_1}}, {{CDD|node|3|node_1|4|node}}, {{CDD|node_1|3|node_1|3|node}}<br>[[Truncated octahedron|t{3,4}]], [[Cuboctahedron|r{4,3}]], [[Truncated tetrahedron|t{3,3}]] 
|}
{{clear}}

=== Runcic cubic honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Runcic cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|nodes_10ru|split2|node|4|node_1}} = {{CDD|node_h1|4|node|3|node|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Rhombicuboctahedron|rr{4,3}]] [[File:Uniform polyhedron-43-t02.png|40px]]<br>[[Cube|{4,3}]] 40px<br>[[Tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcinated alternated cubic honeycomb verf.png|80px]]<br>triangular frustum
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_4</math>, [4,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]||Fm{{overline|3}}m (225)
|-
|bgcolor=#e7dcc3|Dual|| quarter cubille<BR>Cell: [[File:Quarter cubille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''runcic cubic honeycomb''' or '''runcic cubic cellulation''' is a uniform space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[Rhombicuboctahedron|rhombicuboctahedra]], [[Cube|cube]]s, and [[Tetrahedron|tetrahedra]] in a ratio of 1:1:2. Its [[Vertex figure|vertex figure]] is a triangular frustum, with a tetrahedron on one end, cube on the opposite end, and three rhombicuboctahedra around the trapezoidal sides.

[[Biography:John Horton Conway|John Horton Conway]] calls this honeycomb a '''3-RCO-trille''', and its dual quarter cubille.

: [[Image:Runcinated alternated cubic tiling.png|195px]][[File:HC A5-P2-P1.png|115px]]
==== Quarter cubille ====
The dual of a ''runcic cubic honeycomb'' is called a '''quarter cubille''', with Coxeter diagram {{CDD|node_fh|4|node|3|node|4|node_f1}}, with faces in 2 of 4 hyperplanes of the <math>{\tilde{B}}_4</math>, [4,3<sup>1,1</sup>] symmetry fundamental domain. 

Cells can be seen as 1/4 of dissected cube, using 4 vertices and the center. Four cells exist around 6 edges, and 3 cells around 3 edges.
:[[File:Quarter cubille cell.png|160px]]

====Related honeycombs====
It is related to the runcinated cubic honeycomb, with quarter of the cubes [[Alternation (geometry)|alternated]] into tetrahedra, and half [[Expansion (geometry)|expanded]] into rhombicuboctahedra.
{| class=wikitable width=320
|- align=center
|[[File:Runcinated cubic honeycomb.png|160px]]<br>Runcinated cubic<br>{{CDD|node_1|4|node|3|node|4|node_1}}
|[[File:Runcic cubic honeycomb.png|160px]]<br>Runcic cubic<br>{{CDD|node_h1|4|node|3|node|4|node_1}} = {{CDD|nodes_10ru|split2|node|4|node_1}}
|- align=center
|[[Cube|{4,3}]], [[Cube|{4,3}]], [[Cube|{4,3}]], [[Cube|{4,3}]]<br>{{CDD|node_1|4|node|3|node}}, {{CDD|node_1|4|node|2|node_1}}, {{CDD|node_1|2|node|4|node_1}}, {{CDD|node|3|node|4|node_1}}
|[[Tetrahedron|h{4,3}]], [[Rhombicuboctahedron|rr{4,3}]], [[Cube|{4,3}]]<br>{{CDD|node_h1|4|node|3|node}}, {{CDD|node_1|3|node|4|node_1}}, {{CDD|node|3|node|4|node_1}}
|}

This honeycomb can be divided on [[Truncated square tiling|truncated square tiling]] planes, using the octagons centers of the rhombicuboctahedra, creating [[Square cupola|square cupola]]e. This scaliform honeycomb is represented by Coxeter diagram {{CDD|node_h|2x|node_h|4|node|4|node_1}}, and symbol s<sub>3</sub>{2,4,4}, with coxeter notation symmetry [2<sup>+</sup>,4,4].
: [[File:Runcic_snub_244_honeycomb.png|320px]]. 
{{-}}
<HR>

