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 edit Rectified 8-simplex Type uniform 8-polytope Coxeter symbol 061 Schläfli symbol t1 {37 }7 } = {36,1 } { 3 , 3 , 3 , 3 , 3 , 3 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3,3,3,3\\3\end{array}}\right\}} Coxeter-Dynkin diagrams 7-faces 18 6-faces 108 5-faces 336 4-faces 630 Cells 756 Faces 588 Edges 252 Vertices 36 Vertex figure 7-simplex prism, {}×{3,3,3,3,3} Petrie polygon enneagon Coxeter group A8 , [37 ], order 362880 Properties convex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1 8 06,1 for its branching Coxeter-Dynkin diagram, shown as
Coordinates edit 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 edit
Birectified 8-simplex edit Birectified 8-simplex Type uniform 8-polytope Coxeter symbol 052 Schläfli symbol t2 {37 }7 } = {35,2 } or { 3 , 3 , 3 , 3 , 3 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3,3,3\\3,3\end{array}}\right\}} Coxeter-Dynkin diagrams 7-faces 18 6-faces 144 5-faces 588 4-faces 1386 Cells 2016 Faces 1764 Edges 756 Vertices 84 Vertex figure {3}×{3,3,3,3} Coxeter group A8 , [37 ], order 362880 Properties convex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2 8 05,2 for its branching Coxeter-Dynkin diagram, shown as
The birectified 8-simplex is the vertex figure of the 152 honeycomb .
Coordinates edit 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 edit Trirectified 8-simplex edit Trirectified 8-simplex Type uniform 8-polytope Coxeter symbol 043 Schläfli symbol t3 {37 }7 } = {34,3 } or { 3 , 3 , 3 , 3 3 , 3 , 3 } {\displaystyle \left\{{\begin{array}{l}3,3,3,3\\3,3,3\end{array}}\right\}} Coxeter-Dynkin diagrams 7-faces 9 + 9 6-faces 36 + 72 + 36 5-faces 84 + 252 + 252 + 84 4-faces 126 + 504 + 756 + 504 Cells 630 + 1260 + 1260 Faces 1260 + 1680 Edges 1260 Vertices 126 Vertex figure {3,3}×{3,3,3} Petrie polygon enneagon Coxeter group A7 , [37 ], order 362880 Properties convex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3 8 04,3 for its branching Coxeter-Dynkin diagram, shown as
Coordinates edit 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 edit Related polytopes edit Notes edit References edit 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 Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1] (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. Klitzing, Richard. "8D Uniform polytopes (polyzetta)". External links edit Multi-dimensional Glossary
rectified, simplexes, simplex, rectified, simplexbirectified, simplex, trirectified, simplexorthogonal, projections, coxeter, planein, eight, dimensional, geometry, rectified, simplex, convex, uniform, polytope, being, rectification, regular, simplex, there, u. 8 simplex Rectified 8 simplexBirectified 8 simplex Trirectified 8 simplexOrthogonal projections in A8 Coxeter planeIn 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 Contents 1 Rectified 8 simplex 1 1 Coordinates 1 2 Images 2 Birectified 8 simplex 2 1 Coordinates 2 2 Images 3 Trirectified 8 simplex 3 1 Coordinates 3 2 Images 4 Related polytopes 5 Notes 6 References 7 External linksRectified 8 simplex editRectified 8 simplexType uniform 8 polytopeCoxeter symbol 061Schlafli symbol t1 37 r 37 36 1 or 3 3 3 3 3 3 3 displaystyle left begin array l 3 3 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 faces 186 faces 1085 faces 3364 faces 630Cells 756Faces 588Edges 252Vertices 36Vertex figure 7 simplex prism 3 3 3 3 3 Petrie polygon enneagonCoxeter group A8 37 order 362880Properties convexE L Elte identified it in 1912 as a semiregular polytope labeling it as S18 It is also called 06 1 for its branching Coxeter Dynkin diagram shown as nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Coordinates edit 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 edit orthographic projections Ak Coxeter plane A8 A7 A6 A5Graph nbsp nbsp nbsp nbsp Dihedral symmetry 9 8 7 6 Ak Coxeter plane A4 A3 A2Graph nbsp nbsp nbsp Dihedral symmetry 5 4 3 Birectified 8 simplex editBirectified 8 simplexType uniform 8 polytopeCoxeter symbol 052Schlafli symbol t2 37 2r 37 35 2 or 3 3 3 3 3 3 3 displaystyle left begin array l 3 3 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 faces 186 faces 1445 faces 5884 faces 1386Cells 2016Faces 1764Edges 756Vertices 84Vertex figure 3 3 3 3 3 Coxeter group A8 37 order 