9-orthoplex

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Regular 9-orthoplex
9-orthoplex.svg
Orthogonal projection
inside Petrie polygon
Type Regular 9-polytope
Family orthoplex
Schläfli symbol {37,4}
{36,31,1}
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 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 split1.pngCDel nodes.png
8-faces 512 {37}8-simplex t0.svg
7-faces 2304 {36}7-simplex t0.svg
6-faces 4608 {35}6-simplex t0.svg
5-faces 5376 {34}5-simplex t0.svg
4-faces 4032 {33}4-simplex t0.svg
Cells 2016 {3,3}3-simplex t0.svg
Faces 672 {3}2-simplex t0.svg
Edges 144
Vertices 18
Vertex figure Octacross
Petrie polygon Octadecagon
Coxeter groups C9, [37,4]
D9, [36,1,1]
Dual 9-cube
Properties convex

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.

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 {36,1,1} or Coxeter symbol 611.

Alternate names[edit]

  • Enneacross, derived from combining the family name cross polytope with ennea for nine (dimensions) in Greek
  • Pentacosidodecayotton as a 512-facetted 9-polytope (polyyotton)

Construction[edit]

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[edit]

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.

Images[edit]

orthographic projections
B9 B8 B7
9-cube t8.svg 9-cube t8 B8.svg 9-cube t8 B7.svg
[18] [16] [14]
B6 B5
9-cube t8 B6.svg 9-cube t8 B5.svg
[12] [10]
B4 B3 B2
9-cube t8 B4.svg 9-cube t8 B3.svg 9-cube t8 B2.svg
[8] [6] [4]

Related polytopes[edit]

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.

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.
  • Richard Klitzing, 9D uniform polytopes (polyyotta), x3o3o3o3o3o3o3o4o - vee

External links[edit]