=== Runcicantic cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Runcicantic cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h<sub>2,3</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|nodes_10ru|split2|node_1|4|node_1}} = {{CDD|node_h1|4|node|3|node_1|4|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Truncated cuboctahedron|tr{4,3}]] [[File:Uniform polyhedron-43-t012.png|40px]]<BR>[[Truncated cube|t{4,3}]] 40px<BR>[[Truncated tetrahedron|t{3,3}]] [[File:Uniform polyhedron-33-t01.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}<BR>[[Hexagon|hexagon]] {6}<BR>[[Octagon|octagon]] {8}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Runcitruncated alternate cubic honeycomb verf.png|80px]]<BR>mirrored sphenoid
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_4</math>, [4,3<sup>1,1</sup>]
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]||Fm{{overline|3}}m (225)
|-
|bgcolor=#e7dcc3|Dual||half pyramidille<BR>Cell: [[File:Half pyramidille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''runcicantic cubic honeycomb''' or '''runcicantic cubic cellulation''' is a uniform space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[Truncated cuboctahedron|truncated cuboctahedra]], [[Truncated cube|truncated cube]]s and [[Truncated tetrahedron|truncated tetrahedra]] in a ratio of 1:1:2, with a mirrored sphenoid [[Vertex figure|vertex figure]]. It is related to the runcicantellated cubic honeycomb.

[[Biography:John Horton Conway|John Horton Conway]] calls this honeycomb a '''f-tCO-trille''', and its dual half pyramidille.
: [[Image:Cantitruncated alternated cubic tiling.png|190px]][[File:HC A6-A2-A1.png|120px]]
==== Half pyramidille ====

The dual to the ''runcitruncated cubic honeycomb'' is called a '''half pyramidille''', with Coxeter diagram {{CDD|node_fh|4|node|3|node_f1|4|node_f1}}. Faces exist in 3 of 4 hyperplanes of the [4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math> Coxeter group.

Cells are irregular pyramids and can be seen as 1/12 of a [[Cube|cube]], or 1/24 of a [[Rhombic dodecahedron|rhombic dodecahedron]], each defined with three corner and the cube center.
:[[File:Half pyramidille cell.png|160px]]

==== Related skew apeirohedra====
A related uniform [[Skew apeirohedron|skew apeirohedron]] exists with the same [[Vertex arrangement|vertex arrangement]], but triangles and square removed. It can be seen as truncated tetrahedra and truncated cubes augmented together.
:[[File:Runcicantic_cubic_honeycomb_apeirohedron_6688.png|200px]]

====Related honeycombs====
{| class=wikitable
|- align=center
|[[Image:Cantitruncated alternated cubic honeycomb.png|240px]]<br>Runcicantic cubic<br>{{CDD|node_h1|4|node|3|node_1|4|node_1}}
|[[File:Runcitruncated cubic honeycomb.jpg|240px]]<br>Runcicantellated cubic<br>{{CDD|node_1|4|node|3|node_1|4|node_1}}
|}
{{clear}}

===Gyrated tetrahedral-octahedral honeycomb===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Gyrated tetrahedral-octahedral honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||width=200|[[Convex uniform honeycomb|convex uniform honeycomb]]
|-
|bgcolor=#e7dcc3|Coxeter diagrams||{{CDD|node|3|node|6|node_h|2x|node_h|infin|node}}<BR>{{CDD|node|6|node_h|3|node_h|2x|node_h|infin|node}}<BR>{{CDD|branch_hh|split2|node_h|2x|node_h|infin|node}}
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]s||h{4,3,4}:g<BR>h{6,3}h{∞}<BR>s{3,6}h{∞}<BR>s{3<sup>[3]</sup>}h{∞}  
|-
|bgcolor=#e7dcc3|Cells||[[Tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]<BR>[[Octahedron|{3,4}]] [[File:Uniform polyhedron-43-t2.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Gyrated alternated cubic honeycomb verf.png|80px]]<br>[[Triangular orthobicupola|triangular orthobicupola]] ''G3.4.3.4''
|-
|bgcolor=#e7dcc3|[[Physics:Hexagonal crystal system|Space group]]||P6<sub>3</sub>/mmc (194)<BR>[3,6,2<sup>+</sup>,&infin;]
|-
|bgcolor=#e7dcc3|Dual||trapezo-rhombic dodecahedral honeycomb
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''gyrated tetrahedral-octahedral honeycomb''' or '''gyrated alternated cubic honeycomb''' is a space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space made up of [[Octahedron|octahedra]] and [[Tetrahedron|tetrahedra]] in a ratio of 1:2.

It is vertex-uniform with 8 tetrahedra and 6 octahedra around each vertex.