362880Properties convexE L Elte identified it in 1912 as a semiregular polytope labeling it as S28 It is also called 05 2 for its branching Coxeter Dynkin diagram shown as nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp The birectified 8 simplex is the vertex figure of the 152 honeycomb Coordinates edit 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 edit orthographic projections Ak Coxeter plane A8 A7 A6 A5Graph nbsp nbsp nbsp nbsp Dihedral symmetry 9 8 7 6 Ak Coxeter plane A4 A3 A2Graph nbsp nbsp nbsp Dihedral symmetry 5 4 3 Trirectified 8 simplex editTrirectified 8 simplexType uniform 8 polytopeCoxeter symbol 043Schlafli symbol t3 37 3r 37 34 3 or 3 3 3 3 3 3 3 displaystyle left begin array l 3 3 3 3 3 3 3 end array right nbsp Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 faces 9 96 faces 36 72 365 faces 84 252 252 844 faces 126 504 756 504Cells 630 1260 1260Faces 1260 1680Edges 1260Vertices 126Vertex figure 3 3 3 3 3 Petrie polygon enneagonCoxeter group A7 37 order 362880Properties convexE L Elte identified it in 1912 as a semiregular polytope labeling it as S38 It is also called 04 3 for its branching Coxeter Dynkin diagram shown as nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Coordinates edit 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 edit orthographic projections Ak Coxeter plane A8 A7 A6 A5Graph nbsp nbsp nbsp nbsp Dihedral symmetry 9 8 7 6 Ak Coxeter plane A4 A3 A2Graph nbsp nbsp nbsp Dihedral symmetry 5 4 3 Related polytopes editThis polytope is the vertex figure of the 9 demicube and the edge figure of the uniform 261 honeycomb It is also one of 135 uniform 8 polytopes with A8 symmetry A8 polytopes nbsp t0 nbsp t1 nbsp t2 nbsp t3 nbsp t01 nbsp t02 nbsp t12 nbsp t03 nbsp t13 nbsp t23 nbsp t04 nbsp t14 nbsp t24 nbsp t34 nbsp t05 nbsp t15 nbsp t25 nbsp t06 nbsp t16 nbsp t07 nbsp t012 nbsp t013 nbsp t023 nbsp t123 nbsp t014 nbsp t024 nbsp t124 nbsp t034 nbsp t134 nbsp t234 nbsp t015 nbsp t025 nbsp t125 nbsp t035 nbsp t135 nbsp t235 nbsp t045 nbsp t145 nbsp t016 nbsp t026 nbsp t126 nbsp t036 nbsp t136 nbsp t046 nbsp t056 nbsp t017 nbsp t027 nbsp t037 nbsp t0123 nbsp t0124 nbsp t0134 nbsp t0234 nbsp t1234 nbsp t0125 nbsp t0135 nbsp t0235 nbsp t1235 nbsp t0145 nbsp t0245 nbsp t1245 nbsp t0345 nbsp t1345 nbsp t2345 nbsp t0126 nbsp t0136 nbsp t0236 nbsp t1236 nbsp t0146 nbsp t0246 nbsp t1246 nbsp t0346 nbsp t1346 nbsp t0156 nbsp t0256 nbsp t1256 nbsp t0356 nbsp t0456 nbsp t0127 nbsp t0137 nbsp t0237 nbsp t0147 nbsp t0247 nbsp t0347 nbsp t0157 nbsp t0257 nbsp t0167 nbsp t01234 nbsp t01235 nbsp t01245 nbsp t01345 nbsp t02345 nbsp t12345 nbsp t01236 nbsp t01246 nbsp t01346 nbsp t02346 nbsp t12346 nbsp t01256 nbsp t01356 nbsp t02356 nbsp t12356 nbsp t01456 nbsp t02456 nbsp t03456 nbsp t01237 nbsp t01247 nbsp t01347 nbsp t02347 nbsp t01257 nbsp t01357 nbsp t02357 nbsp t01457 nbsp t01267 nbsp t01367 nbsp t012345 nbsp t012346 nbsp t012356 nbsp t012456 nbsp t013456 nbsp t023456 nbsp t123456 nbsp t012347 nbsp t012357 nbsp t012457 nbsp t013457 nbsp t023457 nbsp t012367 nbsp t012467 nbsp t013467 nbsp t012567 nbsp t0123456 nbsp t0123457 nbsp t0123467 nbsp t0123567 nbsp t01234567Notes editReferences editH 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 Ivic Weiss Wiley Interscience Publication 1995 ISBN 978 0 471 01003 6 1 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 Klitzing Richard 8D Uniform polytopes polyzetta o3x3o3o3o3o3o3o rene o3o3x3o3o3o3o3o brene o3o3o3x3o3o3o3o treneExternal links editPolytopes of Various Dimensions Multi dimensional GlossaryvteFundamental convex regular and uniform polytopes in dimensions 2 10Family An Bn I2 p Dn E6 E7 E8 F4 G2 HnRegular polygon Triangle Square p gon Hexagon PentagonUniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron IcosahedronUniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cellUniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicubeUniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicubeUniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicubeUniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytopeTopics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Rectified 8 simplexes amp oldid 1148114089 Birectified 8 simplex, wikipedia, wiki , book, books, library,
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