Regular octaexon

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

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]


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

Model created using straws (edges) and plasticine balls (vertices) 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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
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

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / 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
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