It is not edge-uniform. All edges have 2 tetrahedra and 2 octahedra, but some are alternating, and some are paired.
: [[Image:Gyrated alternated cubic.png|240px]][[File:Gyrated alternated cubic honeycomb.png|240px]]

It can be seen as reflective layers of this layer honeycomb:
{| class=wikitable width=240
!{{CDD|node_h|2x|node_h|6|node|3|node}}
|-
|[[File:Tetroctahedric semicheck.png|240px]]
|}


==== Construction by gyration ====
This is a less symmetric version of another honeycomb, tetrahedral-octahedral honeycomb, in which each edge is surrounded by alternating tetrahedra and octahedra.  Both can be considered as consisting of layers one cell thick, within which the two kinds of cell strictly alternate.  Because the faces on the planes separating these layers form a [[Triangular tiling|regular pattern of triangles]], adjacent layers can be placed so that each octahedron in one layer meets a tetrahedron in the next layer, '''or''' so that each cell meets a cell of its own kind (the layer boundary thus becomes a [[Reflection (mathematics)|reflection]] plane).  The latter form is called ''gyrated''.

The vertex figure is called a [[Triangular orthobicupola|triangular orthobicupola]], compared to the tetrahedral-octahedral honeycomb whose vertex figure [[Cuboctahedron|cuboctahedron]] in a lower symmetry is called a ''triangular gyrobicupola'', so the gyro- prefix is reversed in usage.
{| class=wikitable
|+ [[Vertex figure]]s
|- align=center
!Honeycomb
!Gyrated tet-oct
!Reflective tet-oct
|- align=center
!Image
|[[Image:Triangular orthobicupola.png|150px]]
|[[Image:Cuboctahedron.jpg|150px]]
|- align=center
!Name
!triangular orthobicupola
!triangular gyrobicupola
|- align=center
![[Vertex figure]]
|[[File:Gyrated alternated cubic honeycomb verf.png|100px]]
|[[File:Uniform t0 3333 honeycomb verf.png|100px]]
|- align=center
![[List of spherical symmetry groups|Symmetry]]
| D<sub>3h</sub>, order 12<br>
| D<sub>3d</sub>, order 12<br>(O<sub>h</sub>, order 48)
|}

==== Construction by alternation ====
[[File:Ditetrahedral-octahedral honeycomb verf.png|thumb|[[Vertex figure|Vertex figure]] with nonplanar 3.3.3.3 [[Vertex configuration|vertex configuration]] for the triangular bipyramids]]
The geometry can also be constructed with an [[Alternation (geometry)|alternation]] operation applied to a hexagonal prismatic honeycomb. The [[Hexagonal prism|hexagonal prism]] cells become [[Octahedron|octahedra]] and the voids create [[Triangular bipyramid|triangular bipyramid]]s which can be divided into pairs of [[Tetrahedron|tetrahedra]] of this honeycomb. This honeycomb with bipyramids is called a '''ditetrahedral-octahedral honeycomb'''. There are 3 Coxeter-Dynkin diagrams, which can be seen as 1, 2, or 3 colors of octahedra:
#{{CDD|node|3|node|6|node_h|2x|node_h|infin|node}}
#{{CDD|node|6|node_h|3|node_h|2x|node_h|infin|node}}
#{{CDD|branch_hh|split2|node_h|2x|node_h|infin|node}}
{{clear}}

=== Gyroelongated alternated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Gyroelongated alternated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,4}:ge<BR>{3,6}h<sub>1</sub>{&infin;}
|-
|bgcolor=#e7dcc3|Coxeter diagram||{{CDD|node|3|node|6|node_h|2x|node_h|infin|node_1}}<BR>{{CDD|node|6|node_h|3|node_h|2x|node_h|infin|node_1}}<BR>{{CDD|branch_hh|split2|node_h|2x|node_h|infin|node_1}}
|-
|bgcolor=#e7dcc3|Cells||[[Tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]<BR>[[Octahedron|{3,4}]] 40px<BR>[[Triangular prism|(3.4.4)]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Gyroelongated alternated cubic honeycomb verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Physics:Trigonal crystal system|Space group]]||P6<sub>3</sub>/mmc (194)<BR>[3,6,2<sup>+</sup>,&infin;]
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''gyroelongated alternated cubic honeycomb''' or '''elongated triangular antiprismatic cellulation''' is a space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[Octahedron|octahedra]], [[Triangular prism|triangular prism]]s, and [[Tetrahedron|tetrahedra]] in a ratio of 1:2:2.

It is vertex-transitive with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex.

It is one of 28 [[Convex uniform honeycomb|convex uniform honeycomb]]s.

