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| | The writer's name is Christy Brookins. My day occupation is an invoicing officer but I've currently applied for another 1. What me and my family members love is bungee jumping but I've been using on new things lately. North Carolina is the location he loves most but now he is considering other choices.<br><br>Here is my blog; free online tarot card readings [[http://conniecolin.com/xe/community/24580 conniecolin.com]] |
| !bgcolor=#e7dcc3 colspan=2|{{PAGENAME}}
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| |bgcolor=#ffffff align=center colspan=2|[[File:Cantellated cubic tiling.png|190px]] [[File:HC A5-A3-P2.png|112px]]
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| |bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||rr{4,3,4}<BR>t<sub>0,2</sub>{4,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node|3|node_1|4|node}}
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| |bgcolor=#e7dcc3|Cells||[[rhombicuboctahedron|rr{4,3}]] [[File:Uniform_polyhedron-43-t02.png|40px]]<BR>[[cuboctahedron|r{4,3}]] [[File:Uniform_polyhedron-43-t1.png|40px]]<BR>[[cube|{4,3}]] [[File:Uniform_polyhedron-43-t0.png|40px]]
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| |bgcolor=#e7dcc3|[[Euler characteristic]]||0
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| |bgcolor=#e7dcc3|Vertex figure||[[File:Cantellated cubic honeycomb verf.png|75px]]<BR>([[Wedge (geometry)|Wedge]])
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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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| |bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
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| |bgcolor=#e7dcc3|Dual|| [[quarter oblate octahedrille]]
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
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| |}
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| The '''cantellated cubic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. It is composed of [[rhombicuboctahedron|rhombicuboctahedra]], [[cuboctahedron|cuboctahedra]], and [[cube]]s in a ratio of 1:1:3. | |
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| [[John Horton Conway]] calls this honeycomb a '''2-RCO-trille''', and its dual [[quarter oblate octahedrille]].
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| == Images ==
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| {| class=wikitable width=400
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| |[[File:Cantellated cubic honeycomb.png|200px]]
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| |[[File:Perovskite.jpg|200px]]<BR>It is closely related to the [[Perovskite (structure)|perovskite structure]], shown here with cubic symmetry, with atoms placed at the center of the cells of this honeycomb.
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| |}
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| == Symmetry ==
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| There is a second [[uniform coloring]]s by reflectional symmetry of the [[Coxeter group]]s, the second seen with alternately colored rhombicuboctahedral cells.
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| {| class="wikitable" width=360
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| |+ Vertex uniform colorings by cell
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| !Construction
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| !Truncated cubic honeycomb
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| !Bicantellated alternate cubic
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| |- valign=top
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| ![[Coxeter group]]
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| ![4,3,4], <math>{\tilde{C}}_3</math><BR>=<[4,3<sup>1,1</sup>]>
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| ![4,3<sup>1,1</sup>], <math>{\tilde{B}}_3</math>
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| |-
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| ![[Space group]]||Pm{{overline|3}}m||Fm{{overline|3}}m
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| ![[Coxeter-Dynkin diagram]]
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| !{{CDD|node_1|4|node|3|node_1|4|node}}
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| !{{CDD|node_1|4|node|split1|nodes_11}}
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| |- align=center
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| !Coloring
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| |[[File:Cantellated cubic honeycomb.png|120px]]
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| |[[File:Cantellated cubic honeycomb2.png|120px]]
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| ![[Vertex figure]]
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| |[[File:Cantellated cubic honeycomb verf.png|120px]]
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| |[[File:Runcicantellated alternate cubic honeycomb verf.png|120px]]
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| |- align=center
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| !Vertex<BR>figure<BR>symmetry
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| |[ ]<BR>order 2
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| |[ ]<sup>+</sup><BR>order 1
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| |}
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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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| == See also==
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| *[[Architectonic and catoptric tessellation]]
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| == References ==
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| {{Commons category|Cantellated cubic honeycomb}}
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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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| * [[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|x4o3x4o - srich - O17}}
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| * [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 04-Srich]
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| {{polychora-stub}}
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| [[Category:Honeycombs (geometry)]]
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The writer's name is Christy Brookins. My day occupation is an invoicing officer but I've currently applied for another 1. What me and my family members love is bungee jumping but I've been using on new things lately. North Carolina is the location he loves most but now he is considering other choices.
Here is my blog; free online tarot card readings [conniecolin.com]