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]
{\displaystyle {\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}}}
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: 251, and 521 with respective Coxeter-Dynkin diagrams: This polytope is one of 135 uniform 8-polytopes with A8 symmetry.