The elongated alternated cubic honeycomb has the same arrangement of cells at each vertex, but the overall arrangement differs.  In the ''elongated'' form, each prism meets a tetrahedron at one of its triangular faces and an octahedron at the other; in the ''gyroelongated'' form, the prism meets the same kind of [[Deltahedron|deltahedron]] at each end.

:[[Image:Gyroelongated alternated cubic tiling.png|240px]] [[Image:Gyroelongated alternated cubic honeycomb.png|180px]]
{{clear}}

=== Elongated alternated cubic honeycomb ===
{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Elongated alternated cubic honeycomb
<!--|-
|bgcolor=#ffffff align=center colspan=2|-->
|-
|bgcolor=#e7dcc3|Type||width=200|[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,4}:e<BR>{3,6}g<sub>1</sub>{&infin;}
|-
|bgcolor=#e7dcc3|Cells||[[Tetrahedron|{3,3}]] [[File:Uniform polyhedron-33-t0.png|40px]]<BR>[[Octahedron|{3,4}]] 40px<BR>[[Triangular prism|(3.4.4)]] [[File:Triangular prism.png|40px]]
|-
|bgcolor=#e7dcc3|Faces||[[Triangle|triangle]] {3}<BR>[[Square|square]] {4}
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Gyroelongated alternated cubic honeycomb verf.png|80px]]<br>[[Triangular cupola|triangular cupola]] joined to isosceles [[Hexagonal pyramid|hexagonal pyramid]]
|-
|bgcolor=#e7dcc3|[[Space group|Symmetry group]]|| [6,(3,2<sup>+</sup>,∞,2<sup>+</sup>)] ?
|-
|bgcolor=#e7dcc3|Properties||vertex-transitive
|}
The '''elongated alternated cubic honeycomb''' or '''elongated triangular gyroprismatic cellulation''' is a space-filling [[Tessellation|tessellation]] (or [[Honeycomb (geometry)|honeycomb]]) in Euclidean 3-space.  It is composed of [[Octahedron|octahedra]], [[Triangular prism|triangular prism]]s, and [[Tetrahedron|tetrahedra]] in a ratio of 1:2:2.

It is vertex-transitive with 3 octahedra, 4 tetrahedra, 6 triangular prisms around each vertex.  Each prism meets an octahedron at one end and a tetrahedron at the other.

It is one of 28 [[Convex uniform honeycomb|convex uniform honeycomb]]s.

It has a ''gyrated'' form called the gyroelongated alternated cubic honeycomb with the same arrangement of cells at each vertex.

: [[Image:Elongated alternated cubic tiling.png|240px]][[Image:Elongated alternated cubic honeycomb.png|180px]]

{{-}}

==See also==
* [[Architectonic and catoptric tessellation]]
*[[Cubic honeycomb]]
*[[Engineering:Space frame|Space frame]]

==Notes==
{{reflist}}

== References ==
* [[Biography:John Horton Conway|John H. Conway]], Heidi Burgiel, [[Biography:Chaim Goodman-Strauss|Chaim Goodman-Strauss]], (2008) ''The Symmetries of Things'', {{isbn|978-1-56881-220-5}} (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292–298, includes all the nonprismatic forms)
* George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
* [[Biography:Branko Grünbaum|Branko Grünbaum]], Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.
* [[Biography:Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* {{The Geometrical Foundation of Natural Structure (book)}}
* {{cite book | first=Keith | last=Critchlow |  title=Order in Space: A design source book | publisher=Viking Press| year=1970 | isbn=0-500-34033-1 }}
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, {{isbn|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380–407, MR 2,10] (1.9 Uniform space-fillings)
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* A. Andreini, ''Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative'' (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
* D. M. Y. Sommerville, ''An Introduction to the Geometry of '''n''' Dimensions.'' New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
* {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |publisher=Springer |edition=3rd |isbn=0-387-98585-9 |url-access=registration |url=https://archive.org/details/spherepackingsla0000conw_b8u0 }}

== External links ==
*[http://www.wtcsitememorial.com/ent/entI=706963.html Architectural design made with Tetrahedrons and regular Pyramids based square.(2003) ] {{Webarchive|url=https://web.archive.org/web/20160304142123/http://www.wtcsitememorial.com/ent/entI=706963.html |date=2016-03-04 }}
* {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x3o3o *b4o - octet  - O21}}
* [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 11-Octet]

{{Honeycombs}}

[[Category:Honeycombs (geometry)]]
[[Category:4-polytopes]]

{{Sourceattribution|Tetrahedral-octahedral honeycomb}}