8-simplexRectified 8-simplexBirectified 8-simplexTrirectified 8-simplexOrthogonal projections in A8 Coxeter plane In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.
Rectified 8-simplex Rectified 8-simplexTypeuniform 8-polytopeCoxeter symbol 061Schläfli symbol t1{37}r{37} = {36,1}or Coxeter-Dynkin diagramsor 7-faces186-faces1085-faces3364-faces630Cells756Faces588Edges252Vertices36Vertex figure7-simplex prism, {}×{3,3,3,3,3}Petrie polygonenneagonCoxeter groupA8, , order 362880Propertiesconvex E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 06,1 for its branching Coxeter-Dynkin diagram, shown as . Acronym: rene (Jonathan Bowers)
The rectified 8-simplex is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.
Coordinates The Cartesian coordinates of the vertices of the rectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.
Images
Birectified 8-simplex Birectified 8-simplexTypeuniform 8-polytopeCoxeter symbol 052Schläfli symbol t2{37}2r{37} = {35,2} orCoxeter-Dynkin diagramsor 7-faces186-faces1445-faces5884-faces1386Cells2016Faces1764Edges756Vertices84Vertex figure{3}×{3,3,3,3}Coxeter groupA8, , order 362880Propertiesconvex E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as . Acronym: brene (Jonathan Bowers)
The birectified 8-simplex is the vertex figure of the 152 honeycomb.
Coordinates The Cartesian coordinates of the vertices of the birectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.
Images
Trirectified 8-simplex Trirectified 8-simplexTypeuniform 8-polytopeCoxeter symbol 043Schläfli symbol t3{37}3r{37} = {34,3} orCoxeter-Dynkin diagramsor 7-faces9 + 96-faces 36 + 72 + 365-faces 84 + 252 + 252 + 844-faces 126 + 504 + 756 + 504Cells 630 + 1260 + 1260Faces 1260 + 1680Edges1260Vertices126Vertex figure{3,3}×{3,3,3}Petrie polygonenneagonCoxeter groupA7, , order 362880Propertiesconvex E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 04,3 for its branching Coxeter-Dynkin diagram, shown as . Acronym: trene (Jonathan Bowers)
Coordinates The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.
Images
Related polytopes The three presented polytopes are in the family of 135 uniform 8-polytopes with A8 symmetry.
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. o3x3o3o3o3o3o3o - rene, o3o3x3o3o3o3o3o - brene, o3o3o3x3o3o3o3o - trene
External links Polytopes of Various Dimensions Multi-dimensional Glossary