# 8 simplex

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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°.

## Contents

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.

## Coordinates

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

( 1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   1 / 10 ,   1 / 6 ,   1 / 3 ,   ± 1 ) ( 1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   1 / 10 ,   1 / 6 ,   2 1 / 3 ,   0 ) ( 1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   1 / 10 ,   3 / 2 ,   0 ,   0 ) ( 1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   2 2 / 5 ,   0 ,   0 ,   0 ) ( 1 / 6 ,   1 / 28 ,   1 / 21 ,   5 / 3 ,   0 ,   0 ,   0 ,   0 ) ( 1 / 6 ,   1 / 28 ,   12 / 7 ,   0 ,   0 ,   0 ,   0 ,   0 ) ( 1 / 6 ,   7 / 4 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ) ( 4 / 3 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 )

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.

## 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.

## References

8-simplex Wikipedia

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