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Type uniform polypeton
Schläfli symbol {35}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7

Coxeter group A6, [35], order 5040
Bowers name
and (acronym)
Vertex figure 5-simplex
Circumradius 0.645497
Properties convex, isogonal self-dual

In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.


Alternate namesEdit

It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.[1]

As a configurationEdit

This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.[2][3]



The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:


The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:


This construction is based on facets of the 7-orthoplex.


orthographic projections
Ak Coxeter plane A6 A5 A4
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Dihedral symmetry [4] [3]

Related uniform 6-polytopesEdit


  1. ^ Klitzing, (x3o3o3o3o3o - hop)
  2. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  3. ^ Coxeter, Complex Regular Polytopes, p.117


  • H.S.M. Coxeter:
    • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • 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]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o3o3o - hix".

External linksEdit