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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

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]


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
AkCoxeter 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, 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.

External links

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron o Cube Demicube Dodecahedron o Icosahedron
Uniform 4-polytope 5-cell 16-cell o Tesseract Demitesseract 24-cell 120-cell o 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex o 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex o 6-cube 6-demicube 122 o 221
Uniform 7-polytope 7-simplex 7-orthoplex o 7-cube 7-demicube 132 o 231 o 321
Uniform 8-polytope 8-simplex 8-orthoplex o 8-cube 8-demicube 142 o 241 o 421
Uniform 9-polytope 9-simplex 9-orthoplex o 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex o 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex o n-cube n-demicube 1k2 o 2k1 o k21 n-pentagonal polytope
Topics: Polytope families o Regular polytope o List of regular polytopes and compounds

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



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