# 10 simplex

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In geometry, a 10-simplex is a self-dual regular 10-polytope. It has 11 vertices, 55 edges, 165 triangle faces, 330 tetrahedral cells, 462 5-cell 4-faces, 462 5-simplex 5-faces, 330 6-simplex 6-faces, 165 7-simplex 7-faces, 55 8-simplex 8-faces, and 11 9-simplex 9-faces. Its dihedral angle is cos−1(1/10), or approximately 84.26°.

## Contents

It can also be called a hendecaxennon, or hendeca-10-tope, as an 11-facetted polytope in 10-dimensions. The name hendecaxennon is derived from hendeca for 11 facets in Greek and -xenn (variation of ennea for nine), having 9-dimensional facets, and -on.

## Coordinates

The Cartesian coordinates of the vertices of an origin-centered regular 10-simplex having edge length 2 are:

( 1 / 55 ,   1 / 45 ,   1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   1 / 10 ,   1 / 6 ,   1 / 3 ,   ± 1 ) ( 1 / 55 ,   1 / 45 ,   1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   1 / 10 ,   1 / 6 ,   2 1 / 3 ,   0 ) ( 1 / 55 ,   1 / 45 ,   1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   1 / 10 ,   3 / 2 ,   0 ,   0 ) ( 1 / 55 ,   1 / 45 ,   1 / 6 ,   1 / 28 ,   1 / 21 ,   1 / 15 ,   2 2 / 5 ,   0 ,   0 ,   0 ) ( 1 / 55 ,   1 / 45 ,   1 / 6 ,   1 / 28 ,   1 / 21 ,   5 / 3 ,   0 ,   0 ,   0 ,   0 ) ( 1 / 55 ,   1 / 45 ,   1 / 6 ,   1 / 28 ,   12 / 7 ,   0 ,   0 ,   0 ,   0 ,   0 ) ( 1 / 55 ,   1 / 45 ,   1 / 6 ,   7 / 4 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ) ( 1 / 55 ,   1 / 45 ,   4 / 3 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ) ( 1 / 55 ,   3 1 / 5 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ) ( 20 / 11 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 ,   0 )

More simply, the vertices of the 10-simplex can be positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 11-orthoplex.

The 2-skeleton of the 10-simplex is topologically related to the 11-cell abstract regular polychoron which has the same 11 vertices, 55 edges, but only 1/3 the faces (55).

## References

10-simplex Wikipedia

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