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| |bgcolor=#e7dcc3 align=center colspan=3|'''{{PAGENAME}}'''
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| |bgcolor=#ffffff align=center colspan=3|[[Image:Schlegel half-solid rectified 8-cell.png|280px]]<BR>[[Schlegel diagram]]<BR>Centered on cuboctahedron<BR>tetrahedral 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{4,3,3}<BR>2r{3,3<sup>1,1</sup>}<BR>h<sub>3</sub>{4,3,3}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s
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| |colspan=2|{{CDD|node|4|node_1|3|node|3|node}}<BR>{{CDD|nodes_11|split2|node|3|node}}<BR>{{CDD|nodes_10ru|split2|node|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node_1}}
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| |bgcolor=#e7dcc3|Cells
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| |24
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| |8 [[cuboctahedron|(''3.4.3.4'')]][[Image:Cuboctahedron.png|20px]]<BR>16 [[tetrahedron|(''3.3.3'')]][[Image:Tetrahedron.png|20px]]
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| |bgcolor=#e7dcc3|Faces
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| |88
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| |64 [[triangle|{3}]]<BR>24 [[square (geometry)|{4}]]
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| |-
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| |bgcolor=#e7dcc3|Edges
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| |colspan=2|96
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| |bgcolor=#e7dcc3|Vertices
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| |colspan=2|32
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| |bgcolor=#e7dcc3|[[Vertex figure]]
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| |colspan=2|[[Image:Rectified 8-cell verf.png|60px]][[File:Cantellated demitesseract verf.png|60px]]<BR>(Elongated equilateral-triangular prism)
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| |-
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| |bgcolor=#e7dcc3|[[Coxeter group|Symmetry group]]
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| |colspan=2|BC<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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| |bgcolor=#e7dcc3|Properties
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| |colspan=2|[[Convex polytope|convex]], [[edge-transitive]]
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| |bgcolor=#e7dcc3|Uniform index
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| |colspan=2|''[[Tesseract|10]]'' 11 ''[[16-cell|12]]''
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| |}
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| In [[geometry]], the '''rectified tesseract''', '''rectified 8-cell''', or '''runcic tesseract''' is a [[uniform polychoron]] (4-dimensional [[polytope]]) bounded by 24 [[cell_(mathematics)|cells]]: 8 [[cuboctahedron|cuboctahedra]], and 16 [[tetrahedron|tetrahedra]]. It has half the vertices of a [[runcinated tesseract]], with its {{CDD|node_h|4|node|3|node|3|node_1}} construction.
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| It has two uniform constructions, as a ''rectified 8-cell'' t<sub>1</sub>{4,3,3} and a [[Uniform_polychoron#The_D4_.5B31.2C1.2C1.5D_group_family_.28Demitesseract.29|cantellated demitesseract]], t<sub>0,2</sub>{3<sup>1,1,1</sup>}, the second alternating with two types of tetrahedral cells.
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| ==Construction==
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| The rectified tesseract may be constructed from the [[tesseract]] by [[Rectification (geometry)|truncating]] its vertices at the midpoints of its edges.
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| The [[Cartesian coordinates]] of the vertices of the rectified tesseract with edge length 2 is given by all permutations of:
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| :<math>(0,\ \pm\sqrt{2},\ \pm\sqrt{2},\ \pm\sqrt{2})</math>
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| == Images ==
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| {{4-cube Coxeter plane graphs|t1|100}}
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| {| class="wikitable"
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| |[[Image:Rectified_tesseract1.png|180px]]<BR>Wireframe
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| |[[Image:Rectified_tesseract2.png|180px]]<BR>16 [[tetrahedron|tetrahedral]] cells
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| |}
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| ==Projections==
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| In the cuboctahedron-first parallel projection of the rectified tesseract into 3-dimensional space, the image has the following layout:
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| * The projection envelope is a [[cube]].
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| * A cuboctahedron is inscribed in this cube, with its vertices lying at the midpoint of the cube's edges. The cuboctahedron is the image of two of the cuboctahedral cells.
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| * The remaining 6 cuboctahedral cells are projected to the square faces of the cube.
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| * The 8 tetrahedral volumes lying at the triangular faces of the central cuboctahedron are the images of the 16 tetrahedral cells, two cells to each image.
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| == Alternative names ==
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| *Rit (Jonathan Bowers: for rectified tesseract)
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| *Ambotesseract (Neil Sloane & John Horton Conway)
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| *Rectified tesseract/Runcic tesseract (Norman W. Johnson)
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| **Runcic 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
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| **Rectified 4-hypercube/8-cell/octachoron/4-measure polytope/4-regular orthotope
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| == Related uniform polytopes ==
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| {{Tesseract family}}
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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. (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 11}}
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| * {{KlitzingPolytopes|polychora.htm|4D uniform polytopes (polychora)|o4x3o3o - rit}}
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| {{Polytopes}}
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| [[Category:Four-dimensional geometry]]
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| [[Category:Polychora]]
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