In geometry, a 6simplex is a selfdual regular 6polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5cell 4faces, and 7 5simplex 5faces. Its dihedral angle is cos^{−1}(1/6), or approximately 80.41°.
Alternate names[]
It can also be called a heptapeton, or hepta6tope, as a 7facetted polytope in 6dimensions. The name heptapeton is derived from hepta for seven facets in Greek and peta for having fivedimensional facets, and on. Jonathan Bowers gives a heptapeton the acronym hop.^{[1]}
As a configuration[]
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (fvectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^{[2]}^{[3]}
${\begin{bmatrix}{\begin{matrix}7&6&15&20&15&6\\2&21&5&10&10&5\\3&3&35&4&6&4\\4&6&4&35&3&3\\5&10&10&5&21&2\\6&15&20&15&6&7\end{matrix}}\end{bmatrix}}$
Coordinates[]
The Cartesian coordinates for an origincentered regular heptapeton having edge length 2 are:
 $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)$
 $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ 2{\sqrt {1/3}},\ 0\right)$
 $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {3/2}},\ 0,\ 0\right)$
 $\left({\sqrt {1/21}},\ {\sqrt {1/15}},\ 2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)$
 $\left({\sqrt {1/21}},\ {\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)$
 $\left({\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)$
The vertices of the 6simplex can be more simply positioned in 7space as permutations of:
 (0,0,0,0,0,0,1)
This construction is based on facets of the 7orthoplex.
Images[]
Related uniform 6polytopes[]
The regular 6simplex is one of 35 uniform 6polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A_{6} Coxeter plane orthographic projections.
A6 polytopes

t_{0}

t_{1}

t_{2}

t_{0,1}

t_{0,2}

t_{1,2}

t_{0,3}

t_{1,3}

t_{2,3}

t_{0,4}

t_{1,4}

t_{0,5}

t_{0,1,2}

t_{0,1,3}

t_{0,2,3}

t_{1,2,3}

t_{0,1,4}

t_{0,2,4}

t_{1,2,4}

t_{0,3,4}

t_{0,1,5}

t_{0,2,5}

t_{0,1,2,3}

t_{0,1,2,4}

t_{0,1,3,4}

t_{0,2,3,4}

t_{1,2,3,4}

t_{0,1,2,5}

t_{0,1,3,5}

t_{0,2,3,5}

t_{0,1,4,5}

t_{0,1,2,3,4}

t_{0,1,2,3,5}

t_{0,1,2,4,5}

t_{0,1,2,3,4,5}

Notes[]
 ^ Klitzing, (x3o3o3o3o3o  hop)
 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
 ^ Coxeter, Complex Regular Polytopes, p.117
References[]
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd ion, 1973), Dover ion, ISBN 0486614808, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (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 ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, ed by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (Chapter 26. pp. 409: Hemicubes: 1_{n1})
 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 links[]