6-simplex
f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7
(χ=0)
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°.
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.
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.
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 A6Coxeter plane orthographic projections.
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