World Library  
Flag as Inappropriate
Email this Article


Article Id: WHEBN0010083518
Reproduction Date:

Title: 6-orthoplex  
Author: World Heritage Encyclopedia
Language: English
Subject: 6-cube, 7-orthoplex, Six-dimensional space, Uniform k 21 polytope, Coxeter group
Publisher: World Heritage Encyclopedia



Orthogonal projection
inside Petrie polygon|-
Type Regular 6-polytope
Family orthoplex
Schläfli symbols {3,3,3,3,4}
Coxeter-Dynkin diagrams node_1|3|node|3|node|3|node|3|node|4|node}}
5-faces 64 {34}
4-faces 192 {33}
Cells 240 {3,3}
Faces 160 {3}
Edges 60
Vertices 12
Vertex figure 5-orthoplex
Petrie polygon dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Dual 6-cube
Properties convex

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

It has two constructed forms, the first being regular with Schläfli symbol {34,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,31,1} or Coxeter symbol 311.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 6-hypercube, or hexeract.

Alternate names


There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or [4,3,3,3,3] Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or [33,1,1] Coxeter group. A lowest symmetry construction is based on a dual of a 6-orthotope, called a 6-fusil.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure
Alternate 6-orthoplex node_1|3|node|3|node|3|node|3|node|4|node}} {3,3,3,3,4} [3,3,3,3,4] 46080 node_1|3|node|3|node|3|node|4|node}}
regular 6-orthoplex node_1|3|node|3|node|3|node|split1|nodes}} {3,3,3,31,1} [3,3,3,31,1] 23040 node_1|3|node|3|node|split1|nodes}}
6-fusil node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}} 6{} [25] 64 node_f1|2|node_f1|2|node_f1|2|node_f1|2|node_f1}}

Cartesian coordinates

Cartesian coordinates for the vertices of a 6-orthoplex, centered at the origin are

(±1,0,0,0,0,0), (0,±1,0,0,0,0), (0,0,±1,0,0,0), (0,0,0,±1,0,0), (0,0,0,0,±1,0), (0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.


Related polytopes

The 6-orthoplex can be projected down to 3-dimensions into the vertices of a regular icosahedron, as seen in this 2D projection:

H3 Coxeter plane

D6 Coxeter plane
This construction can be geometrically seen as the 12 vertices of the 6-orthoplex projected to 3 dimensions. This represents a geometric folding of the D6 to H3 Coxeter groups:

Seen by these 2D Coxeter plane orthogonal projections, the two overlapping central vertices define the third axis in this mapping.

It is in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 3k1 series. (A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral hosohedron.)

This polytope is one of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.


  • 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

External links

  • 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.