View source for Bitruncated cubic honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Bitruncated cubic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Bitruncated cubic tiling.png|180px]]&nbsp;[[File:HC-A4.png|120px]]
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2t{4,3,4}<BR>t<sub>1,2</sub>{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD||node|4|node_1|3|node_1|4|node}}
|-
|bgcolor=#e7dcc3|Cell type||[[truncated octahedron|(''4.6.6'')]]
|-
|bgcolor=#e7dcc3|Face types||[[square (geometry)|square]] {4}<BR>[[hexagon]] {6}
|-
|bgcolor=#e7dcc3|Edge figure||[[isosceles triangle]] {3}
|-
|bgcolor=#e7dcc3|Vertex figure||[[Image:Bitruncated cubic honeycomb verf2.png|80px]]<BR>([[tetragonal disphenoid]])
|-
|bgcolor=#e7dcc3|[[Space group]]<BR>[[Fibrifold notation]]<BR>[[Coxeter notation#Space groups|Coxeter notation]]||[[Cubic crystal system|Im{{overline|3}}m (229)]]<BR>8<sup>o</sup>:2<BR><nowiki>[[</nowiki>4,3,4]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
|-
|bgcolor=#e7dcc3|Dual||Oblate tetrahedrille<BR>[[Disphenoid tetrahedral honeycomb]]<BR>Cell: [[File:Oblate tetrahedrille cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]], [[Isochoric figure|isochoric]]
|}
[[File:Cubes-A4 ani.gif|thumb|The bitruncated cubic honeycomb shown here in relation to a cubic honeycomb]]
The '''bitruncated cubic honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]] made up of [[truncated octahedron|truncated octahedra]] (or, equivalently, [[Bitruncation (geometry)|bitruncated]] cubes). It has 4 truncated octahedra around each vertex. Being composed entirely of [[truncated octahedron|truncated octahedra]], it is [[cell-transitive]]. It is also [[edge-transitive]], with 2 hexagons and one square on each edge, and [[vertex-transitive]]. It is one of 28 [[Convex uniform honeycomb|uniform honeycombs]].

[[John Horton Conway]] calls this honeycomb a '''truncated octahedrille''' in his [[Architectonic and catoptric tessellation]] list, with its dual called an ''oblate tetrahedrille'', also called a [[disphenoid tetrahedral honeycomb]]. Although a regular [[tetrahedron]] can not tessellate space alone, this dual has identical [[disphenoid tetrahedron]] cells with [[isosceles triangle]] faces.

== Geometry==
It can be realized as the [[Voronoi tessellation]] of the [[body-centred cubic]] lattice. [[Lord Kelvin]] conjectured that a variant of the ''bitruncated cubic honeycomb'' (with curved faces and edges, but the same combinatorial structure) was the optimal soap bubble foam. However, a number of less symmetrical structures have later been found to be more efficient foams of soap bubbles, among which the [[Weaire–Phelan structure]] appears to be the best.

The honeycomb represents the [[permutohedron]] tessellation for 3-space. The coordinates of the vertices for one octahedron represent a [[hyperplane]] of integers in 4-space, specifically [[permutation]]s of (1,2,3,4). The tessellation is formed by translated copies within the hyperplane. 
:[[File:Symmetric group 4; permutohedron 3D; l-e factorial numbers.svg|240px]]

The tessellation is the highest tessellation of [[parallelohedron]]s in 3-space.

=== Projections ===
The ''bitruncated cubic honeycomb'' can be orthogonally projected into the euclidean plane with various symmetry arrangements. The highest (hexagonal) symmetry form projects into a nonuniform [[rhombitrihexagonal tiling]]. A square symmetry projection forms two overlapping [[truncated square tiling]], which combine together as a [[chamfered square tiling]]. 
{| class=wikitable
|+ Orthogonal projections
!Symmetry
!p6m (*632)
!p4m (*442)
!colspan=3|pmm (*2222)
|-
!Solid
|[[File:Bitruncated_cubic_honeycomb_ortho2.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho4.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho1.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho3.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_ortho5.png|120px]]
|-
!Frame
|[[File:Bitruncated_cubic_honeycomb_orthoframe2.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe4.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe1.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe3.png|120px]]
|[[File:Bitruncated_cubic_honeycomb_orthoframe5.png|120px]]
|}

== Symmetry ==
The vertex figure for this honeycomb is a [[disphenoid tetrahedron]], and it is also the [[Goursat tetrahedron]] ([[fundamental domain]]) for the <math>{\tilde{A}}_3</math> [[Coxeter group]]. This honeycomb has four uniform constructions, with the truncated octahedral cells having different [[Coxeter group]]s and [[Wythoff construction]]s.  These uniform symmetries can be represented by coloring differently the cells in each construction.

