10-orthoplex

10-orthoplex Decacross
Orthogonal projection inside Petrie polygon
Type	Regular 10-polytope
Family	Orthoplex
Schläfli symbol	{38,4} {37,31,1}
Coxeter-Dynkin diagrams
9-faces	1024 {38}
8-faces	5120 {37}
7-faces	11520 {36}
6-faces	15360 {35}
5-faces	13440 {34}
4-faces	8064 {33}
Cells	3360 {3,3}
Faces	960 {3}
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, Hanner polytope

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 (checker-boarded) facets, with Schläfli symbol {3⁷,3^1,1} or Coxeter symbol 7₁₁.

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

Alternate names

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

Construction

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 [3^7,1,1] symmetry group.

Cartesian coordinates

Cartesian coordinates for the vertices of a 10-orthoplex, centred 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.

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. (1966)
• Klitzing, Richard. "10D uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o4o - ka". https://bendwavy.org/klitzing/dimensions/polyxenna.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