10-orthoplexRectified 10-orthoplexBirectified 10-orthoplexTrirectified 10-orthoplexQuadrirectified 10-orthoplexQuadrirectified 10-cubeTrirectified 10-cubeBirectified 10-cubeRectified 10-cube10-cubeOrthogonal projections in BC10 Coxeter plane
In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a 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 cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex.
These polytopes are part of a family of 1023 uniform 10-polytopes with BC10 symmetry.
Rectified 10-cube Rectified 10-cubeTypeuniform 10-polytopeSchläfli symbol t1{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdges46080Vertices5120Vertex figure8-simplex prismCoxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Rectified dekeract (Acronym: rade) (Jonathan Bowers)
Cartesian coordinates Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,±1,±1,±1,0)
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Birectified 10-cube Birectified 10-cubeTypeuniform 10-polytopeCoxeter symbol 0711Schläfli symbol t2{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdges184320Vertices11520Vertex figure{4}x{36}Coxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Birectified dekeract (Acronym: brade) (Jonathan Bowers)
Cartesian coordinates Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,±1,±1,0,0)
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Trirectified 10-cube Trirectified 10-cubeTypeuniform 10-polytopeSchläfli symbol t3{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdges322560Vertices15360Vertex figure{4,3}x{35}Coxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Trirectified dekeract (Acronym: trade) (Jonathan Bowers)Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)
Cartesian coordinates Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,±1,0,0,0)
Images
Quadrirectified 10-cube Quadrirectified 10-cubeTypeuniform 10-polytopeSchläfli symbol t4{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdges322560Vertices13440Vertex figure{4,3,3}x{34}Coxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Quadrirectified dekeract Quadrirectified decacross (Acronym: terade) (Jonathan Bowers)Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)
Cartesian coordinates Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,±1,0,0,0,0)
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Notes
References H.S.M. Coxeter: H.S.M. Coxeter, Regular Polytopes, 3rd edition, Dover, New York, 1973 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, (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 Uniform Polytopes, Manuscript (1991) N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966) o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3x3o3o3o4o - terade
External links Polytopes of Various Dimensions Multi-dimensional Glossary