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# 7-simplex: Wikis

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# Encyclopedia

(Redirected to Octaexon article)

Regular octaexon
(7-simplex)

Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family simplex
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
Schläfli symbol {3,3,3,3,3,3}
Coxeter-Dynkin diagram
Coxeter group A7 [3,3,3,3,3,3]
Dual Self-dual
Properties convex

In geometry, an octexon, or octa-7-tope is a 7-simplex, a self-dual regular 7-polytope with 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.

The name octaexon is derived from octa for eight facets in Greek and -ex for having six-dimensional facets, and -on.

## Coordinates

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

$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right)$
$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right)$
$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right)$
$\left(\sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right)$
$\left(\sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right)$
$\left(\sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right)$
$\left(-\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

Regular octaexon
(7-simplex)

Orthogonal projection
inside Petrie polygon
TypeRegular 7-polytope
Familysimplex
6-faces8 6-simplex
5-faces28 5-simplex
4-faces56 5-cell
Cells70 tetrahedron
Faces56 triangle
Edges28
Vertices8
Vertex figure6-simplex
Petrie polygonoctagon
Schläfli symbol {3,3,3,3,3,3}
Coxeter-Dynkin diagram
Coxeter group A7 [3,3,3,3,3,3]
DualSelf-dual
Propertiesconvex

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 a octaexon the acronym oca.[1]

## Coordinates

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

$\left\left(\sqrt\left\{1/28\right\},\ \sqrt\left\{1/21\right\},\ \sqrt\left\{1/15\right\},\ \sqrt\left\{1/10\right\},\ \sqrt\left\{1/6\right\},\ \sqrt\left\{1/3\right\},\ \pm1\right\right)$
$\left\left(\sqrt\left\{1/28\right\},\ \sqrt\left\{1/21\right\},\ \sqrt\left\{1/15\right\},\ \sqrt\left\{1/10\right\},\ \sqrt\left\{1/6\right\},\ -2\sqrt\left\{1/3\right\},\ 0\right\right)$
$\left\left(\sqrt\left\{1/28\right\},\ \sqrt\left\{1/21\right\},\ \sqrt\left\{1/15\right\},\ \sqrt\left\{1/10\right\},\ -\sqrt\left\{3/2\right\},\ 0,\ 0\right\right)$
$\left\left(\sqrt\left\{1/28\right\},\ \sqrt\left\{1/21\right\},\ \sqrt\left\{1/15\right\},\ -2\sqrt\left\{2/5\right\},\ 0,\ 0,\ 0\right\right)$
$\left\left(\sqrt\left\{1/28\right\},\ \sqrt\left\{1/21\right\},\ -\sqrt\left\{5/3\right\},\ 0,\ 0,\ 0,\ 0\right\right)$
$\left\left(\sqrt\left\{1/28\right\},\ -\sqrt\left\{12/7\right\},\ 0,\ 0,\ 0,\ 0,\ 0\right\right)$
$\left\left(-\sqrt\left\{7/4\right\},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right\right)$

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

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 one of 71 uniform 7-polytopes with A7 symmetry.

## Notes

1. ^ Richard Klitzing 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons x3o3o3o3o3o - oca