Regular enneazetton
8-simplex t0.svg
Orthogonal projection
inside Petrie polygon
Type Regular 8-polytope
Family simplex
Schläfli symbol {3,3,3,3,3,3,3}
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.pngCDel 3.pngCDel node.png
7-faces 9 7-simplex7-simplex t0.svg
6-faces 36 6-simplex6-simplex t0.svg
5-faces 84 5-simplex5-simplex t0.svg
4-faces 126 5-cell4-simplex t0.svg
Cells 126 tetrahedron3-simplex t0.svg
Faces 84 triangle2-simplex t0.svg
Edges 36
Vertices 9
Vertex figure 7-simplex
Petrie polygon enneagon
Coxeter group A8 [3,3,3,3,3,3,3]
Dual Self-dual
Properties convex

In geometry, an 8-simplex is a self-dual regular 8-polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5-cell 4-faces, 84 5-simplex 5-faces, 36 6-simplex 6-faces, and 9 7-simplex 7-faces. Its dihedral angle is cos−1(1/8), or approximately 82.82°.

It can also be called an enneazetton, or ennea-8-tope, as a 9-facetted polytope in eight-dimensions. The name enneazetton is derived from ennea for nine facets in Greek and -zetta for having seven-dimensional facets, and -on.

As a configurationEdit

This configuration matrix represents the 8-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-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.[1][2]



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


More simply, the vertices of the 8-simplex can be positioned in 9-space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9-orthoplex.

Another origin-centered construction uses (1,1,1,1,1,1,1,1)/3 and permutations of (1,1,1,1,1,1,1,-11)/12 for edge length √2.


orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Dihedral symmetry [9] [8] [7] [6]
Ak Coxeter plane A4 A3 A2
Dihedral symmetry [5] [4] [3]

Related polytopes and honeycombsEdit

This polytope is a facet in the uniform tessellations: 251, and 521 with respective Coxeter-Dynkin diagrams:


This polytope is one of 135 uniform 8-polytopes with A8 symmetry.


  1. ^ Coxeter 1973, §1.8 Configurations
  2. ^ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.

External linksEdit

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