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| | Greetings. The author's name is Phebe and she feels comfortable when individuals use the full title. Playing baseball is the pastime he will never stop doing. My day job is a meter reader. For a while she's been in South Dakota.<br><br>Here is my web blog ... [http://chatmast.com/index.php?do=/BookerSipessk/info/ over the counter 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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| |bgcolor=#e7dcc3|Face types||[[square (geometry)|square]] {4}<BR>[[hexagon]] {6}
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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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| |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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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
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| |bgcolor=#e7dcc3|Dual||Oblate tetrahedrille<BR>[[Disphenoid tetrahedral honeycomb]]
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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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