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| !bgcolor=#e7dcc3 colspan=2|Cubic honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Cubic honeycomb.png|155px]][[Image:Partial cubic honeycomb.png|155px]]
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| |-
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| |bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Tessellations_of_Euclidean_3-space|Regular honeycomb]]
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| |-
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| |bgcolor=#e7dcc3|Family||[[Hypercube 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>11,15</sub>, A<sub>1</sub><BR>W<sub>1</sub>, G<sub>22</sub>
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3,4}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node|3|node|4|node}}
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| |-
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| |bgcolor=#e7dcc3|Cell type||[[cube|{4,3}]]
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| |-
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| |bgcolor=#e7dcc3|Face type||[[square (geometry)|{4}]]
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| |bgcolor=#e7dcc3|Vertex figure||[[Image:Cubic honeycomb verf.png|80px]]<BR>([[octahedron]])
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| |bgcolor=#e7dcc3|[[Space group]]<BR>[[Fibrifold notation]]||Pm{{overline|3}}m (221)<BR>4<sup>−</sup>:2
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
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| |bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]]
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| |-
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[Quasiregular honeycomb]]
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| |}
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| The '''cubic honeycomb''' is the only regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space, made up of [[cube|cubic]] cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its [[vertex figure]] is a regular [[octahedron]].
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| [[John Horton Conway]] calls this self-dual honeycomb a '''cubille'''.
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| It is a [[Self-dual tessellation|self-dual]] tessellation with [[Schläfli symbol]] {4,3,4}.
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| ==Cartesian coordinates==
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| [[File:Kubisches_Kristallsystem.jpg|100px|left|thumb|[[Simple cubic]]]]
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| The [[Cartesian coordinates]] of the vertices are:
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| ::(i, j, k)
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| :for all integral values: i,j,k, with edges parallel to the axes and with an edge length of 1.
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| == Related honeycombs==
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| It is part of a multidimensional family of [[hypercube honeycomb]]s, with [[Schläfli symbol]]s of the form {4,3,...,3,4}, starting with the [[square tiling]], {4,4} in the plane.
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| It is one of 28 [[Convex uniform honeycomb|uniform honeycombs]] using [[uniform polyhedron|convex uniform polyhedral]] cells.
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| == Isometries of simple cubic lattices==
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| Simple cubic lattices can be distorted into lower symmetries, represented by lower crystal systems:
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| {| class=wikitable
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| ![[Crystal system]]
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| ![[Monoclinic_crystal_system|Monoclinic]]<BR>[[Triclinic_crystal_system|Triclinic]]
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| ![[Orthorhombic_crystal_system|Orthorhombic]]
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| ![[Tetragonal_crystal_system|Tetragonal]]
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| ![[rhombohedral lattice system|Rhombohedral]]
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| ![[Cubic crystal system|Cubic]]
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| |- align=center
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| ![[Unit cell]]
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| |[[Parallelopiped]]
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| |colspan=2|[[Rectangular cuboid|Cuboid]]
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| |Trigonal<BR>[[trapezohedron]]
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| |[[Cube]]
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| |- valign=top align=center
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| ![[Point group]]<BR>Order<BR>Rotation subgroup
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| |[ ], (*)<BR>Order 2<BR>[ ]<sup>+</sup>, (1)
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| |[2,2], (*222)<BR>Order 8<BR>[2,2]<sup>+</sup>, (222)
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| |[4,2], (*422)<BR>Order 16<BR>[4,2]<sup>+</sup>, (422)
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| |[3], (*33)<BR>Order 6<BR>[3]<sup>+</sup>, (33)
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| |[4,3], (*432)<BR>Order 48<BR>[4,3]<sup>+</sup>, (432)
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| |- align=center
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| !Diagram
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| |[[File:Monoclinic.svg|80px]]
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| |[[File:Orthorhombic.svg|80px]]
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| |[[File:Tetragonal.svg|60px]]
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| |[[File:Hexagonal latticeR.svg|100px]]
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| |[[File:Lattic_simple_cubic.svg|100px]]
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| |- align=center
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| ![[Space group]]<BR>Rotation subgroup
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| |Pm (6)<BR>P1 (1)
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| |Pmmm (47)<BR>P222 (16)
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| |P4/mmm (123)<BR>P422 (89)
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| |R3m (160)<BR>R3 (146)
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| |Pm{{overline|3}}m (221)<BR>P432 (207)
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| |- align=center
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| ![[Coxeter notation]]
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| | -
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| | [∞]<sub>a</sub>×[∞]<sub>b</sub>×[∞]<sub>c</sub>
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| | [4,4]<sub>a</sub>×[∞]<sub>c</sub>
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| | -
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| | [4,3,4]<sub>a</sub>
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| |- align=center
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| ![[Coxeter diagram]]
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| | -
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| | {{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}
