10-orthoplexRectified 10-orthoplexBirectified 10-orthoplexTrirectified 10-orthoplexQuadrirectified 10-orthoplexQuadrirectified 10-cubeTrirectified 10-cubeBirectified 10-cubeRectified 10-cube10-cubeOrthogonal projections in A10 Coxeter plane
In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex.
There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex.
These polytopes are part of a family of 1023 uniform 10-polytopes with BC10 symmetry.
Rectified 10-orthoplex Rectified 10-orthoplexTypeuniform 10-polytopeSchläfli symbol t1{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdges2880Vertices180Vertex figure8-orthoplex prismPetrie polygonicosagonCoxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex.
The rectified 10-orthoplex is the vertex figure of the demidekeractic honeycomb. or
Alternate names Rectified decacross (Acronym: rake) (Jonathan Bowers)
Construction There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or [4,38] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or [37,1,1] Coxeter group.
Cartesian coordinates Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0,0,0,0,0,0)
Root vectors Its 180 vertices represent the root vectors of the simple Lie group D10. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B10.
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Birectified 10-orthoplex Birectified 10-orthoplexTypeuniform 10-polytopeSchläfli symbol t2{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Birectified decacross (Acronym: brake) (Jonathan Bowers)
Cartesian coordinates Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,0,0,0,0,0,0,0)
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Trirectified 10-orthoplex Trirectified 10-orthoplexTypeuniform 10-polytopeSchläfli symbol t3{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Trirectified decacross (Acronym: trake) (Jonathan Bowers)Klitzing, (o3o3o3x3o3o3o3o3o4o - trake)
Cartesian coordinates Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,0,0,0,0,0,0)
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Quadrirectified 10-orthoplex Quadrirectified 10-orthoplexTypeuniform 10-polytopeSchläfli symbol t4{38,4}Coxeter-Dynkin diagrams9-faces8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figureCoxeter groupsC10, [4,38]D10, [37,1,1]Propertiesconvex
Alternate names Quadrirectified decacross (Acronym: terake) (Jonthan Bowers)Klitzing, (o3o3o3o3x3o3o3o3o4o - terake)
Cartesian coordinates Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,0,0,0,0,0)
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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) o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake
External links Polytopes of Various Dimensions Multi-dimensional Glossary