In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex. There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex. These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry. Rectified 10-orthoplex In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex. Rectified 10-orthoplex The rectified 10-orthoplex is the vertex figure for the demidekeractic honeycomb. or Alternate names rectified decacross (Acronym rake) (Jonathan Bowers) Construction There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or [4,38] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or [37,1,1] Coxeter group. Cartesian coordinates Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,0,0,0,0,0,0,0,0) Root vectors Its 180 vertices represent the root vectors of the simple Lie group D10. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B10. Images Birectified 10-orthoplex Alternate names Birectified decacross Cartesian coordinates Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,0,0,0,0,0,0,0) Images Trirectified 10-orthoplex Alternate names Trirectified decacross (Acronym trake) (Jonathan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,0,0,0,0,0,0) Images Quadrirectified 10-orthoplex Alternate names Quadrirectified decacross (Acronym brake) (Jonthan Bowers) Cartesian coordinates Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length are all permutations of: (±1,±1,±1,±1,±1,0,0,0,0,0) 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 Ivic Weiss, Wiley-Interscience Publication, 1995, (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