# 7-simplex

Regular octaexon
(7-simplex)

Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family simplex
Schläfli symbol {3,3,3,3,3,3}
Coxeter-Dynkin diagram
6-faces 8 6-simplex
5-faces 28 5-simplex
4-faces 56 5-cell
Cells 70 tetrahedron
Faces 56 triangle
Edges 28
Vertices 8
Vertex figure 6-simplex
Petrie polygon octagon
Coxeter group A7 [3,3,3,3,3,3]
Dual Self-dual
Properties convex

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.[1]

## As a configuration

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors), the nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[2][3]

${\displaystyle {\begin{bmatrix}{\begin{matrix}8&7&21&35&35&21&7\\2&28&6&15&20&15&6\\3&3&56&5&10&10&5\\4&6&4&70&4&6&4\\5&10&10&5&56&3&3\\6&15&20&15&6&28&2\\7&21&35&35&21&7&8\end{matrix}}\end{bmatrix}}}$

## Coordinates

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

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

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.

## Images

 7-Simplex in 3D Ball and stick model in triakis tetrahedral envelope 7-Simplex as an Amplituhedron Surface 7-simplex to 3D with camera perspective showing hints of its 2D Petrie projection
orthographic projections
Ak Coxeter plane A7 A6 A5
Graph
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry [5] [4] [3]

## 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.

## Notes

1. ^ Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o - oca".
2. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
3. ^ Coxeter, Complex Regular Polytopes, p.117