6-simplex

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6-simplex
Type uniform polypeton
Schläfli symbol {35}
Coxeter-Dynkin 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
Elements

f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7
(χ=0)

Coxeter group A6, [35], order 5040
Bowers name
and (acronym)
Heptapeton
(hop)
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 names[edit]

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]

Coordinates[edit]

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

\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)
\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)
\left(\sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)
\left(\sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)
\left(\sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)
\left(-\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)

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

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

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

Images[edit]

orthographic projections
Ak Coxeter plane A6 A5 A4
Graph 6-simplex t0.svg 6-simplex t0 A5.svg 6-simplex t0 A4.svg
Dihedral symmetry [7] [6] [5]
Ak Coxeter plane A3 A2
Graph 6-simplex t0 A3.svg 6-simplex t0 A2.svg
Dihedral symmetry [4] [3]

Related uniform 6-polytopes[edit]

The regular 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.

6-simplex t0.svg
t0
6-simplex t1.svg
t1
6-simplex t2.svg
t2
6-simplex t01.svg
t0,1
6-simplex t02.svg
t0,2
6-simplex t12.svg
t1,2
6-simplex t03.svg
t0,3
6-simplex t13.svg
t1,3
6-simplex t23.svg
t2,3
6-simplex t04.svg
t0,4
6-simplex t14.svg
t1,4
6-simplex t05.svg
t0,5
6-simplex t012.svg
t0,1,2
6-simplex t013.svg
t0,1,3
6-simplex t023.svg
t0,2,3
6-simplex t123.svg
t1,2,3
6-simplex t014.svg
t0,1,4
6-simplex t024.svg
t0,2,4
6-simplex t124.svg
t1,2,4
6-simplex t034.svg
t0,3,4
6-simplex t015.svg
t0,1,5
6-simplex t025.svg
t0,2,5
6-simplex t0123.svg
t0,1,2,3
6-simplex t0124.svg
t0,1,2,4
6-simplex t0134.svg
t0,1,3,4
6-simplex t0234.svg
t0,2,3,4
6-simplex t1234.svg
t1,2,3,4
6-simplex t0125.svg
t0,1,2,5
6-simplex t0135.svg
t0,1,3,5
6-simplex t0235.svg
t0,2,3,5
6-simplex t0145.svg
t0,1,4,5
6-simplex t01234.svg
t0,1,2,3,4
6-simplex t01235.svg
t0,1,2,3,5
6-simplex t01245.svg
t0,1,2,4,5
6-simplex t012345.svg
t0,1,2,3,4,5

Notes[edit]

  1. ^ Klitzing, (x3o3o3o3o3o - hop)

References[edit]

  • 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)
  • Richard Klitzing, 6D uniform polytopes (polypeta), x3o3o3o3o - hix

External links[edit]