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.

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

7-Simplex in 3D
280px Model created using straws (edges) and plasticine balls (vertices) in triakis tetrahedral envelope	280px 7-Simplex as an Amplituhedron Surface	280px 7-simplex to 3D with camera perspective showing hints of its 2D Petrie projection

orthographic projections

Ak Coxeter plane	A7	A6	A5
Graph		150px	150px
Dihedral symmetry	[8]	[7]	[6]
Ak Coxeter plane	A4	A3	A2
Graph	150px	150px	150px
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.

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary