7-simplex

﻿
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 a octaexon the acronym oca.[1]

Coordinates

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

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

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.

 t0 t1 t2 t3 t0,1 t0,2 t1,2 t0,3 t1,3 t2,3 t0,4 t1,4 t2,4 t0,5 t1,5 t0,6 t0,1,2 t0,1,3 t0,2,3 t1,2,3 t0,1,4 t0,2,4 t1,2,4 t0,3,4 t1,3,4 t2,3,4 t0,1,5 t0,2,5 t1,2,5 t0,3,5 t1,3,5 t0,4,5 t0,1,6 t0,2,6 t0,3,6 t0,1,2,3 t0,1,2,4 t0,1,3,4 t0,2,3,4 t1,2,3,4 t0,1,2,5 t0,1,3,5 t0,2,3,5 t1,2,3,5 t0,1,4,5 t0,2,4,5 t1,2,4,5 t0,3,4,5 t0,1,2,6 t0,1,3,6 t0,2,3,6 t0,1,4,6 t0,2,4,6 t0,1,5,6 t0,1,2,3,4 t0,1,2,3,5 t0,1,2,4,5 t0,1,3,4,5 t0,2,3,4,5 t1,2,3,4,5 t0,1,2,3,6 t0,1,2,4,6 t0,1,3,4,6 t0,2,3,4,6 t0,1,2,5,6 t0,1,3,5,6 t0,1,2,3,4,5 t0,1,2,3,4,6 t0,1,2,3,5,6 t0,1,2,4,5,6 t0,1,2,3,4,5,6

Notes

1. ^ Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o3o3o3o - oca