9-simplexRectified 9-simplexBirectified 9-simplexTrirectified 9-simplexQuadrirectified 9-simplexOrthogonal projections in A9 Coxeter plane In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.
These polytopes are part of a family of 271 uniform 9-polytopes with A9 symmetry.
There are unique 4 degrees of rectifications. Vertices of the rectified 9-simplex are located at the edge-centers of the 9-simplex. Vertices of the birectified 9-simplex are located in the triangular face centers of the 9-simplex. Vertices of the trirectified 9-simplex are located in the tetrahedral cell centers of the 9-simplex. Vertices of the quadrirectified 9-simplex are located in the 5-cell centers of the 9-simplex.
Rectified 9-simplex Rectified 9-simplexTypeuniform 9-polytopeSchläfli symbol t1{3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams8-faces207-faces1356-faces4805-faces10504-faces1512Cells1470Faces960Edges360Vertices45Vertex figure8-simplex prismPetrie polygondecagonCoxeter groupsA9, [3,3,3,3,3,3,3,3]Propertiesconvex
The rectified 9-simplex is the vertex figure of the 10-demicube.
Alternate names Rectified decayotton (reday) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the rectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 10-orthoplex.
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Birectified 9-simplex Birectified 9-simplexTypeuniform 9-polytopeSchläfli symbol t2{3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams8-faces7-faces6-faces5-faces4-facesCellsFacesEdges1260Vertices120Vertex figure{3}×{3,3,3,3,3}Coxeter groupsA9, [3,3,3,3,3,3,3,3]Propertiesconvex
This polytope is the vertex figure for the 162 honeycomb. Its 120 vertices represent the kissing number of the related hyperbolic 9-dimensional sphere packing.
Alternate names Birectified decayotton (breday) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the birectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 10-orthoplex.
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Trirectified 9-simplex Trirectified 9-simplexTypeuniform 9-polytopeSchläfli symbol t3{3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{3,3}×{3,3,3,3}Coxeter groupsA9, [3,3,3,3,3,3,3,3]Propertiesconvex
Alternate names Trirectified decayotton (treday) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the trirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 10-orthoplex.
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Quadrirectified 9-simplex Quadrirectified 9-simplexTypeuniform 9-polytopeSchläfli symbol t4{3,3,3,3,3,3,3,3}Coxeter-Dynkin diagramsor 8-faces7-faces6-faces5-faces4-facesCellsFacesEdgesVerticesVertex figure{3,3,3}×{3,3,3}Coxeter groupsA9×2, []Propertiesconvex
Alternate names Quadrirectified decayotton Icosayotton (icoy) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the quadrirectified 9-simplex can be most simply positioned in 10-space as permutations of (0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 10-orthoplex.
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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) o3x3o3o3o3o3o3o3o - reday, o3o3x3o3o3o3o3o3o - breday, o3o3o3x3o3o3o3o3o - treday, o3o3o3o3x3o3o3o3o - icoy
External links Polytopes of Various Dimensions Multi-dimensional Glossary