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| | Hi there, I am Alyson Pomerleau and I believe it sounds fairly good when you say it. To perform lacross is 1 of the issues she loves most. He works as a bookkeeper. Her family lives in Ohio.<br><br>My website ... psychic solutions by lynne, [http://kpupf.com/xe/talk/735373 relevant resource site], |
| !bgcolor=#e7dcc3 colspan=2|Demipenteractic honeycomb
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| |bgcolor=#ffffff align=center colspan=2|(No image)
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| |bgcolor=#e7dcc3|Type||[[Uniform_polypeton#Regular and uniform honeycombs|uniform honeycomb]]
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| |bgcolor=#e7dcc3|Family||[[Alternated hypercubic honeycomb]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||h{4,3,3,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter diagram]]||
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| {{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} or {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}
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| <BR>{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} or {{CDD|node_h1|4|node|3|node|3|node|split1|nodes}}
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| <BR>{{CDD|node_h|4|node|3|node|3|node|3|node|4|node_h}}
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| <BR>{{CDD|node_h|4|node|3|node|3|node|4|node|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|3|node|split1|nodes|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|3|node|3|node|4|node_h|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|3|node|4|node|2|node_h|infin|node|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|split1|nodes|2|node_h|infin|node|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|3|node|4|node_h|2|node_h|infin|node|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|4|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|4|node|4|node_h|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}
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| <BR>{{CDD|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node|2|node_h|infin|node}}
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| |bgcolor=#e7dcc3|[[Facet (geometry)|Facets]]||[[5-orthoplex|{3,3,3,4}]] [[File:5-cube t4.svg|25px]]<BR>[[5-demicube|h{4,3,3,3}]] [[File:5-demicube t0 D5.svg|25px]]
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[Rectified 5-orthoplex|t<sub>1</sub>{3,3,3,4}]] [[File:Rectified pentacross.svg|25px]]
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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{B}}_5</math> [4,3,3,3<sup>1,1</sup>]<BR><math>{\tilde{D}}_5</math> [3<sup>1,1</sup>,3,3<sup>1,1</sup>]
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| |}
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| The '''5-demicube honeycomb''', or '''demipenteractic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 5-space. It is constructed as an [[Alternation (geometry)|alternation]] of the regular [[5-cube honeycomb]].
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| It is the first tessellation in the [[Cubic_honeycomb#Alternated_hypercube_tessellations|demihypercube honeycomb]] family which, with all the next ones, is not regular, being composed of two different types of [[Uniform polytope|uniform]] [[Facet (mathematics)|facet]]s. The [[5-cube]]s become alternated into [[5-demicube]]s h{4,3,3,3} and the alternated vertices create [[5-orthoplex]] {3,3,3,4} facets.
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| == D5 lattice ==
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| The [[vertex arrangement]] of the '''5-demicubic honeycomb''' is the '''D<sub>5</sub> lattice''' which is the densest known [[sphere packing]] in 5 dimensions.<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/D5.html</ref> The 40 vertices of the [[rectified 5-orthoplex]] [[vertex figure]] of the ''5-demicubic honeycomb'' reflect the [[kissing number]] 40 of this lattice.<ref>''Sphere packings, lattices, and groups'', by [[John Horton Conway]], Neil James Alexander Sloane, Eiichi Bannai
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| [http://books.google.com/books?id=upYwZ6cQumoC&lpg=PP1&dq=Sphere%20Packings%2C%20Lattices%20and%20Groups&pg=PR19#v=onepage&q=&f=false]</ref>
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| The D{{sup sub|+|5}} packing (also called D{{sup sub|2|5}}) can be constructed by the union of two D<sub>5</sub> lattices. The analogous packings form lattices only in even dimensions. The kissing number is 2<sup>4</sup>=16 (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|nodes_10ru|split2|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|split1|nodes_10lu}}
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| The D{{sup sub|*|5}}<ref>http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds5.html</ref> lattice (also called D{{sup sub|4|5}} and C{{sup sub|2|5}}) can be constructed by the union of all four 5-demicubic lattices:<ref>Conway (1998), p. 120</ref> It is also the 5-dimensional [[body centered cubic]], the union of two 5-cube honeycombs in dual positions.
