# 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*.

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.^{[1]}^{[2]}

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

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.

This polytope is a facet in the uniform tessellations: 2_{51}, and 5_{21} with respective Coxeter-Dynkin diagrams: