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8-simplex

Convex regular 8-polytope

Convex regular 8-polytope

Regular enneazetton
(8-simplex)
[[File:8-simplex t0.svg280px]]
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

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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