# 9-simplex

Regular decayotton
(9-simplex) Orthogonal projection
inside Petrie polygon
Type Regular 9-polytope
Family simplex
Schläfli symbol {3,3,3,3,3,3,3,3}
Coxeter-Dynkin diagram                 8-faces 10 8-simplex 7-faces 45 7-simplex 6-faces 120 6-simplex 5-faces 210 5-simplex 4-faces 252 5-cell Cells 210 tetrahedron Faces 120 triangle Edges 45
Vertices 10
Vertex figure 8-simplex
Petrie polygon decagon
Coxeter group A9 [3,3,3,3,3,3,3,3]
Dual Self-dual
Properties convex

In geometry, a 9-simplex is a self-dual regular 9-polytope. It has 10 vertices, 45 edges, 120 triangle faces, 210 tetrahedral cells, 252 5-cell 4-faces, 210 5-simplex 5-faces, 120 6-simplex 6-faces, 45 7-simplex 7-faces, and 10 8-simplex 8-faces. Its dihedral angle is cos−1(1/9), or approximately 83.62°.

It can also be called a decayotton, or deca-9-tope, as a 10-facetted polytope in 9-dimensions.. The name decayotton is derived from deca for ten facets in Greek and yotta (a variation of "oct" for eight), having 8-dimensional facets, and -on.

## Coordinates

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

$\left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ {\sqrt {1/3}},\ \pm 1\right)$
$\left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ {\sqrt {1/6}},\ -2{\sqrt {1/3}},\ 0\right)$
$\left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ {\sqrt {1/10}},\ -{\sqrt {3/2}},\ 0,\ 0\right)$
$\left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ {\sqrt {1/15}},\ -2{\sqrt {2/5}},\ 0,\ 0,\ 0\right)$
$\left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ {\sqrt {1/21}},\ -{\sqrt {5/3}},\ 0,\ 0,\ 0,\ 0\right)$
$\left({\sqrt {1/45}},\ 1/6,\ {\sqrt {1/28}},\ -{\sqrt {12/7}},\ 0,\ 0,\ 0,\ 0,\ 0\right)$
$\left({\sqrt {1/45}},\ 1/6,\ -{\sqrt {7/4}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$
$\left({\sqrt {1/45}},\ -4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$
$\left(-3{\sqrt {1/5}},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

More simply, the vertices of the 9-simplex can be positioned in 10-space as permutations of (0,0,0,0,0,0,0,0,0,1). This construction is based on facets of the 10-orthoplex.