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| | Oscar is what my spouse loves to contact me and I completely dig that title. For years at home [http://ehealthforum.com/health/sore-throat-after-oral-sex-t138861.html std test] he's been working as a meter reader and it's some thing he truly appreciate. One of [http://geekyplasta.com/blogs/post/3926 over the counter std test] home [http://www.reachoutsociety.info/user/JCondon std home test] test issues she loves most is to [http://www.Askdoctork.com/what-should-i-know-about-living-with-genital-herpes-201211073624 read comics] and she'll be starting something else along with it. My family members life in Minnesota and my family members loves it.<br><br>Look into my page: [http://glskating.com/groups/stay-yeast-infection-free-with-these-helpful-suggestions/ glskating.com] |
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| [[Image:Cantitruncated cubic tiling.png|190px]] [[File:HC A6-A4-P2.png|110px]]
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| |bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||tr{4,3,4}<BR>t<sub>0,1,2</sub>{4,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node_1|3|node_1|4|node}}
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| |bgcolor=#e7dcc3|Vertex figure||[[Image:Cantitruncated cubic honeycomb verf.png|50px]][[File:Omnitruncated alternated cubic honeycomb verf.png|50px]]<BR>(Irreg. [[tetrahedron]])
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| |bgcolor=#e7dcc3|[[Coxeter group]]||[4,3,4], <math>{\tilde{C}}_3</math>
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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|Dual||[[triangular pyramidille]]
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
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| |}
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| The '''cantitruncated cubic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space, made up of [[truncated cuboctahedron|truncated cuboctahedra]], [[truncated octahedron|truncated octahedra]], and [[cube]]s in a ratio of 1:1:3.
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| [[John Horton Conway]] calls this honeycomb a '''n-tCO-trille''', and its dual [[triangular pyramidille]].
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| == Images==
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| [[Image:Cantitruncated Cubic Honeycomb.svg|250px]] | |
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| == Symmetry ==
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| Cells can be shown in two different symmetries. The linear [[Coxeter-Dynkin diagram]] form can be drawn with one color for each cell type. The bifurcating diagram form can be drawn with two types (colors) of [[truncated cuboctahedron]] cells alternating.
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| {| class="wikitable" width=320
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| !Construction
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| !Cantitruncated cubic
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| !Omnitruncated 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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| ![[Space group]]||Pm{{overline|3}}m (221)||Fm{{overline|3}}m (225)
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| ![[Fibrifold]]||4<sup>−</sup>:2||2<sup>−</sup>:2
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| |- align=center
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| !Coloring
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| |[[Image:Cantitruncated_Cubic_Honeycomb.svg|80px]]
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| |[[Image:Cantitruncated_Cubic_Honeycomb2.svg|80px]]
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| ![[Coxeter-Dynkin diagram]]
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| !{{CDD|node_1|4|node_1|3|node_1|4|node}}
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| !{{CDD|node_1|4|node_1|split1|nodes_11}}
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| ![[Vertex figure]]
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| |[[Image:Cantitruncated cubic honeycomb verf.png|80px]]
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| |[[File:Omnitruncated alternated cubic honeycomb verf.png|80px]]
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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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| === Alternation ===
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| [[File:Alternated_cantitruncated_cubic_honeycomb_verf.png|150px|thumb|Vertex figure for alternated bitruncated cubic honeycomb, with 5 tetrahedral, one icosahedral, and two snub cube cells, but edge-lengths can't be made equal.]]
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| {| width=400 class=wikitable
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| |[[Image:Alternated cantitruncated cubic honeycomb.png|400px]]<BR>
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| This image shows a partial honeycomb of the [[Alternation (geometry)|alternation]] of the ''cantitruncated cubic honeycomb''. It contains three types of cells: [[snub cube]]s, [[Icosahedron|icosahedra]] (snub tetrahedron), and [[Tetrahedron|tetrahedra]]. In addition the gaps created at the alternated vertices form tetrahedral cells.<BR>This honeycomb exists in two mirror image forms. Although it is not uniform, constructionally it can be given as [[Coxeter-Dynkin diagram]]s {{CDD|node_h|4|node_h|split1|nodes_hh}} or {{CDD|node_h|4|node_h|3|node_h|4|node}}.
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| |}
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| ==See also==
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| {{Commons category|Cantitruncated cubic honeycomb}}
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| *[[Architectonic and catoptric tessellation]]
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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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| * {{The Geometrical Foundation of Natural Structure (book)|page=197}} Chapter 5 (Polyhedral packing and spacing filling): Fig. 5-13, p.176 shows this honeycomb. Fig. 5-34 shows a partial honeycomb of the alternation with only snub cube cells show.
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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|x4x3x4o - grich - O18}}
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| * [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 06-Grich]
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| [[Category:Honeycombs (geometry)]]
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| {{polychora-stub}}
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