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°.

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.

As a configuration
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.

$$\begin{bmatrix}\begin{matrix}7 & 6 & 15 & 20 & 15 & 6 \\ 2 & 21 & 5 & 10 & 10 & 5 \\ 3 & 3 & 35 & 4 & 6 & 4 \\ 4 & 6 & 4 & 35 & 3 & 3 \\ 5 & 10 & 10 & 5 & 21 & 2 \\ 6 & 15 & 20 & 15 & 6 & 7 \end{matrix}\end{bmatrix}$$

Coordinates
The Cartesian coordinates for an 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.

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.