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In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.
Contents
It has two constructed forms, the first being regular with Schläfli symbol {37,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {36,31,1} or Coxeter symbol 611.
It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.
Alternate names
Construction
There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or [4,37] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D9 or [36,1,1] symmetry group.
Cartesian coordinates
Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are
(±1,0,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0,0), (0,0,0,±1,0,0,0,0,0), (0,0,0,0,±1,0,0,0,0), (0,0,0,0,0,±1,0,0,0), (0,0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1,0), (0,0,0,0,0,0,0,0,±1)Every vertex pair is connected by an edge, except opposites.