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

This configuration matrix represents the 7-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.[2][3]

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

## Symmetry

 7-simplex as a join of two orthogonal tetrahedra in a symmetric 2D orthographic project: 2⋅{3,3} or {3,3}∨{3,3}, 6 red edges, 6 blue edges, and 16 yellow cross edges. 7-simplex as a join of 4 orthogonal segments, projected into a 3D cube: 4⋅{ } = { }∨{ }∨{ }∨{ }. The 28 edges are shown as 12 yellow edges of the cube, 12 cube face diagonals in light green, and 4 full diagonals in red. This partition can be considered a tetradisphenoid, or a join of two disphenoid.

There are many lower symmetry constructions of the 7-simplex.

Some are expressed as join partitions of two or more lower simplexes. The symmetry order of each join is the product of the symmetry order of the elements, and raised further if identical elements can be interchanged.

Join Symbol Symmetry Order Extended f-vectors
(factorization)
Regular 7-simplex {3,3,3,3,3,3} [3,3,3,3,3,3] 8! = 40320 (1,8,28,56,70,56,28,8,1)
6-simplex-point join (pyramid) {3,3,3,3,3}∨( ) [3,3,3,3,3,1] 7!×1! = 5040 (1,7,21,35,35,21,7,1)*(1,1)
5-simplex-segment join {3,3,3,3}∨{ } [3,3,3,3,2,1] 6!×2! = 1440 (1,6,15,20,15,6,1)*(1,2,1)
5-cell-triangle join {3,3,3}∨{3} [3,3,3,2,3,1] 5!×3! = 720 (1,5,10,10,5,1)*(1,3,3,1)
triangle-triangle-segment join {3}∨{3}∨{ } [[3,2,3],2,1,1] ((3!)2×2!)×2! = 144 (1,3,3,1)2*(1,2,1)
Tetrahedron-tetrahedron join 2⋅{3,3} = {3,3}∨{3,3} [[3,3,2,3,3],1] (4!)2×2! = 1052 (1,4,6,4,1)2
4 segment join 4⋅{ } = { }∨{ }∨{ }∨{ } [4[2,2,2],1,1,1] (2!)4×4! = 384 (1,2,1)4
8 point join 8⋅( ) [8[1,1,1,1,1,1]] (1!)8×8! = 40320 (1,1)8

## 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

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.

A7 polytopes

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. ^ Klitzing, Richard. "7D uniform polytopes (polyexa) x3o3o3o3o3o3o — oca".
2. ^ Coxeter, H.S.M. (1973). "§1.8 Configurations". Regular Polytopes (3rd ed.). Dover. ISBN 0-486-61480-8.
3. ^ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds