Regular octaexon
7-simplex t0.svg
Orthogonal projection
inside Petrie polygon
Type Regular 7-polytope
Family simplex
Schläfli symbol {3,3,3,3,3,3}
Coxeter-Dynkin diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
6-faces 8 6-simplex6-simplex t0.svg
5-faces 28 5-simplex5-simplex t0.svg
4-faces 56 5-cell4-simplex t0.svg
Cells 70 tetrahedron3-simplex t0.svg
Faces 56 triangle2-simplex t0.svg
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]


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

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.


7-Simplex in 3D
Uniform polytope 3,3,3,3,3,3 t0.jpg
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 7-simplex t0.svg 7-simplex t0 A6.svg 7-simplex t0 A5.svg
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Graph 7-simplex t0 A4.svg 7-simplex t0 A3.svg 7-simplex t0 A2.svg
Dihedral symmetry [5] [4] [3]

Related polytopes[]

This polytope is a facet in the uniform tessellation 331 with Coxeter-Dynkin diagram:

CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

This polytope is one of 71 uniform 7-polytopes with A7 symmetry.


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

External links[]

Fundamental convex regular and uniform polytopes in dimensions 2–10
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 4-polytope 5-cell 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