10-orthoplex

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10-orthoplex
Decacross
10-orthoplex.svg
Orthogonal projection
inside Petrie polygon
Type Regular 10-polytope
Family orthoplex
Schläfli symbol {38,4}
{37,31,1}
Coxeter-Dynkin diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
9-faces 1024 {38} 9-simplex t0.svg
8-faces 5120 {37} 8-simplex t0.svg
7-faces 11520 {36}7-simplex t0.svg
6-faces 15360 {35}6-simplex t0.svg
5-faces 13440 {34}5-simplex t0.svg
4-faces 8064 {33}4-simplex t0.svg
Cells 3360 {3,3}3-simplex t0.svg
Faces 960 {3}2-simplex t0.svg
Edges 180
Vertices 20
Vertex figure 9-orthoplex
Petrie polygon Icosagon
Coxeter groups C10, [38,4]
D10, [37,1,1]
Dual 10-cube
Properties convex

In 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 Schläfli symbol {38,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {37,1,1} or Coxeter symbol 711.

Alternate names[edit]

  • Decacross is derived from combining the family name cross polytope with deca for ten (dimensions) in Greek
  • Chilliaicositetra-xennon as a 1024-facetted 10-polytope (polyxennon).

Related polytopes[edit]

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 10-hypercube or 10-cube.

Construction[edit]

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.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a 10-orthoplex, 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[edit]

orthographic projections
B10 B9 B8
10-cube t9.svg 10-cube t9 B9.svg 10-cube t9 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t9 B7.svg 10-cube t9 B6.svg 10-cube t9 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t9 B4.svg 10-cube t9 B3.svg 10-cube t9 B2.svg
[8] [6] [4]

References[edit]

  • 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, 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)
  • Richard Klitzing, 10D uniform polytopes (polyxenna), x3o3o3o3o3o3o3o3o4o - ka

External links[edit]