{| class=wikitable
|+ Five uniform colorings by cell
|-
![[Space group]]||Im{{overline|3}}m (229)||Pm{{overline|3}}m (221)||Fm{{overline|3}}m (225)||F{{overline|4}}3m (216)||Fd{{overline|3}}m (227)
|-
![[Fibrifold]]||8<sup>o</sup>:2||4<sup>−</sup>:2||2<sup>−</sup>:2||1<sup>o</sup>:2||2<sup>+</sup>:2
|- valign=top
! valign=center|[[Coxeter group]]
! <math>{\tilde{C}}_3</math>×2<BR><nowiki>[[</nowiki>4,3,4]]<BR>=[4[3<sup>[4]</sup>]]<BR>{{CDD|node|4|node_c1|3|node_c1|4|node}} = {{CDD|branch_c1|3ab|branch_c1}}
! <math>{\tilde{C}}_3</math><BR>[4,3,4]<BR>=[2[3<sup>[4]</sup>]]<BR>{{CDD|node|4|node_c1|3|node_c2|4|node}} = {{CDD|branch_c1-2|3ab|branch_c2-1}}
! <math>{\tilde{B}}_3</math><BR>[4,3<sup>1,1</sup>]<BR>=<[3<sup>[4]</sup>]><BR>{{CDD|nodeab_c1-2|split2|node_c3|4|node}} = {{CDD|node_c3|split1|nodeab_c1-2|split2|node_c3}}
! <math>{\tilde{A}}_3</math><BR>[3<sup>[4]</sup>]<BR>&nbsp;<BR>{{CDD|node_c3|split1|nodeab_c1-2|split2|node_c4}}
! <math>{\tilde{A}}_3</math>×2<BR><nowiki>[[</nowiki>3<sup>[4]</sup>]]<BR>=<nowiki>[[</nowiki>3<sup>[4]</sup>]]<BR>{{CDD|branch_c1|3ab|branch_c2}}
|-
![[Coxeter diagram]]
!{{CDD||branch_11|4a4b|nodes}}
!{{CDD||node|4|node_1|3|node_1|4|node}}
!{{CDD|nodes_11|split2|node_1|4|node}}
!{{CDD|node_1|split1|nodes_11|split2|node_1}}
!{{CDD|branch_11|3ab|branch_11}}
|- align=center
![[truncated octahedron|truncated octahedra]]
! 1<BR>[[File:Uniform polyhedron-43-t12.svg|25px]]
! 1:1<BR>[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-43-t12.svg|25px]]
! 2:1:1<BR>[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
! 1:1:1:1<BR>[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
! 1:1<BR>[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
|- align=center
![[Vertex figure]]
|[[Image:Bitruncated cubic honeycomb verf2.png|80px]]
|[[Image:Bitruncated cubic honeycomb verf.png|80px]]
|[[File:Cantitruncated alternate cubic honeycomb verf.png|80px]]
|[[File:Omnitruncated 3-simplex honeycomb verf.png|80px]]
|[[File:Omnitruncated 3-simplex honeycomb verf2.png|80px]]
|- align=center
!Vertex<BR>figure<BR>symmetry
|[2<sup>+</sup>,4]<BR>(order 8)
|[2]<BR>(order 4)
|[ ]<BR>(order 2)
|[ ]<sup>+</sup><BR>(order 1)
|[2]<sup>+</sup><BR>(order 2)
|-
!Image<BR>Colored by<BR>cell
|[[Image:Bitruncated Cubic Honeycomb1.svg|100px]]
|[[Image:Bitruncated Cubic Honeycomb.svg|100px]]
|[[Image:Bitruncated cubic honeycomb3.png|100px]]
|[[Image:Bitruncated cubic honeycomb2.png|100px]]
|[[Image:Bitruncated Cubic Honeycomb1.svg|100px]]
|}

== Related polyhedra and honeycombs ==
[[File:Muoctahedron.png|thumb|The [[regular skew apeirohedron]] {6,4{{pipe}}4} contains the hexagons of this honeycomb.]]
The [4,3,4], {{CDD|node|4|node|3|node|4|node}}, [[Coxeter group]] generates 15 permutations of uniform tessellations, 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.
{{C3 honeycombs}}

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

This honeycomb is one of [[Uniform 5-polytope#Regular and uniform honeycombs|five distinct uniform honeycombs]]<ref>[http://mathworld.wolfram.com/Necklace.html], [http://oeis.org/A000029 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 diagram]]s: 
{{A3 honeycombs}}