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| | {{CDD|node_1|4|node|4|node|2|node_1|infin|node}}
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| | -
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| | {{CDD|node_1|4|node|3|node|4|node}}
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| |}
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| == Uniform colorings ==
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| There is a large number of [[uniform coloring]]s, derived from different symmetries. These include:
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| {| class="wikitable"
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| ![[Coxeter notation]]<BR>[[Space group]]
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| ![[Coxeter diagram]]
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| ![[Schläfli symbol]]
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| !Partial<BR>honeycomb
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| !Colors by letters
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| |-
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| ![4,3,4]<BR>Pm{{overline|3}}m (221)
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| |{{CDD|node_1|4|node|3|node|4|node}}
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| |{4,3,4}
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| | [[Image:Partial cubic honeycomb.png|50px]]
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| | 1: aaaa/aaaa
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| |-
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| ![4,3<sup>1,1</sup>]<BR>Fm{{overline|3}}m (225)
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| |{{CDD|node_1|4|node|split1|nodes}}
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| |{4,3<sup>1,1</sup>}
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| | [[Image:Bicolor cubic honeycomb.png|50px]]
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| | 2: abba/baab
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| |-
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| ![4,3,4]<BR>Pm{{overline|3}}m (221)
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| |{{CDD|node_1|4|node|3|node|4|node_1}}
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| |t<sub>0,3</sub>{4,3,4}
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| | [[Image:Runcinated cubic honeycomb.png|50px]]
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| | 4: abbc/bccd
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| |-
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| !<nowiki>[[</nowiki>4,3,4]]<BR>Pm{{overline|3}}m (229)
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| |{{CDD|node_1|4|node|3|node|4|node_1}}
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| |t<sub>0,3</sub>{4,3,4}
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| | 4: abbb/bbba
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| |-
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| ![4,3,4,2,∞]
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| |{{CDD|node_1|4|node|4|node|2|node_1|infin|node_1}}
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| |{4,4}×t{∞}
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| | [[Image:Square prismatic honeycomb.png|50px]]
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| | 2: aaaa/bbbb
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| |-
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| ![4,3,4,2,∞]
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| |{{CDD|node|4|node_1|4|node|2|node_1|infin|node}}
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| |t<sub>1</sub>{4,4}×{∞}
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| | [[Image:Square prismatic 2-color honeycomb.png|50px]]
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| | 2: abba/abba
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| |-
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| ![∞,2,∞,2,∞]
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| |{{CDD|node_1|infin|node_1|2|node_1|infin|node_1|2|node_1|infin|node}}
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| |t{∞}×t{∞}×{∞}
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| | [[Image:Square 4-color prismatic honeycomb.png|50px]]
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| | 4: abcd/abcd
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| |-
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| ![∞,2,∞,2,∞]
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| |{{CDD|node_1|infin|node_1|2|node_1|infin|node_1|2|node_1|infin|node_1}}
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| |t{∞}×t{∞}×t{∞}
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| | [[Image:Cubic 8-color honeycomb.png|50px]]
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| | 8: abcd/efgh
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| |}
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| == Related 3-space tesellations ==
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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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| === Related polytopes and honeycombs===
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| It is related to the regular [[4-polytope]] [[tesseract]], [[Schläfli symbol]] {4,3,3}, which exists in 4-space, and only has ''3'' cubes around each edge. It's also related to the [[order-5 cubic honeycomb]], Schläfli symbol {4,3,5}, of [[hyperbolic space]] with 5 cubes around each edge.
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| It is in a sequence of polychora and honeycomb with [[octahedron|octahedral]] [[vertex figure]]s.
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| {{Octahedral_vertex_figure_tessellations}}
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| It in a sequence of [[regular polychora]] and honeycombs with [[cube|cubic]] [[cell (geometry)|cells]].
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| {{Cubic cell tessellations}}
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| ==See also ==
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| {{Commonscat|Cubic honeycomb}}
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| *[[Architectonic and catoptric tessellation]]
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| *[[Alternated cubic honeycomb]]
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| *[[List of regular polytopes]]
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| * [[Order-5 cubic honeycomb]] A hyperbolic cubic honeycomb with 5 cubes per edge
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| == References ==
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| {{reflist}}
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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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| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
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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|x4o3o4o - chon - O1}}
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| * [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 01-Chon]
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| {{Honeycombs}}
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| [[Category:Honeycombs (geometry)]]
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| [[Category:Polychora]]
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