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| | Let me initial start by introducing myself. My title is Boyd Butts although it is not the name on my birth certification. North Dakota is exactly where me and my spouse live. For many years I've been operating as a payroll clerk. It's not a common factor but what she likes doing is foundation jumping and now she is attempting to make cash with it.<br><br>Here is my website ... [http://wixothek.com/user/MBuckmast home std test kit] |
| !bgcolor=#e7dcc3 colspan=2|Omnitruncated cubic honeycomb
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| |bgcolor=#ffffff align=center colspan=2|[[Image:Omnitruncated cubic tiling.png|190px]] [[File:HC A6-Pr8.png|110px]]
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| |bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1,2,3</sub>{4,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node_1|4|node_1|3|node_1|4|node_1}}
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| |bgcolor=#e7dcc3|Vertex figure||[[Image:Omnitruncated cubic honeycomb verf.png|80px]]<BR>[[Phyllic disphenoid]]
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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]]||[4,3,4], <math>{\tilde{C}}_3</math>
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| |bgcolor=#e7dcc3|Dual||[[eighth pyramidille]]
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
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| |}
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| The '''omnitruncated cubic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. It is composed of [[truncated cuboctahedron|truncated cuboctahedra]] and [[octagonal prism]]s in a ratio of 1:3.
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| [[John Horton Conway]] calls this honeycomb a '''b-tCO-trille''', and its dual [[eighth pyramidille]].
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| == Symmetry ==
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| Cells can be shown in two different symmetries. The [[Coxeter diagram]] form has two colors of [[truncated cuboctahedron|truncated cuboctahedra]] and [[octahedral prism]]s. The symmetry can be doubled by relating the first and last branches of the Coxeter diagram, which can be shown with one color for all the truncated cuboctahedral and octahedral prism cells.
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| |+ Two uniform colorings
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| ![[Coxeter notation|Symmetry]]
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| !<math>{\tilde{C}}_3</math>, [4,3,4]
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| !<math>{\tilde{C}}_3</math>×2, [<span/>[4,3,4]]
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| ![[Space group]]||Pm{{overline|3}}m (221)||Im{{overline|3}}m (229)
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| ![[Fibrifold]]||4<sup>−</sup>:2||8<sup>o</sup>:2
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| |- align=center
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| !Coloring
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| |[[Image:Omnitruncated cubic honeycomb1.png|150px]]
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| |[[Image:Omnitruncated cubic honeycomb2.png|150px]]
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| ![[Coxeter diagram]]
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| !{{CDD|node_1|4|node_1|3|node_1|4|node_1}}
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| !{{CDD|branch_11|4a4b|nodes_11}}
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| |- align=center
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| ![[Vertex figure]]
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| |[[File:Omnitruncated cubic honeycomb verf.png|100px]]
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| |[[Image:Omnitruncated cubic honeycomb verf2.png|100px]]
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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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| === Alternation===
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| A '''snub cubic honeycomb''' can be constructed by [[Alternation (geometry)|alternation]] of the omnitruncated cubic honeycomb, although it can not be made uniform, but it can be given [[Coxeter-Dynkin diagram]]: {{CDD|node_h|4|node_h|3|node_h|4|node_h}} and has symmetry [4,3,4]<sup>+</sup>. It makes [[snub cube]]s from the [[truncated cuboctahedron|truncated cuboctahedra]], [[square antiprism]]s from the [[octagonal prism]]s and with new tetrahedral cells created in the gaps.
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| ==See also==
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| {{Commons category|Omnitruncated 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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| * [[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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| * [[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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| * {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x4x3x4x - otch - O20}}
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| * [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 08-Otch]
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| [[Category:Honeycombs (geometry)]]
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| {{polychora-stub}}
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Let me initial start by introducing myself. My title is Boyd Butts although it is not the name on my birth certification. North Dakota is exactly where me and my spouse live. For many years I've been operating as a payroll clerk. It's not a common factor but what she likes doing is foundation jumping and now she is attempting to make cash with it.
Here is my website ... home std test kit