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| !bgcolor=#e7dcc3 colspan=2|Alternated cubic honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Alternated cubic tiling.png|210px]] [[File:HC P1-P3.png|130px]]
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| |bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
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| |bgcolor=#e7dcc3|Family||[[Alternated hypercubic honeycomb]]
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| |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>
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| |J<sub>21,31,51</sub>, A<sub>2</sub><BR>W<sub>9</sub>, G<sub>1</sub>
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,4}<BR>{3<sup>[4]</sup>}<BR>ht<sub>0,3</sub>{4,3,4}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|nodes_10ru|split2|node|4|node}} or {{CDD|node_h1|4|node|3|node|4|node}}<BR>{{CDD|node_1|split1|nodes|split2|node}} or {{CDD|nodes|split2|node|4|node_h1}} or {{CDD|nodes_hh|4a4b|branch}}<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}}
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| |-
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| |bgcolor=#e7dcc3|Cell types||[[tetrahedron|{3,3}]], [[octahedron|{3,4}]]
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| |-
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| |bgcolor=#e7dcc3|Face types||[[triangle]] {3}
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| |-
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| |bgcolor=#e7dcc3|Edge figure||[{3,3}.{3,4}]<sup>2</sup><BR>([[rectangle]])
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| |-
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| |bgcolor=#e7dcc3|Vertex figure||[[Image:Alternated cubic honeycomb verf.svg|80px]][[File:Uniform t0 3333 honeycomb verf.png|80px]]<BR>[[File:Cuboctahedron.png|80px]][[File:Cantellated tetrahedron.png|80px]]<BR>([[cuboctahedron]])
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| |-
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| |bgcolor=#e7dcc3|Cells/edge||[{3,3}.{3,4}]<sup>2</sup>
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| |bgcolor=#e7dcc3|Faces/edge||4 {3}
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| |-
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| |bgcolor=#e7dcc3|Cells/vertex||[[tetrahedron|{3,3}]]<sup>8</sup>+[[octahedron|{3,4}]]<sup>6</sup>
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| |-
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| |bgcolor=#e7dcc3|Faces/vertex||24 {3}
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| |bgcolor=#e7dcc3|Edges/vertex||12
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| |-
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| |bgcolor=#e7dcc3|[[Space group|Symmetry group]]||Fm{{overline|3}}m (225)
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| |bgcolor=#e7dcc3|[[Coxeter notation|Symmetry]]||½<math>{\tilde{C}}_3</math>, [1<sup>+</sup>,4,3,4]<BR><math>{\tilde{B}}_3</math>, [4,3<sup>1,1</sup>]<BR><math>{\tilde{A}}_3</math>×2, <[3<sup>[4]</sup>]>
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| |bgcolor=#e7dcc3|Dual||Dodecahedrille<BR>[[rhombic dodecahedral honeycomb]]
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[quasiregular honeycomb]]
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| |}
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| The '''tetrahedral-octahedral honeycomb''' or '''alternated cubic honeycomb''' or '''half cubic honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]]. It is composed of alternating [[octahedron|octahedra]] and [[tetrahedron|tetrahedra]] in a ratio of 1:2. | |
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| 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.
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| [[John Horton Conway]] calls this honeycomb a '''Tetroctahedrille''', and its dual [[Rhombic dodecahedral honeycomb|dodecahedrille]].
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| It is part of an infinite family of [[uniform tessellation]]s called [[alternated hypercubic honeycomb]]s, formed as an [[Alternation (geometry)|alternation]] of a hypercubic honeycomb and being composed of [[demihypercube]] and [[cross-polytope]] facets.
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| In this case of 3-space, the [[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.
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| There's 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.
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| ==Cartesian coordinates==
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| For an ''alternted 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]])
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| :(i, j, k)
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| == Images ==
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| {| class="wikitable" width=200
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| |align=center valign=top|[[Image:TetraOctaHoneycomb-VertexConfig.svg|180px]]<BR>This diagram shows an [[exploded view]] of the cells surrounding each vertex.