{{-}}
=== Alternated form ===
{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Alternated bitruncated cubic honeycomb
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Convex honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||2s{4,3,4}<BR>2s{4,3<sup>1,1</sup>}<BR>sr{3<sup>[4]</sup>}
|-
|bgcolor=#e7dcc3|[[Coxeter diagram]]s||{{CDD|node|4|node_h|3|node_h|4|node}}<br>{{CDD|node|4|node_h|split1|nodes_hh}} = {{CDD|node|4|node_h|3|node_h|4|node_h0}}<br>{{CDD|nodes_hh|split2|node_h|4|node}} = {{CDD|node_h0|4|node_h|3|node_h|4|node}}<br>{{CDD|node_h|split1|nodes_hh|split2|node_h}} = {{CDD|node_h0|4|node_h|3|node_h|4|node_h0}}
|-
|bgcolor=#e7dcc3|Cells||[[tetrahedron]]<br>[[icosahedron]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:Alternated bitruncated cubic honeycomb verf.png|80px]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||[[4,3<sup>+</sup>,4]], <math>{\tilde{C}}_3</math>
|-
|bgcolor=#e7dcc3|Dual||[[Ten-of-diamonds decahedron|Ten-of-diamonds honeycomb]]<BR>Cell: [[File:Alternated bitruncated cubic honeycomb dual cell.png|80px]]
|-
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
|}
This honeycomb can be [[Alternation (geometry)|alternated]], creating pyritohedral [[icosahedra]] from the truncated octahedra with disphenoid tetrahedral cells created in the gaps. There are three constructions from three related [[Coxeter-Dynkin diagram]]s: {{CDD|node|4|node_h|3|node_h|4|node}}, {{CDD|node|4|node_h|split1|nodes_hh}}, and {{CDD|node_h|split1|nodes_hh|split2|node_h}}. These have symmetry [4,3<sup>+</sup>,4], [4,(3<sup>1,1</sup>)<sup>+</sup>] and [3<sup>[4]</sup>]<sup>+</sup> respectively. The first and last symmetry can be doubled as <nowiki>[[</nowiki>4,3<sup>+</sup>,4]] and <nowiki>[[</nowiki>3<sup>[4]</sup>]]<sup>+</sup>.

The dual honeycomb is made of cells called [[Ten_of_diamonds_decahedron#Honeycomb|ten-of-diamonds decahedra]].

{| class=wikitable
|+ Five uniform colorings
|-
![[Space group]]||I{{overline|3}} (204) ||Pm{{overline|3}} (200) ||Fm{{overline|3}} (202)||Fd{{overline|3}} (203) || F23 (196)
|-
![[Fibrifold]]||8<sup>−o</sup>||4<sup>−</sup>||2<sup>−</sup>||2<sup>o+</sup> ||1<sup>o</sup>
|-
![[Coxeter group]]||<nowiki>[[</nowiki>4,3<sup>+</sup>,4]]|| [4,3<sup>+</sup>,4]|| [4,(3<sup>1,1</sup>)<sup>+</sup>]|| <nowiki>[[</nowiki>3<sup>[4]</sup>]]<sup>+</sup>||[3<sup>[4]</sup>]<sup>+</sup>
|- align=center
![[Coxeter diagram]]
|{{CDD|branch_hh|4a4b|nodes}}
|{{CDD|node|4|node_h|3|node_h|4|node}}
|{{CDD|node|4|node_h|split1|nodes_hh}}
|{{CDD|branch_hh|3ab|branch_hh}}
|{{CDD|node_h|split1|nodes_hh|split2|node_h}}
|- align=center
!Order
|double
|full
|half
|quarter<BR>double
|quarter
|-
!Image<BR>colored by cells
|[[Image:Alternated bitruncated cubic honeycomb1.png|100px]]
|[[Image:Alternated bitruncated cubic honeycomb2.png|100px]]
|[[Image:Alternated bitruncated cubic honeycomb3.png|100px]]
|[[Image:Alternated bitruncated cubic honeycomb1.png|100px]]
|[[Image:Alternated bitruncated cubic honeycomb4.png|100px]]
|}

This honeycomb is represented in the boron atoms of the [[Allotropes of boron#.CE.B1-rhombohedral boron|&alpha;-rhombohedral crystal]]. The centers of the icosahedra are located at the fcc positions of the lattice.<ref>Williams, 1979, p 199, Figure 5-38.</ref>
:[[File:Alfaboron.jpg|160px]]

=== Related polytopes ===
Nonuniform variants with [4,3,4] symmetry and two types of truncated octahedra can be doubled by placing the two types of truncated octahedra to produce a nonuniform honeycomb with [[truncated octahedron|truncated octahedra]] and [[hexagonal prism]]s (as ditrigonal trapezoprisms). Its vertex figure is a ''C<sub>2v</sub>''-symmetric [[triangular bipyramid]].

This honeycomb can then be alternated to produce another nonuniform honeycomb with [[icosahedron|pyritohedral icosahedra]], [[octahedron|octahedra]] (as triangular antiprisms), and [[tetrahedron|tetrahedra]] (as sphenoids). Its vertex figure has ''C<sub>2v</sub>'' symmetry and consists of 2 [[pentagon]]s, 4 [[rectangle]]s, 4 [[isosceles triangle]]s (divided into two sets of 2), and 4 [[triangle|scalene triangle]]s.

==See also==
{{Commons category|Bitruncated cubic honeycomb}}
*[[Architectonic and catoptric tessellation]]
* [[Cubic honeycomb]]
* [[Brillouin zone]]

== Notes==
{{reflist}}

== References ==
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, [[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)''
* [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
* '''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)
* [[Alfredo Andreini|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.
* {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|o4x3x4o - batch - O16}}
* [http://polyhedra.doskey.com/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 05-Batch]
* {{The Geometrical Foundation of Natural Structure (book)}}

==External links ==
* {{mathworld | urlname = Space-FillingPolyhedron  | title = Space-filling polyhedron}}

[[Category:3-honeycombs]]
[[Category:Bitruncated tilings]]
[[Category:Uniform 4-polytopes]]