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| | Hello, my name is Andrew and my wife doesn't like it at all. What me and my family members love is to climb but I'm considering on starting some thing new. He functions as a bookkeeper. North Carolina is the location he enjoys most but now he is considering other choices.<br><br>Here is my web blog :: online psychic chat ([http://www.khuplaza.com/dent/14869889 http://www.khuplaza.com/dent/14869889]) |
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| |bgcolor=#e7dcc3 align=center colspan=3|'''Rectified 24-cell'''
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| |-
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| |bgcolor=#ffffff align=center colspan=3|[[File:Schlegel half-solid cantellated 16-cell.png|280px]]<BR>[[Schlegel diagram]]<BR>8 of 24 cuboctahedral cells shown
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| |bgcolor=#e7dcc3|Type
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| |colspan=2|[[Uniform polychoron]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]
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| |colspan=2|r{3,4,3}<BR>rr{3,3,4}<BR>r{3<sup>1,1,1</sup>}
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
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| |colspan=2|{{CDD|node|3|node_1|4|node|3|node}}<BR>{{CDD|node_1|3|node|3|node_1|4|node}}<BR>{{CDD|node_1|3|node|split1|nodes_11}} or {{CDD|node|splitsplit1|branch3_11|node_1}}
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| |-
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| |bgcolor=#e7dcc3|Cells
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| |48
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| |24 [[Cuboctahedron|''3.4.3.4'']] [[File:Cuboctahedron.png|20px]]<BR>24 [[cube|''4.4.4'']] [[File:Hexahedron.png|20px]]
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| |bgcolor=#e7dcc3|Faces
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| |240
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| |96 [[triangle|{3}]]<br>144 [[square (geometry)|{4}]]
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|288
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| |-
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|96
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[File:rectified 24-cell verf.png|50px]][[File:Cantellated 16-cell verf.png|50px]][[File:Runcicantellated demitesseract verf.png|50px]]<BR>[[Triangular prism]]
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry groups]]
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| |colspan=2|F<sub>4</sub> [3,4,3], order 1152<BR>B<sub>4</sub> [3,3,4], order 384<BR>D<sub>4</sub> [3<sup>1,1,1</sup>], order 192
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| |-
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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]], [[edge-transitive]]
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| |-
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[24-cell|22]]'' 23 ''[[truncated 24-cell|24]]''
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| |}
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| In [[geometry]], the '''rectified 24-cell''' is a uniform 4-dimensional polytope (or [[uniform polychoron]]), which is bounded by 48 [[cell (mathematics)|cells]]: 24 [[cube]]s, and 24 [[cuboctahedron|cuboctahedra]]. It can be obtained by reducing the [[icositetrachoron]]'s cells to cubes or cuboctahedra.
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| It can also be considered a '''cantellated 16-cell''' with the lower symmetries B<sub>4</sub> = [3,3,4]. B<sub>4</sub> would lead to a bicoloring of the [[cuboctahedron|cuboctahedral]] cells into 8 and 16 each. It is also called a '''runcicantellated demitesseract''' in a D<sub>4</sub> symmetry, giving 3 colors of cells, 8 for each.
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| == Cartesian coordinates ==
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| A rectified 24-cell having an edge length of √2 has vertices given by all permutations and sign permutations of the following [[Cartesian coordinate]]s:
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| : (0,1,1,2) [4!/2!×2<sup>3</sup> = 96 vertices]
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| The dual configuration with edge length 2 has all coordinate and sign permutations of:
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| : (0,2,2,2) [4×2<sup>3</sup> = 32 vertices] | |
| : (1,1,1,3) [4×2<sup>4</sup> = 64 vertices] | |
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| == Images ==
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| {{24-cell_4-cube_Coxeter_plane_graphs|t1|200|t2}}
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| {| class="wikitable" width=360
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| ![[Stereographic projection]]
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| |colspan=2 align=center|[[File:Rectified 24cell.png|360px]]<BR>
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| |Center of [[stereographic projection]]<BR>with 96 triangular faces blue
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| |}
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| == Symmetry constructions ==
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| There are three different symmetry constructions of this polytope. The lowest <math>{D}_3</math> construction can be doubled into <math>{C}_3</math> by adding a mirror that maps the bifurcating nodes onto each other. <math>{D}_3</math> can be mapped up to <math>{F}_3</math> symmetry by adding two mirror that map all three end nodes together.
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| The [[vertex figure]] is a [[triangular prism]], containing two cubes and three cuboctahedra. The three symmetries can be seen with 3 colored cuboctahedra in the lowest <math>{D}_3</math> construction, and two colors (1:2 ratio) in <math>{C}_3</math>, and all identical cuboctahedra in <math>{F}_3</math>.
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| {| class='wikitable'
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| ![[Coxeter group]]
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| !<math>{F}_3</math> = [3,4,3]
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| !<math>{C}_3</math> = [4,3,3]
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| !<math>{D}_3</math> = [3,3<sup>1,1</sup>]
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| |-
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| !Order
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| |1152
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| |384
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| |192
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| |-
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| !Full<BR>symmetry<BR>group
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| |[3,4,3]
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| |[4,3,3]
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| |<[3,3<sup>1,1</sup>]> = [4,3,3]<BR>[3[3<sup>1,1,1</sup>]] = [3,4,3]
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| |-
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| ![[Coxeter diagram]]
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| |{{CDD|node|3|node_1|4|node|3|node}}
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| |{{CDD|node|4|node_1|3|node|3|node_1}}
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| |{{CDD|nodes_11|split2|node|3|node_1}}
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| ![[Facet (geometry)|Facets]]
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| |'''3:''' {{CDD|node|3|node_1|4|node}}<BR>'''2:''' {{CDD|node_1|4|node|3|node}}
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| |'''2,2:''' {{CDD|node_1|3|node|3|node_1}}<BR>'''2:''' {{CDD|node|4|node_1|2|node_1}}
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| |'''1,1,1:''' {{CDD|node_1|3|node|3|node_1}}<BR>'''2:''' {{CDD|node_1|2|node_1|2|node_1}}
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| |-
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| ![[Vertex figure]]
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| |[[File:Rectified 24-cell verf.png|80px]]
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| |[[File:Cantellated 16-cell verf.png|80px]]
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| |[[File:Runcicantellated demitesseract verf.png|80px]]
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| |}
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| ==Alternate names==
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| * Rectified 24-cell, Cantellated 16-cell ([[Norman Johnson (mathematician)|Norman Johnson]])
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| * Rectified icositetrachoron (Acronym rico) ([[George Olshevsky]], Jonathan Bowers)
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| ** Cantellated hexadecachoron
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| * Disicositetrachoron
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| * Amboicositetrachoron ([[John Horton Conway|Neil Sloane & John Horton Conway]])
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| == Related uniform polytopes ==
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| {{Demitesseract family}}
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| {{24-cell_family}}
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| The ''rectified 24-cell'' can also be derived as a ''cantellated 16-cell'':
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| {{Tesseract family}}
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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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| ** Coxeter, ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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| ** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
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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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| * [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strass, ''The Symmetries of Things'' 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1<sub>n1</sub>)
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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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| * {{PolyCell | urlname = section2.html| title = 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Model 23}}
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| ** {{PolyCell | urlname = section3.html| title = 3. Convex uniform polychora based on the icositetrachoron (24-cell) - Model 23}}
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| ** {{PolyCell | urlname = section7.html| title = 7. Uniform polychora derived from glomeric tetrahedron B4 - Model 23 }}
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| * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o3x4o3o - rico}}
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora]]
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