There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.
These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.
There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or [4,37] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or [36,1,1] Coxeter group.
Cartesian coordinatesedit
Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length are all permutations of:
(±1,±1,0,0,0,0,0,0,0)
Root vectorsedit
Its 144 vertices represent the root vectors of the simple Lie group D9. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups.
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, ISBN978-0-471-01003-6[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]
rectified, orthoplexes, orthoplex, rectified, orthoplex, birectified, orthoplex, trirectified, orthoplex, quadrirectified, cube, trirectified, cube, birectified, cube, rectified, cube, cube, orthogonal, projections, coxeter, plane, nine, dimensional, geometry,. 9 orthoplex Rectified 9 orthoplex Birectified 9 orthoplex Trirectified 9 orthoplex Quadrirectified 9 cube Trirectified 9 cube Birectified 9 cube Rectified 9 cube 9 cube Orthogonal 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 orthoplex There are 9 rectifications of the 9 orthoplex Vertices of the rectified 9 orthoplex are located at the edge centers of the 9 orthoplex Vertices of the birectified 9 orthoplex are located in the triangular face centers of the 9 orthoplex Vertices of the trirectified 9 orthoplex are located in the tetrahedral cell centers of the 9 orthoplex These polytopes are part of a family 511 uniform 9 polytopes with BC9 symmetry Contents 1 Rectified 9 orthoplex 1 1 Alternate names 1 2 Construction 1 3 Cartesian coordinates 1 3 1 Root vectors 1 4 Images 2 Birectified 9 orthoplex 2 1 Alternate names 2 2 Images 3 Trirectified 9 orthoplex 3 1 Alternate names 3 2 Images 4 Notes 5 References 6 External linksRectified 9 orthoplex editRectified 9 orthoplex Type uniform 9 polytope Schlafli symbol t1 37 4 Coxeter Dynkin diagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 faces 6 faces 5 faces 4 faces Cells Faces Edges 2016 Vertices 144 Vertex figure 7 orthoplex prism Petrie polygon octakaidecagon Coxeter groups C9 4 37 D9 36 1 1 Properties convex The rectified 9 orthoplex is the vertex figure for the demienneractic honeycomb 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 nbsp nbsp nbsp nbsp Alternate names edit rectified enneacross Acronym riv Jonathan Bowers 1 Construction edit There are two Coxeter groups associated with the rectified 9 orthoplex one with the C9 or 4 37 Coxeter group and a lower symmetry with two copies of 8 orthoplex facets alternating with the D9 or 36 1 1 Coxeter group Cartesian coordinates edit Cartesian coordinates for the vertices of a rectified 9 orthoplex centered at the origin edge length 2 displaystyle sqrt 2 nbsp are all permutations of 1 1 0 0 0 0 0 0 0 Root vectors edit Its 144 vertices represent the root vectors of the simple Lie group D9 The vertices can be seen in 3 hyperplanes with the 36 vertices rectified 8 simplexs cells on opposite sides and 72 vertices of an expanded 8 simplex passing through the center When combined with the 18 vertices of the 9 orthoplex these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups Images edit orthographic projections B9 B8 B7 nbsp nbsp nbsp 18 16 14 B6 B5 nbsp nbsp 12 10 B4 B3 B2 nbsp nbsp nbsp 8 6 4 A7 A5 A3 8 6 4 Birectified 9 orthoplex editAlternate names edit Rectified 9 demicube Birectified enneacross Acronym brav Jonathan Bowers 2 Images edit orthographic projections B9 B8 B7 nbsp nbsp nbsp 18 16 14 B6 B5 nbsp nbsp 12 10 B4 B3 B2 nbsp nbsp nbsp 8 6 4 A7 A5 A3 8 6 4 Trirectified 9 orthoplex editAlternate names edit trirectified enneacross Acronym tarv Jonathan Bowers 3 Images edit orthographic projections B9 B8 B7 nbsp nbsp nbsp 18 16 14 B6 B5 nbsp nbsp 12 10 B4 B3 B2 nbsp nbsp nbsp 8 6 4 A7 A5 A3 8 6 4 Notes edit Klitzing o3x3o3o3o3o3o3o4o riv Klitzing o3o3x3o3o3o3o3o4o brav Klitzing o3o3o3x3o3o3o3o4o tarv References 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 1966 Klitzing Richard 9D uniform polytopes polyyotta x3o3o3o3o3o3o3o4o vee o3x3o3o3o3o3o3o4o riv o3o3x3o3o3o3o3o4o brav o3o3o3x3o3o3o3o4o tarv o3o3o3o3x3o3o3o4o nav o3o3o3o3o3x3o3o4o tarn o3o3o3o3o3o3x3o4o barn o3o3o3o3o3o3o3x4o ren o3o3o3o3o3o3o3o4x enneExternal links editPolytopes of Various Dimensions Multi dimensional Glossary vteFundamental convex regular and uniform polytopes in dimensions 2 10 Family An Bn I2 p Dn E6 E7 E8 F4 G2 Hn Regular polygon Triangle Square p gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron Cube Demicube Dodecahedron Icosahedron Uniform polychoron Pentachoron 16 cell Tesseract Demitesseract 24 cell 120 cell 600 cell Uniform 5 polytope 5 simplex 5 orthoplex 5 cube 5 demicube Uniform 6 polytope 6 simplex 6 orthoplex 6 cube 6 demicube 122 221 Uniform 7 polytope 7 simplex 7 orthoplex 7 cube 7 demicube 132 231 321 Uniform 8 polytope 8 simplex 8 orthoplex 8 cube 8 demicube 142 241 421 Uniform 9 polytope 9 simplex 9 orthoplex 9 cube 9 demicube Uniform 10 polytope 10 simplex 10 orthoplex 10 cube 10 demicube Uniform n polytope n simplex n orthoplex n cube n demicube 1k2 2k1 k21 n pentagonal polytope Topics Polytope families Regular polytope List of regular polytopes and compounds Retrieved from https en wikipedia org w index php title Rectified 9 orthoplexes amp oldid 1199305822 Birectified 9 orthoplex, wikipedia, wiki, book, books, library,
