It has two constructed forms, the first being regular with Schläfli symbol {38,4}, and the second with alternately labeled (checker-boarded) facets, with Schläfli symbol {37,31,1} or Coxeter symbol711.
There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or [4,38] symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or [37,1,1] symmetry group.
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]
orthoplex, decacrossorthogonal, projectioninside, petrie, polygontype, regular, polytopefamily, orthoplexschläfli, symbol, coxeter, dynkin, diagrams9, faces, 1024, faces, 5120, faces, 11520, faces, 15360, faces, 13440, faces, 8064, cells, 3360, faces, edges, 1. 10 orthoplexDecacrossOrthogonal projectioninside Petrie polygonType Regular 10 polytopeFamily OrthoplexSchlafli symbol 38 4 37 31 1 Coxeter Dynkin diagrams9 faces 1024 38 8 faces 5120 37 7 faces 11520 36 6 faces 15360 35 5 faces 13440 34 4 faces 8064 33 Cells 3360 3 3 Faces 960 3 Edges 180Vertices 20Vertex figure 9 orthoplexPetrie polygon IcosagonCoxeter groups C10 38 4 D10 37 1 1 Dual 10 cubeProperties Convex Hanner polytopeIn geometry a 10 orthoplex or 10 cross polytope is a regular 10 polytope with 20 vertices 180 edges 960 triangle faces 3360 octahedron cells 8064 5 cells 4 faces 13440 5 faces 15360 6 faces 11520 7 faces 5120 8 faces and 1024 9 faces It has two constructed forms the first being regular with Schlafli symbol 38 4 and the second with alternately labeled checker boarded facets with Schlafli symbol 37 31 1 or Coxeter symbol 711 It is one of an infinite family of polytopes called cross polytopes or orthoplexes The dual polytope is the 10 hypercube or 10 cube Contents 1 Alternate names 2 Construction 3 Cartesian coordinates 4 Images 5 References 6 External linksAlternate names EditDecacross is derived from combining the family name cross polytope with deca for ten dimensions in Greek Chilliaicositetraxennon as a 1024 facetted 10 polytope polyxennon Construction EditThere are two Coxeter groups associated with the 10 orthoplex one regular dual of the 10 cube with the C10 or 4 38 symmetry group and a lower symmetry with two copies of 9 simplex facets alternating with the D10 or 37 1 1 symmetry group Cartesian coordinates EditCartesian coordinates for the vertices of a 10 orthoplex centred at the origin are 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 Every vertex pair is connected by an edge except opposites Images EditOrthographic projections B10 B9 B8 20 18 16 B7 B6 B5 14 12 10 B4 B3 B2 8 6 4 A9 A5 10 6 A7 A3 8 4 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 10D uniform polytopes polyxenna x3o3o3o3o3o3o3o3o4o ka External links EditOlshevsky George Cross polytope Glossary for Hyperspace Archived from the original on 4 February 2007 Polytopes 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 10 orthoplex amp oldid 1122323530, wikipedia, wiki, book, books, library,
