 10orthoplex

10orthoplex
Decacross
Orthogonal projection
inside Petrie polygonType Regular 10polytope Family orthoplex Schläfli symbol {3^{8},4}
{3^{7,1,1}}CoxeterDynkin diagrams
9faces 1024 {3^{8}} 8faces 5120 {3^{7}} 7faces 11520 {3^{6}} 6faces 15360 {3^{5}} 5faces 13440 {3^{4}} 4faces 8064 {3^{3}} Cells 3360 {3,3} Faces 960 {3} Edges 180 Vertices 20 Vertex figure 9orthoplex Petrie polygon Icosagon Coxeter groups C_{10}, [3^{8},4]
D_{10}, [3^{7,1,1}]Dual 10cube Properties convex In geometry, a 10orthoplex or 10cross polytope, is a regular 10polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5cells 4faces, 13440 5faces, 15360 6faces, 11520 7faces, 5120 8faces, and 1024 9faces.
It has two constructed forms, the first being regular with Schläfli symbol {3^{8},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3^{7,1,1}} or Coxeter symbol 7_{11}.
Contents
Alternate names
 Decacross is derived from combining the family name cross polytope with deca for ten (dimensions) in Greek
 Chilliaicositetraxennon as a 1024facetted 10polytope (polyxennon).
Related polytopes
It is one of an infinite family of polytopes, called crosspolytopes or orthoplexes. The dual polytope is the 10hypercube or 10cube.
Construction
There are two Coxeter groups associated with the 10orthoplex, one regular, dual of the 10cube with the C_{10} or [4,3^{8}] symmetry group, and a lower symmetry with two copies of 9simplex facets, alternating, with the D_{10} or [3^{7,1,1}] symmetry group.
Cartesian coordinates
Cartesian coordinates for the vertices of a 10orthoplex, centered 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
orthographic projections B_{10} B_{9} B_{8} [20] [18] [16] B_{7} B_{6} B_{5} [14] [12] [10] B_{4} B_{3} B_{2} [8] [6] [4] 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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
 Richard Klitzing, 10D uniform polytopes (polyxenna), x3o3o3o3o3o3o3o3o4o  ka
External links
 Olshevsky, George, Cross polytope at Glossary for Hyperspace.
 Polytopes of Various Dimensions
 Multidimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 Family A_{n} BC_{n} D_{n} E_{6} / E_{7} / E_{8} / F_{4} / G_{2} H_{n} Regular polygon Triangle Square Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron 5cell 16cell • Tesseract Demitesseract 24cell 120cell • 600cell Uniform 5polytope 5simplex 5orthoplex • 5cube 5demicube Uniform 6polytope 6simplex 6orthoplex • 6cube 6demicube 1_{22} • 2_{21} Uniform 7polytope 7simplex 7orthoplex • 7cube 7demicube 1_{32} • 2_{31} • 3_{21} Uniform 8polytope 8simplex 8orthoplex • 8cube 8demicube 1_{42} • 2_{41} • 4_{21} Uniform 9polytope 9simplex 9orthoplex • 9cube 9demicube Uniform 10polytope 10simplex 10orthoplex • 10cube 10demicube npolytopes nsimplex northoplex • ncube ndemicube 1_{k2} • 2_{k1} • k_{21} pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes This geometryrelated article is a stub. You can help Wikipedia by expanding it.