# 7 simplex

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In 7-dimensional geometry, a 7-simplex is a self-dual regular 7-polytope. It has 8 vertices, 28 edges, 56 triangle faces, 70 tetrahedral cells, 56 5-cell 5-faces, 28 5-simplex 6-faces, and 8 6-simplex 7-faces. Its dihedral angle is cos−1(1/7), or approximately 81.79°.

## Alternate names

It can also be called an octaexon, or octa-7-tope, as an 8-facetted polytope in 7-dimensions. The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on. Jonathan Bowers gives an octaexon the acronym oca.

## Coordinates

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

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

More simply, the vertices of the 7-simplex can be positioned in 8-space as permutations of (0,0,0,0,0,0,0,1). This construction is based on facets of the 8-orthoplex.

## Related polytopes

This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram:

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.

## References

7-simplex Wikipedia

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