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| | The writer is recognized by the name of Figures Lint. To collect cash is what his family members and him enjoy. Hiring has been my profession for some time but I've already utilized for an additional 1. Years in the past he moved to North Dakota and his family enjoys it.<br><br>My weblog ... [http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/ at home std test] |
| !bgcolor=#e7dcc3 colspan=2|Bitruncated cubic honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Bitruncated cubic tiling.png|180px]] [[File:HC-A4.png|128px]]
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| |bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||2t{4,3,4}<BR>t<sub>1,2</sub>{4,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD||node|4|node_1|3|node_1|4|node}}
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| |-
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| |bgcolor=#e7dcc3|Cell type||[[truncated octahedron|(''4.6.6'')]]
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| |-
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| |bgcolor=#e7dcc3|Face types||[[square (geometry)|square]] {4}<BR>[[hexagon]] {6}
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| |-
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| |bgcolor=#e7dcc3|Edge figure||[[isosceles triangle]] {3}
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| |bgcolor=#e7dcc3|Vertex figure||[[Image:Bitruncated cubic honeycomb verf2.png|80px]]<BR>([[disphenoid tetrahedron]])
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| |-
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| |bgcolor=#e7dcc3|Cells/edge|| (4.6.6)<sup>3</sup>
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| |-
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| |bgcolor=#e7dcc3|Cells/vertex|| (4.6.6)<sup>4</sup>
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| |-
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| |bgcolor=#e7dcc3|Faces/edge|| ''4.6.6''
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| |-
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| |bgcolor=#e7dcc3|Faces/vertex|| 4<sup>2</sup>.6<sup>4</sup>
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| |bgcolor=#e7dcc3|Edges/vertex|| 4
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| |-
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| |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>[</span>[4,3,4]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
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| |-
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| |bgcolor=#e7dcc3|Dual||Oblate tetrahedrille<BR>[[Disphenoid tetrahedral honeycomb]]
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| |-
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| |bgcolor=#e7dcc3|Properties||[[Isogonal figure|isogonal]], [[Isotoxal figure|isotoxal]], [[Isochoric figure|isochoric]]
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| |}
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| The '''[[Bitruncation (geometry)|bitruncated]] [[cubic honeycomb]]''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]] made up of [[truncated octahedron|truncated octahedra]]. It has 4 [[truncated octahedron|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]]. | |
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| [[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.
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| 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) is the optimal soap bubble foam. However, the [[Weaire–Phelan structure]] is a less symmetrical, but more efficient, foam of soap bubbles.
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| == Symmetry ==
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| 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.
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| {| class=wikitable
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| |+ Five uniform colorings by cell
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| |-
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| ![[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)
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| |-
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| ![[Fibrifold]]||8<sup>o</sup>:2||4<sup>−</sup>:2||2<sup>−</sup>:2||1<sup>o</sup>:2||2<sup>+</sup>:2
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| |- valign=top
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| ! valign=center|[[Coxeter group]]
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| ! <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}}
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| ! <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}}
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| ! <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}}
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| ! <math>{\tilde{A}}_3</math><BR>[3<sup>[4]</sup>]<BR> <BR>{{CDD|node_c3|split1|nodeab_c1-2|split2|node_c4}}
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| ! <math>{\tilde{A}}_3</math>×2<BR>[</span>[3<sup>[4]</sup>]]<BR>=[</span>[3<sup>[4]</sup>]]<BR>{{CDD|branch_c1|3ab|branch_c2}}
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| |-
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| ![[Coxeter diagram]]
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| !{{CDD||branch_11|4a4b|nodes}}
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| !{{CDD||node|4|node_1|3|node_1|4|node}}
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| !{{CDD|nodes_11|split2|node_1|4|node}}
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| !{{CDD|node_1|split1|nodes_11|split2|node_1}}
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| !{{CDD|branch_11|3ab|branch_11}}
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| |- align=center
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| ![[truncated octahedron|truncated octahedra]]
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| ! 1<BR>[[File:Uniform polyhedron-43-t12.svg|25px]]
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| ! 1:1<BR>[[File:Uniform polyhedron-43-t12.svg|25px]]:[[File:Uniform polyhedron-43-t12.svg|25px]]
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| ! 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]]
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| ! 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]]
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| ! 1:1<BR>[[File:Uniform polyhedron-33-t012.png|25px]]:[[File:Uniform polyhedron-33-t012.png|25px]]
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| |- align=center
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| ![[Vertex figure]]
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| |[[Image:Bitruncated cubic honeycomb verf2.png|80px]]
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| |[[Image:Bitruncated cubic honeycomb verf.png|80px]]
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| |[[File:Cantitruncated alternate cubic honeycomb verf.png|80px]]
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| |[[File:Omnitruncated 3-simplex honeycomb verf.png|80px]]
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| |[[File:Omnitruncated 3-simplex honeycomb verf2.png|80px]]
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| |- align=center
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| !Vertex<BR>figure<BR>symmetry
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| |[2<sup>+</sup>,4]<BR>(order 8)
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| |[2]<BR>(order 4)
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| |[ ]<BR>(order 2)
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| |[ ]<sup>+</sup><BR>(order 1)
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| |[2]<sup>+</sup><BR>(order 2)
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| |-
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| !Image<BR>Colored by<BR>cell
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| |[[Image:Bitruncated Cubic Honeycomb1.svg|100px]]
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| |[[Image:Bitruncated Cubic Honeycomb.svg|100px]]
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| |[[Image:Bitruncated cubic honeycomb3.png|100px]]
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| |[[Image:Bitruncated cubic honeycomb2.png|100px]]
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| |[[Image:Bitruncated Cubic Honeycomb1.svg|100px]]
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| |}
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| == Related polyhedra and honeycombs ==
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| [[File:Four-hexagon skew polyhedron.png|thumb|The [[regular skew polyhedron]] {6,4|4} contains the hexagons of this honeycomb.]]