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| :{{CDD|nodes_10ru|split2||node|3|node|split1|nodes}} + {{CDD|nodes_01rd|split2|node|3|node|split1|nodes}} + {{CDD|nodes|split2|node|3|node|split1|nodes_10lu}} + {{CDD|nodes|split2|node|3|node|split1|nodes_01ld}} = {{CDD|node_1|4|node|3|node|3|node|3|node|4|node}} + {{CDD|node|4|node|3|node|3|node|3|node|4|node_1}}
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| The [[kissing number]] of the D{{sup sub|*|5}} lattice is 10 (''2n'' for n≥5) and it [[Voronoi tessellation]] is a [[tritruncated 5-cubic honeycomb]], {{CDD|branch_11|3ab|nodes|4a4b|nodes}}, containing all with [[bitruncated 5-orthoplex]], {{CDD|node|4|node|3|node_1|3|node_1|3|node}} [[Voronoi cell]]s.<ref>Conway (1998), p. 466</ref>
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| == Symmetry constructions ==
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| There are three uniform construction symmetries of this tessellation. Each symmetry can be represented by arrangements of differened colors on the 32 [[5-demicube]] facets around each vertex.
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| {|class='wikitable'
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| ![[Coxeter group]]
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| ![[Schläfli symbol]]
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| ![[Coxeter-Dynkin diagram]]
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| ![[Vertex figure]]<BR>Symmetry
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| ![[Facet (geometry)|Facets]]/verf
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| |<math>{\tilde{B}}_5</math> = [3<sup>1,1</sup>,3,3,4]<BR>= [1<sup>+</sup>,4,3,3,4]||{3<sup>1,1</sup>,3,3,4}<BR> = h{4,3,3,3,4}||{{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node}}||{{CDD|node|3|node_1|3|node|3|node|4|node}}<BR>[3,3,3,4]
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| ||32: [[5-demicube]]<BR>10: [[5-orthoplex]]
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| |<math>{\tilde{D}}_5</math> = [3<sup>1,1</sup>,3,3<sup>1,1</sup>]<BR>= [1<sup>+</sup>,4,3,3<sup>1,1</sup>]||{3<sup>1,1</sup>,3,3<sup>1,1</sup>}||{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} = {{CDD|node_h1|4|node|3|node|3|node|split1|nodes}}||{{CDD|node|3|node_1|3|node|split1|nodes}}<BR>[3<sup>2,1,1</sup>]
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| ||16+16: [[5-demicube]]<BR>10: [[5-orthoplex]]
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| |-
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| |<math>{\tilde{C}}_5</math> = ([[4,3,3,4,2<sup>+</sup>]])||ht<sub>0,4</sub>{4,3,3,4}||{{CDD|node_h|4|node|3|node|3|node|3|node|4|node_h}}||
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| ||16+8+8: [[5-demicube]]<BR>10: [[5-orthoplex]]
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| |}
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| == Related honeycombs==
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| {{D5 honeycombs}}
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| == See also ==
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| *[[Uniform polytope]]
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| Regular and uniform honeycombs in 5-space:
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| *[[5-cube honeycomb]]
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| *[[5-demicube honeycomb]]
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| * [[5-simplex honeycomb]]
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| * [[Truncated 5-simplex honeycomb]]
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| * [[Omnitruncated 5-simplex honeycomb]]
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| == References ==
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| {{reflist}}
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| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
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| ** pp. 154–156: Partial truncation or alternation, represented by ''h'' prefix: h{4,4}={4,4}; h{4,3,4}={3<sup>1,1</sup>,4}, h{4,3,3,4}={3,3,4,3}, ...
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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 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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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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| *{{GlossaryForHyperspace | anchor=half | title=Half measure polytope }}
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| {{Honeycombs}}
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| {{DEFAULTSORT:Demipenteractic Honeycomb}}
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| [[Category:Honeycombs (geometry)]]
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| [[Category:6-polytopes]]
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