# 6 simplex

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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

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.

## Coordinates

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

( 1 / 21 ,   1 / 15 ,   1 / 10 ,   1 / 6 ,   1 / 3 ,   ± 1 ) ( 1 / 21 ,   1 / 15 ,   1 / 10 ,   1 / 6 ,   2 1 / 3 ,   0 ) ( 1 / 21 ,   1 / 15 ,   1 / 10 ,   3 / 2 ,   0 ,   0 ) ( 1 / 21 ,   1 / 15 ,   2 2 / 5 ,   0 ,   0 ,   0 ) ( 1 / 21 ,   5 / 3 ,   0 ,   0 ,   0 ,   0 ) ( 12 / 7 ,   0 ,   0 ,   0 ,   0 ,   0 )

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.

## Related uniform 6-polytopes

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.

## References

6-simplex Wikipedia

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