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| | I'm Laverne and was born on 6 October 1970. My hobbies are Vintage clothing and Chainmail making.<br><br>Here is my weblog :: [http://www.freshwidewallpapers.com/profile/romaum Here is your mountain bike sizing.] |
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| |[[File:5-cube t4.svg|100px]]<BR>[[5-orthoplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}}
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| |[[File:5-cube t3.svg|100px]]<BR>Rectified 5-orthoplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}}
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| |[[File:5-cube t2.svg|100px]]<BR>[[Birectified 5-cube]]<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|4|node}}
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| |[[File:5-cube t1.svg|100px]]<BR>[[Rectified 5-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|4|node}}
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| |[[File:5-cube t0.svg|100px]]<BR>[[5-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
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| !colspan=5|[[Orthogonal projection]]s in B<sub>5</sub> [[Coxeter plane]]
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| |}
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| In five-dimensional [[geometry]], a '''rectified 5-orthoplex''' is a convex [[uniform 5-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[5-orthoplex]].
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| There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the [[5-orthoplex]] itself, and the 4th and last being the [[5-cube]]. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.
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| == Rectified 5-orthoplex==
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| {| class="wikitable" align="right" style="margin-left:10px" width="250"
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| !bgcolor=#e7dcc3 colspan=2|Rectified pentacross
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| |bgcolor=#e7dcc3|Type||[[uniform polyteron]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|split1|nodes}}
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| |bgcolor=#e7dcc3|Hypercells||42 total:<BR>10 [[16-cell|{3,3,4}]]<BR>32 [[Rectified 5-cell|t<sub>1</sub>{3,3,3}]]
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| |-
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| |bgcolor=#e7dcc3|Cells||240 total:<BR>80 [[octahedron|{3,4}]]<BR>160 [[tetrahedron|{3,3}]]
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| |bgcolor=#e7dcc3|Faces||400 total:<BR>80+320 [[triangle|{3}]]
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| |bgcolor=#e7dcc3|Edges||240
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| |bgcolor=#e7dcc3|Vertices||40
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified pentacross verf.png|40px]]<BR>[[Octahedral prism]]
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| |bgcolor=#e7dcc3|[[Petrie polygon]]||[[Decagon]]
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| |bgcolor=#e7dcc3|[[Coxeter group]]s||BC<sub>5</sub>, [3,3,3,4]<BR>D<sub>5</sub>, [3<sup>2,1,1</sup>]
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| |bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
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| |}
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| Its 40 vertices represent the root vectors of the [[simple Lie group]] D<sub>5</sub>. The vertices can be seen in 3 [[hyperplane]]s, with the 10 vertices [[rectified 5-cell]]s cells on opposite sides, and 20 vertices of a [[runcinated 5-cell]] passing through the center. When combined with the 10 vertices of the [[5-orthoplex]], these vertices represent the 50 root vectors of the B<sub>5</sub> and C<sub>5</sub> simple Lie groups.
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| === Alternate names===
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| * rectified pentacross
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| * rectified triacontiditeron (32-faceted polyteron)
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| === Construction ===
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| There are two [[Coxeter group]]s associated with the ''rectified pentacross'', one with the C<sub>5</sub> or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of ''16-cell'' facets, alternating, with the D<sub>5</sub> or [3<sup>2,1,1</sup>] Coxeter group.
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| == Cartesian coordinates ==
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| [[Cartesian coordinates]] for the vertices of a rectified pentacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
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| : (±1,±1,0,0,0)
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| ===Images===
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| {{5-cube Coxeter plane graphs|t3|150}}
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| == Related polytopes ==
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| The ''rectified 5-orthoplex'' is the vertex figure for the [[5-demicube honeycomb]]:
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| :{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} | |
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| This polytope is one of 31 [[Uniform_polyteron#Uniform_polyteron|uniform polytera]] generated from the regular [[5-cube]] or [[5-orthoplex]].
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| {{Penteract family}}
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| == Notes==
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| {{reflist}}
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| == References ==
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| * [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
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| ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
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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]
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| *** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
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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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| * [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
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| ** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
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| * {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} o3x3o3o4o - rat
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| == External links ==
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| *{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
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| * [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
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| * [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
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| {{Polytopes}}
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| [[Category:5-polytopes]]
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I'm Laverne and was born on 6 October 1970. My hobbies are Vintage clothing and Chainmail making.
Here is my weblog :: Here is your mountain bike sizing.