8-simplex

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Convex regular 8-polytope

title: "8-simplex" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["8-polytopes"] description: "Convex regular 8-polytope" topic_path: "general/8-polytopes" source: "https://en.wikipedia.org/wiki/8-simplex" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Convex regular 8-polytope ::

::data[format=table] | Regular enneazetton (8-simplex) | |---| | [[File:8-simplex t0.svg|280px]] Orthogonal projection inside Petrie polygon | | Type | | Family | | Schläfli symbol | | Coxeter-Dynkindiagram | | 7-faces | | 6-faces | | 5-faces | | 4-faces | | Cells | | Faces | | Edges | | Vertices | | Vertex figure | | Petrie polygon | | Coxeter group | | Dual | | Properties | ::

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, with suffix -on.

Jonathan Bowers gives it the acronym ene.

As a configuration

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.

\begin{bmatrix}\begin{matrix} 9 & 8 & 28 & 56 & 70 & 56 & 28 & 8 \ 2 & 36 & 7 & 21 & 35 & 35 & 21 & 7 \ 3 & 3 & 84 & 6 & 15 & 20 & 15 & 6 \ 4 & 6 & 4 & 126 & 5 & 10 & 10 & 5 \ 5 & 10 & 10 & 5 & 126 & 4 & 6 & 4 \ 6 & 15 & 20 & 15 & 6 & 84 & 3 & 3 \ 7 & 21 & 35 & 35 & 21 & 7 & 36 & 2 \ 8 & 28 & 56 & 70 & 56 & 28 & 8 & 9 \end{matrix}\end{bmatrix}

Coordinates

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

:\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ \sqrt{1/3},\ \pm1\right) :\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ \sqrt{1/6},\ -2\sqrt{1/3},\ 0\right) :\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ \sqrt{1/10},\ -\sqrt{3/2},\ 0,\ 0\right) :\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ \sqrt{1/15},\ -2\sqrt{2/5},\ 0,\ 0,\ 0\right) :\left(1/6,\ \sqrt{1/28},\ \sqrt{1/21},\ -\sqrt{5/3},\ 0,\ 0,\ 0,\ 0\right) :\left(1/6,\ \sqrt{1/28},\ -\sqrt{12/7},\ 0,\ 0,\ 0,\ 0,\ 0\right) :\left(1/6,\ -\sqrt{7/4},\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right) :\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)

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.

Images

::figure[src="https://upload.wikimedia.org/wikipedia/commons/8/89/K9-gyroelongated_square_pyramid.gif" caption="The skeleton can be projected into the 9 vertices of a [[gyroelongated square pyramid]], edges colored by length."] ::

Related polytopes and honeycombs

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.

References

  • Coxeter, H.S.M.:

      • (Paper 22)
      • (Paper 23)
      • (Paper 24)
  • (x3o3o3o3o3o3o3o – ene)

References

  1. {{harvnb. Klitzing
  2. {{harvnb. Coxeter. 1973
  3. Coxeter, H.S.M.. (1991). "Regular Complex Polytopes". Cambridge University Press.

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