8-simplex

In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°.

It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.

As a configuration
This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-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.

$$\begin{bmatrix}\begin{matrix} 9 & 8 & 28 &  56 &  70 & 56 & 28 & 8 \\ 2 & 36 &  7 &  21 &  35 & 35 & 21 & 7 \\ 3 &  3 & 84 &   6 &  15 & 20 & 15 & 6 \\ 4 &  6 &  4 & 126 &   5 & 10 & 10 & 5 \\ 5 & 10 & 10 &   5 & 126 &  4 &  6 & 4 \\ 6 & 15 & 20 &  15 &   6 & 84 &  3 & 3 \\ 7 & 21 & 35 &  35 &  21 &  7 & 36 & 2 \\ 8 & 28 & 56 &  70 &  56 & 28 &  8 & 9 \end{matrix}\end{bmatrix}$$

Coordinates
The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:


 * $$\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)$$
 * $$\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)$$
 * $$\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)$$
 * $$\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)$$
 * $$\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)$$
 * $$\left(1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)$$
 * $$\left(1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$$
 * $$\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$$

More simply, the vertices of the 8-simplex can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.

Related polytopes and honeycombs
This polytope is a facet in the uniform tessellations: 251, and 521 with respective Coxeter-Dynkin diagrams:

This polytope is one of 135 uniform 8-polytopes with A8 symmetry.