8-simplex

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.

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

Coordinates

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

:$\left(1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \pm1\right)$
:$\left(1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0\right)$
:$\left(1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0\right)$
:$\left(1/6,\ \sqrt,\ \sqrt,\ \sqrt,\ -2\sqrt,\ 0,\ 0,\ 0\right)$
:$\left(1/6,\ \sqrt,\ \sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0\right)$
:$\left(1/6,\ \sqrt,\ -\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0\right)$
:$\left(1/6,\ -\sqrt,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$
:$\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

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

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)
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*
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* (x3o3o3o3o3o3o3o – ene)

External links

*

Polytopes of Various Dimensions



{{Polytopes

8-polytopes