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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 namesEdit

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 configurationEdit

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
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
Dihedral symmetry [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Dihedral symmetry [5] [4] [3]

Related polytopesEdit


  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 linksEdit