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| {{no footnotes|date=November 2011}}
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| {| class="wikitable" align="right" style="margin-left:10px" width="250"
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| |bgcolor=#e7dcc3 align=center colspan=3|'''{{PAGENAME}}'''
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| |-
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| |bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid rectified 5-cell.png|220px]]<BR>[[Schlegel diagram]] with the 5 tetrahedral cells shown.
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| |-
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| |bgcolor=#e7dcc3|Type
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| |colspan=2|[[Uniform polychoron]]
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| |-
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]
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| |colspan=2| t<sub>1</sub>{3,3,3}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]
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| |colspan=2|{{CDD|node|3|node_1|3|node|3|node}}
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |10
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| |5 [[tetrahedron|{3,3}]] [[Image:Tetrahedron.png|20px]]<BR>5 [[octahedron|''3.3.3.3'']] [[Image:Uniform polyhedron-33-t1.png|20px]]
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| |-
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| |bgcolor=#e7dcc3|Faces
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| |colspan=2|30 [[Triangle|{3}]]
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|30
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|10
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| |-
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[Image:Rectified 5-cell verf.png|80px]]<BR>[[Triangular prism]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
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| |colspan=2|A<sub>4</sub>, [3,3,3], order 120
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| |-
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| |bgcolor=#e7dcc3|[[Petrie Polygon]]
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| |colspan=2|[[Pentagon]]
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]], [[isogonal figure|isogonal]], [[isotoxal figure|isotoxal]]
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| |-
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[Pentachoron|1]]'' 2 ''[[Truncated 5-cell|3]]''
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| |}
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| [[Image:triangular prism.png|220px|thumb|right|[[Vertex figure]]: [[triangular prism]]<BR>5 faces:<BR>
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| [[Image:tetrahedron vertfig.png|100px]][[Image:Octahedron vertfig.png|100px]]<BR>
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| 2 ([[tetrahedron|''3.3.3'']]) and 3 ([[octahedron|''3.3.3.3'']])
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| ]]
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| In [[Four-dimensional space|four dimensional]] [[geometry]], the '''[[Rectification (geometry)|rectified]] [[5-cell]]''' is a [[uniform polychoron]] composed of 5 regular tetrahedral and 5 regular octahedral [[cell (mathematics)|cells]]. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the [[vertex figure]] is a [[triangular prism]].
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| It is one of three [[Semiregular 4-polytopes#Semiregular_polytopes|semiregular polychora]] made of two or more cells which are [[platonic solid]]s, discovered by [[Thorold Gosset]] in his 1900 paper. He called it a '''Tetroctahedric''' for being made of [[tetrahedron]] and [[octahedron]] cells.
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| The [[vertex figure]] of the ''rectified 5-cell'' is a uniform [[triangular prism]], formed by three [[octahedron|octahedra]] around the sides, and two [[tetrahedron|tetrahedra]] on the opposite ends.
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| == Alternate names == | |
| * Dispentachoron
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| * Rectified 5-cell ([[Norman Johnson (mathematician)|Norman W. Johnson]])
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| * Rectified [[Simplex|4-simplex]]
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| * Rectified pentachoron (Acronym: rap) (Jonathan Bowers)
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| * Ambopentachoron (Neil Sloane & [[John Horton Conway]])
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| ==Images ==
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| {{4-simplex Coxeter plane graphs|t1|150}}
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| {| class="wikitable" width=640
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| |align=center|[[Image:Rectified simplex stereographic.png|220px]]<BR>[[stereographic projection]]<BR>(centered on [[octahedron]])
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| |align=center|[[Image:Rectified 5-cell net.png|220px]]<BR>[[Net (polytope)]]
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| |-
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| |align=center|[[Image:Rectified 5cell-perspective-tetrahedron-first-01.gif]]
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| |[[Tetrahedron]]-centered perspective projection into 3D space, with nearest tetrahedron to the 4D viewpoint rendered in red, and the 4 surrounding octahedra in green. Cells lying on the far side of the polytope have been culled for clarity (although they can be discerned from the edge outlines). The rotation is only of the 3D projection image, in order to show its structure, not a rotation in 4D space.
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| |}
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| == Coordinates ==
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| The [[Cartesian coordinate]]s of the vertices of an origin-centered rectified 5-cell having edge length 2 are:
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| {|
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| :<math>\left(\sqrt{\frac{2}{5}},\ \frac{2}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left(\sqrt{\frac{2}{5}},\ \frac{2}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\left(\sqrt{\frac{2}{5}},\ \frac{-2}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\left(\sqrt{\frac{2}{5}},\ \frac{-2}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm1\right)</math>
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| :<math>\left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0 \right)</math>
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| :<math>\left(\frac{-3}{\sqrt{10}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0 \right)</math>
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| |}
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| More simply, the vertices of the ''rectified 5-cell'' can be positioned on a [[hyperplane]] in 5-space as permutations of (0,0,0,1,1) ''or'' (0,0,1,1,1). These construction can be seen as positive [[orthant]] facets of the [[rectified pentacross]] or [[birectified penteract]] respectively.
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| == Related polychora ==
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| This polytope is the [[vertex figure]] of the [[5-demicube]], and the [[edge figure]] of the uniform [[2 21 polytope|2<sub>21</sub> polytope]].
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| It is also one of 9 [[Uniform polychoron|uniform polychora]] constructed from the [3,3,3] [[Coxeter group]].
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| {{Pentachoron family}}
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| === Related polytopes and honeycombs ===
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| The rectified 5-cell is second in a dimensional series of [[Uniform k21 polytope|semiregular polytope]]s. Each progressive [[uniform polytope]] is constructed [[vertex figure]] of the previous polytope. [[Thorold Gosset]] identified this series in 1900 as containing all [[regular polytope]] facets, containing all [[simplex]]es and [[orthoplex]]es ([[tetrahedron]]s and [[octahedron]]s in the case of the rectified 5-cell). In [[Coxeter]]'s notation the rectified 5-cell is given the symbol 0<sub>21</sup>.
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| {{k 21 polytopes}}
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| ==See also==
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| *[[Semiregular k 21 polytope]]
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| == References ==
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| * [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
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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. (1966)
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| ==External links==
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| * [http://www.polytope.de/nr03.html Rectified 5-cell] - data and images
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| ** {{PolyCell | urlname = section1.html| title = 1. Convex uniform polychora based on the pentachoron - Model 2}}
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| * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|x3o3o3o - rap}}
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora]]
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