10-simplexRectified 10-simplexBirectified 10-simplexTrirectified 10-simplexQuadrirectified 10-simplexOrthogonal projections in A9 Coxeter plane In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex.
These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry.
There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.
Rectified 10-simplex Rectified 10-simplexTypeuniform polyxennonSchläfli symbol t1{3,3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams9-faces228-faces1657-faces6606-faces16505-faces27724-faces3234Cells2640Faces1485Edges495Vertices55Vertex figure9-simplex prismPetrie polygondecagonCoxeter groupsA10, [3,3,3,3,3,3,3,3,3]Propertiesconvex
The rectified 10-simplex is the vertex figure of the 11-demicube.
Alternate names Rectified hendecaxennon (acronym: ru) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.
Images
Birectified 10-simplex Birectified 10-simplexTypeuniform 9-polytopeSchläfli symbol t2{3,3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams8-faces7-faces6-faces5-faces4-facesCellsFacesEdges1980Vertices165Vertex figure{3}x{3,3,3,3,3,3}Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]Propertiesconvex
Alternate names Birectified hendecaxennon (acronym: bru) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.
Images
Trirectified 10-simplex Trirectified 10-simplexTypeuniform polyxennonSchläfli symbol t3{3,3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams8-faces7-faces6-faces5-faces4-facesCellsFacesEdges4620Vertices330Vertex figure{3,3}x{3,3,3,3,3}Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]Propertiesconvex
Alternate names Trirectified hendecaxennon (acronym: tru) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.
Images
Quadrirectified 10-simplex Quadrirectified 10-simplexTypeuniform polyxennonSchläfli symbol t4{3,3,3,3,3,3,3,3,3}Coxeter-Dynkin diagrams8-faces7-faces6-faces5-faces4-facesCellsFacesEdges6930Vertices462Vertex figure{3,3,3}x{3,3,3,3}Coxeter groupsA10, [3,3,3,3,3,3,3,3,3]Propertiesconvex
Alternate names Quadrirectified hendecaxennon (acronym: teru) (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.
Images
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) o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru
External links Polytopes of Various Dimensions Multi-dimensional Glossary