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| {| class=wikitable align=right width=450
| | I’m Lenard from Hooton Pagnell studying Continuing Education and Summer Sessions. I did my schooling, secured 72% and hope to find someone with same interests in Racquetball.<br><br>Here is my webpage - [http://www.wallpaperhdquality.com/profile/toliddell Popular mountain bike sizing.] |
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| |[[File:6-cube_t5.svg|150px]]<BR>[[6-orthoplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
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| |[[File:6-cube_t4.svg|150px]]<BR>Rectified 6-orthoplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
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| |[[File:6-cube_t3.svg|150px]]<BR>Birectified 6-orthoplex<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
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| |- align=center
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| |[[File:6-cube_t2.svg|150px]]<BR>[[Birectified 6-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|4|node}}
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| |[[File:6-cube_t1.svg|150px]]<BR>[[Rectified 6-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|4|node}}
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| |[[File:6-cube_t0.svg|150px]]<BR>[[6-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
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| !colspan=4|[[Orthogonal projection]]s in B<sub>6</sub> [[Coxeter plane]]
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| |}
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| In six-dimensional [[geometry]], a '''rectified 6-orthoplex''' is a convex [[uniform 6-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[6-orthoplex]].
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| There are unique 6 degrees of rectifications, the zeroth being the [[6-orthoplex]], and the 6th and last being the [[6-cube]]. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.
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| == Rectified 6-orthoplex==
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| {| class="wikitable" align="right" style="margin-left:10px" width="250"
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| !bgcolor=#e7dcc3 colspan=2|Rectified hexacross
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| |bgcolor=#e7dcc3|Type||[[uniform polypeton]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]|| r{3,3,3,3,4}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|3|node|split1|nodes}}
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| |bgcolor=#e7dcc3|5-faces||76 total:<BR>64 [[rectified 5-simplex]]<BR>12 [[5-orthoplex]]
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| |-
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| |bgcolor=#e7dcc3|4-faces||576 total:<BR>192 [[rectified 5-cell]]<BR>384 [[5-cell]]
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| |-
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| |bgcolor=#e7dcc3|Cells||1200 total:<BR>240 [[octahedron]]<BR>960 [[tetrahedron]]
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| |bgcolor=#e7dcc3|Faces||1120 total:<BR>160 and 960 triangles
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| |bgcolor=#e7dcc3|Edges||480
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| |bgcolor=#e7dcc3|Vertices||60
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]||16-cell prism
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| |-
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| |bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]<BR>D<sub>6</sub>, [3<sup>3,1,1</sup>]
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| |-
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| |bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
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| |}
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| The ''rectified 6-orthoplex'' is the [[vertex figure]] for the [[demihexeractic honeycomb]].
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| :{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}}
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| === Alternate names===
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| * rectified hexacross
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| * rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)
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| === Construction ===
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| There are two [[Coxeter group]]s associated with the ''rectified hexacross'', one with the C<sub>6</sub> or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D<sub>6</sub> or [3<sup>3,1,1</sup>] Coxeter group.
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| === Cartesian coordinates ===
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| [[Cartesian coordinates]] for the vertices of a rectified hexacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
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| : (±1,±1,0,0,0,0)
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| ==== Root vectors ====
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| The 60 vertices represent the root vectors of the [[simple Lie group]] D<sub>6</sub>. The vertices can be seen in 3 [[hyperplane]]s, with the 15 vertices [[rectified 5-simplex]]s cells on opposite sides, and 30 vertices of an [[expanded 5-simplex]] passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B<sub>6</sub> and C<sub>6</sub> simple Lie groups.
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| ===Images===
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| {{6-cube Coxeter plane graphs|t5|150}}
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| == Birectified 6-orthoplex==
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| {| class="wikitable" align="right" style="margin-left:10px" width="250"
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| !bgcolor=#e7dcc3 colspan=2|Birectified 6-orthoplex
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| |-
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| |bgcolor=#e7dcc3|Type||[[uniform polypeton]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r{3,3,3,3,4}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|3|node|split1|nodes}}
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| |-
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| |bgcolor=#e7dcc3|5-faces||76
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| |-
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| |bgcolor=#e7dcc3|4-faces||636
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| |bgcolor=#e7dcc3|Cells||2160
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| |-
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| |bgcolor=#e7dcc3|Faces||2880
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| |-
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| |bgcolor=#e7dcc3|Edges||1440
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| |-
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| |bgcolor=#e7dcc3|Vertices||160
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[equilateral triangle|{3}]]×[[octahedron|{3,4}]] duoprism
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| |-
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| |bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]<BR>D<sub>6</sub>, [3<sup>3,1,1</sup>]
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| |-
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| |bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
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| |}
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| The '''birectified 6-orthoplex''' can tessellation space in the [[trirectified 6-cubic honeycomb]].
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| === Alternate names===
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| * birectified hexacross
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| * birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)
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| === Cartesian coordinates ===
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| [[Cartesian coordinates]] for the vertices of a rectified hexacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
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| : (±1,±1,±1,0,0,0)
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| ===Images===
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| {{6-cube Coxeter plane graphs|t4|150}}
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| == Related polytopes ==
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| These polytopes are a part a family of 63 [[Uniform_polypeton|uniform polypeta]] generated from the B<sub>6</sub> [[Coxeter plane]], including the regular [[6-cube]] or [[6-orthoplex]].
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| {{Hexeract 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''', editied 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|polypeta.htm|6D uniform polytopes (polypeta)| o3x3o3o3o4o - rag}}
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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:6-polytopes]]
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