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| 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.
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| {{C3 honeycombs}}
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| 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.
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| {{B3 honeycombs}}
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| This honeycomb is one of [[Uniform polyteron#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:
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| {{A3 honeycombs}}
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| === Alternated form===
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| [[File:Alternated bitruncated cubic honeycomb verf.png|150px|thumb|Vertex figure for alternated bitruncated cubic honeycomb, with 4 tetrahedral and 4 icosahedral cells. All edges represent triangles in the honeycomb, but edge-lengths can't be made equal.]]
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| This honeycomb can be [[Alternation (geometry)|alternated]], creating regular [[icosahedron]] from the truncated octahedra with irregular 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 [</span>[3<sup>[4]</sup>]]<sup>+</sup>.
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| This honeycomb is represented in the boron atoms of the [[Allotropes_of_boron#.CE.B1-rhombohedral_boron|α-rhombihedral crystal]]. The centers of the icosahedra are located at the fcc positions of the lattice.<ref>Williams, 1979, p 199, Figure 5-38.</ref>
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| {| class=wikitable
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| |+ Five uniform colorings
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| |-
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| ![[Space group]]||I{{overline|3}} (204) ||Pm{{overline|3}} (200) ||Fm{{overline|3}} (202)||Fd{{overline|3}} (203) || F23 (196)
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| |-
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| ![[Fibrifold]]||8<sup>−o</sup>||4<sup>−</sup>||2<sup>−</sup>||2<sup>o+</sup> ||1<sup>o</sup>
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| |-
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| ![[Coxeter group]]|| <nowiki>[[</nowiki>4,3<sup>+</sup>,4]]|| [4,3<sup>+</sup>,4]|| [4,(3<sup>1,1</sup>)<sup>+</sup>]|| [</span>[3<sup>[4]</sup>]]<sup>+</sup>||[3<sup>[4]</sup>]<sup>+</sup>
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| |- align=center
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| |[[Coxeter diagram]]
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| |{{CDD||branch_hh|4a4b|nodes}}
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| |{{CDD||node|4|node_h|3|node_h|4|node}}
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| |{{CDD|node|4|node_h|split1|nodes_hh}}
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| |{{CDD|branch_hh|3ab|branch_hh}}
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| |{{CDD|node_h|split1|nodes_hh|split2|node_h}}
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| |- align=center
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| !Order
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| |double
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| |full
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| |half
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| |quarter<BR>double
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| |quarter
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| |}
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| === Projection by folding ===
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| The ''bitruncated cubic honeycomb'' can be orthogonally projected into the planar [[truncated square tiling]] by a [[Coxeter–Dynkin diagram#Geometric folding|geometric folding]] operation that maps two pairs of mirrors into each other. The projection of the ''bitruncated cubic honeycomb'' creating two offset copies of the truncated square tiling [[vertex arrangement]] of the plane:
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| {|class=wikitable
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| ![[Coxeter group|Coxeter<BR>group]]
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| !<math>{\tilde{A}}_3</math>
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| !<math>{\tilde{C}}_2</math>
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| |- align=center
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| ![[Coxeter–Dynkin diagram#Geometric folding|Coxeter<BR>diagram]]
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| |{{CDD|node_1|split1|nodes_11|split2|node_1}}
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| |{{CDD|node_1|4|node_1|4|node_1}}
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| |- align=center
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| !Graph
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| |[[File:Bitruncated Cubic Honeycomb flat.png|280px]]<BR>Bitruncated cubic honeycomb
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| |[[File:Uniform tiling 44-t012.png|280px]]<BR>[[Truncated square tiling]]
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| |}
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| ==See also==
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| {{Commons category|Bitruncated cubic honeycomb}}
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| *[[Architectonic and catoptric tessellation]]
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| * [[Cubic honeycomb]]
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| * [[Brillouin zone]]
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| == Notes==
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| {{reflist}}
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| == References ==
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| * [[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)
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| * [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
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| * [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
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| * '''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]
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| ** (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)
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| * [[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.
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| * {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|o4x3x4o - batch - O16}}
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| * [http://polyhedra.doskey.com/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 05-Batch]
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| * {{The Geometrical Foundation of Natural Structure (book)}}
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| ==External links ==
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| * {{mathworld | urlname = Space-FillingPolyhedron | title = Space-filling polyhedron}}
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| [[Category:Honeycombs (geometry)]]
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