World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0017979183
Reproduction Date:

Title: 10-orthoplex  
Author: World Heritage Encyclopedia
Language: English
Subject: Uniform 10-polytope, 10-cube, 9-simplex, Coxeter group, List of mathematical shapes
Publisher: World Heritage Encyclopedia



Orthogonal projection
inside Petrie polygon
Type Regular 10-polytope
Family Orthoplex
Schläfli symbol {38,4}
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

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 {37,31,1} 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
  • Chilliaicositetraxennon as a 1024-facetted 10-polytope (polyxennon).


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

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.


Orthographic projections
B10 B9 B8
[20] [18] [16]
B7 B6 B5
[14] [12] [10]
B4 B3 B2
[8] [6] [4]


  • 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

  • Olshevsky, George, Cross polytope at Glossary for Hyperspace.
  • Polytopes of Various Dimensions
  • Multi-dimensional Glossary
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.