en>Mark viking: Added web ref verifying defn, removed unreferenced tag

2013-11-16T01:37:52Z

Added web ref verifying defn, removed unreferenced tag

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{| class=wikitable align=right
|- align=center valign=top
|[[File:10-cube_t8.svg|120px]] [[10-orthoplex]] {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|[[File:10-cube_t7.svg|120px]] [[Rectified 10-orthoplex]] {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}
|[[File:10-cube_t6.svg|120px]] [[Birectified 10-orthoplex]] {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}
|[[File:10-cube_t5.svg|120px]] [[Trirectified 10-orthoplex]] {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
|- align=center valign=top
|[[File:10-cube_t5.svg|120px]] [[Quadirectified 10-orthoplex]] {{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:10-cube_t4.svg|120px]] Quadrirectified 10-cube {{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
|[[File:10-cube_t3.svg|120px]] Trirectified 10-cube {{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|[[File:10-cube_t2.svg|120px]] Birectified 10-cube {{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|- align=center valign=top
|[[File:10-cube_t1.svg|120px]] Rectified 10-cube {{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|[[File:10-cube_t0.svg|120px]] [[10-cube]] {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
!colspan=4|[[Orthogonal projection]]s in BC10 [[Coxeter plane]]
|}
In ten-dimensional [[geometry]], a '''rectified 10-cube''' is a convex [[uniform 10-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[10-cube]].

There are 10 rectifications of the ''10-cube'', with the zeroth being the 10-cube itself. Vertices of the ''rectified 10-cube'' are located at the edge-centers of the ''10-cube''. Vertices of the ''birectified 10-cube'' are located in the square face centers of the ''10-cube''. Vertices of the ''trirectified 10-cube'' are located in the [[cube|cubic]] cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytpoe, the [[10-orthoplex]].

These polytopes are part of a family 1023 [[uniform 10-polytope]]s with BC10 symmetry.
== Rectified 10-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Rectified 10-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 10-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t1{38,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes_11|split2|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|7-faces||
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||46080
|-
|bgcolor=#e7dcc3|Vertices||5120
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||8-simplex prism
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C10, [4,38] D10, [37,1,1]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Rectified dekeract (Acronym rade) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o3o3o3o3o3x4o - rade)</ref>
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a rectified 10-cube, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

===Images===
{{B10 Coxeter plane graphs|t1|150}}

== Birectified 10-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Birectified 10-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 10-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t2{38,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes|split2|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|7-faces||
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||184320
|-
|bgcolor=#e7dcc3|Vertices||11520
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{4}x{36}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C10, [4,38] D10, [37,1,1]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Birectified dekeract (Acronym brade) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o3o3o3o3x3o4o - brade)</ref>
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a birectified 10-cube, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,±1,±1,±1,±1,±1,±1,0,0)

===Images===
{{B10 Coxeter plane graphs|t2|150}}

== Trirectified 10-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Trirectified 10-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 10-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t3{38,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes|split2|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|7-faces||
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||322560
|-
|bgcolor=#e7dcc3|Vertices||15360
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{4,3}x{35}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C10, [4,38] D10, [37,1,1]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Tririrectified dekeract (Acronym trade) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)</ref>
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a triirectified 10-cube, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,±1,±1,±1,±1,±1,0,0,0)
===Images===
{{B10 Coxeter plane graphs|t3|150}}

== Quadrirectified 10-cube==
{| class="wikitable" align="right" style="margin-left:10px" width="280"
!bgcolor=#e7dcc3 colspan=2|Quadrirectified 10-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 10-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t4{38,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}} {{CDD|nodes|split2|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
|-
|bgcolor=#e7dcc3|7-faces||
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||322560
|-
|bgcolor=#e7dcc3|Vertices||13440
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{4,3,3}x{34}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C10, [4,38] D10, [37,1,1]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}

=== Alternate names===
* Quadrirectified dekeract
* Quadrirectified decacross (Acronym trade) (Jonathan Bowers)<ref>Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)</ref>
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a quadrirectified 10-cube, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,±1,±1,±1,±1,0,0,0,0)
===Images===
{{B10 Coxeter plane graphs|t4|150}}

== Notes==
{{reflist}}

== References ==
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
** '''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]
*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* {{KlitzingPolytopes|polyxenna.htm|10D|uniform polytopes (polyxenna)}} x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker

== External links ==
*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
* [http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:10-polytopes]]

Data matrix (multivariate statistics) - Revision history

en>Mark viking: Added web ref verifying defn, removed unreferenced tag