# 8-simplex

Regular enneazetton
(8-simplex) Orthogonal projection
inside Petrie polygon
Type Regular 8-polytope
Family simplex
Schläfli symbol {3,3,3,3,3,3,3}
Coxeter-Dynkin diagram               7-faces 9 7-simplex 6-faces 36 6-simplex 5-faces 84 5-simplex 4-faces 126 5-cell Cells 126 tetrahedron Faces 84 triangle Edges 36
Vertices 9
Vertex figure 7-simplex
Petrie polygon enneagon
Coxeter group A8 [3,3,3,3,3,3,3]
Dual Self-dual
Properties convex

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:

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