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| |}
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| == Symmetry==
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| There is two reflective construction and many alternated [[Cubic_honeycomb#Uniform_colorings|cubic honeycomb]] ones, examples:
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| {| class=wikitable
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| !Symmetry
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| !<math>{\tilde{B}}_3</math>, [4,3<sup>1,1</sup>]<BR>= ½<math>{\tilde{C}}_3</math>, [1<sup>+</sup>,4,3,4]
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| !<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>]
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| !<nowiki>[[</nowiki>(4,3,4,2<sup>+</sup>)]]
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| ![(4,3,4,2<sup>+</sup>)]
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| |-
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| ![[Space group]]
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| !Fm{{overline|3}}m (225)
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| !F{{overline|4}}3m (216)
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| !I{{overline|4}}3m (217)
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| !P{{overline|4}}3m (215)
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| |-
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| !Image
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| |[[Image:Tetrahedral-octahedral honeycomb.png|160px]]
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| |[[Image:Tetrahedral-octahedral honeycomb2.png|160px]]
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| |-
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| !Types of tetrahedra
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| !1
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| !2
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| !2
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| !4
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| |-
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| ![[Coxeter diagram|Coxeter<BR>diagram]]
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| !{{CDD|nodes_10ru|split2|node|4|node}} = {{CDD|node_h1|4|node|3|node|4|node}}
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| ! {{CDD|node_1|split1|nodes|split2|node}} = {{CDD|nodes|split2|node|4|node_h1}} = {{CDD|node_h0|4|node|3|node|4|node_h1}}
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| !{{CDD|branch|4a4b|nodes_hh}}
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| !{{CDD|node_h|4|node|3|node|4|node_h}}
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| |}
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| === Projection by folding ===
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| The ''alternated cubic honeycomb'' can be orthogonally projected into the planar [[square tiling]] by a [[Coxeter–Dynkin diagram#Geometric folding|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]] of the plane:
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| {|class=wikitable
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| ![[Coxeter group|Coxeter<BR>group]]
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| ![[Coxeter–Dynkin diagram#Geometric folding|Coxeter<BR>diagram]]
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| !Graph
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| |- align=center
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| !<math>{\tilde{A}}_3</math>
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| |{{CDD|node_1|split1|nodes|split2|node}}
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| |[[File:Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg|100px]]<BR>alternated cubic honeycomb
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| |- align=center
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| !<math>{\tilde{C}}_2</math>
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| |{{CDD|node_1|4|node|4|node}}
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| |[[File:Uniform tiling 44-t0.png|100px]]<BR>[[square tiling]]
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| |}
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| ==A3/D3 lattice==
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| Its [[vertex arrangement]] represents an [[A3 lattice|A<sub>3</sub> lattice]] or ''D<sub>3</sub> lattice''.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D3.html</ref><ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A3.html</ref> It is the 3-dimensional case of a [[simplectic honeycomb]]. Its Voronoi cell is a [[rhombic dodecahedron]], the dual of the [[cuboctahedron]] vertex figure for the tet-oct honeycomb.
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| 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>
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| :{{CDD|node_1|split1|nodes|split2|node}} + {{CDD|node|split1|nodes|split2|node_1}}
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| 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]] of the [[disphenoid tetrahedral honeycomb]], dual honeycomb of the uniform [[bitruncated cubic honeycomb]]:<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds3.html</ref> It is also the [[body centered cubic]], the union of two [[cubic honeycomb]]s in dual positions.
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| :{{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}}.
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| The [[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]], {{CDD|branch_11|4a4b|nodes}}, containing all [[truncated octahedron|truncated octahedral]] [[Voronoi cell]]s, {{CDD|node|4|node_1|3|node_1}}.<ref>Conway (1998), p. 466</ref>
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| == Related honeycombs==
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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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| ==See also==
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| * [[Architectonic and catoptric tessellation]]
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| *[[Cubic honeycomb]]
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| *[[Space frame]]
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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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| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
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| * {{The Geometrical Foundation of Natural Structure (book)}}
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| * {{cite book | first=Keith | last=Critchlow | authorlink=Keith Critchlow | title=Order in Space: A design source book | publisher=Viking Press| year=1970 | isbn=0-500-34033-1 }}
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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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| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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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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| * [[Duncan MacLaren Young Sommerville|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
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| * {{cite book |author=Conway JH, Sloane NJH |year=1998 |title=Sphere Packings, Lattices and Groups |edition=3rd |isbn=0-387-98585-9}}
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| == External links ==
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| {{Commons category|Tetrahedral-octahedral honeycomb}}
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| *[http://www.wtcsitememorial.com/ent/entI=706963.html Architectural design made with Tetrahedrons and regular Pyramids based square.(2003) ]
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| * {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x3o3o *b4o - octet - O21}}
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| * [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 11-Octet]
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| {{Honeycombs}}
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| [[Category:Honeycombs (geometry)]]
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| [[Category:Polychora]]
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