# 6-simplex

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**.^{[1]}

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:

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 A_{6} Coxeter plane orthographic projections.