9-orthoplex

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

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 {3⁶,3^1,1} 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

• 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

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 [3^6,1,1] 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.

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, 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.
• Klitzing, Richard. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o4o - vee". https://bendwavy.org/klitzing/dimensions/polyyotta.htm. 

External links

• Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007. https://web.archive.org/web/20070204075028/members.aol.com/Polycell/glossary.html#Cross. 
• Polytopes of Various Dimensions
• Multi-dimensional Glossary