8-simplex

Regular enneazetton (8-simplex)
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	simplex
Schläfli symbol	{3,3,3,3,3,3,3}
Coxeter-Dynkin diagram	Template:CDD
7-faces	9 7-simplex
6-faces	36 6-simplex
5-faces	84 5-simplex
4-faces	126 5-cell
Cells	126 tetrahedron
Faces	84 triangle
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 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.

$\left[\begin{array}{c}\begin{array}{cccccccc}9& 8& 28& 56& 70& 56& 28& 8\\ 2& 36& 7& 21& 35& 35& 21& 7\\ 3& 3& 84& 6& 15& 20& 15& 6\\ 4& 6& 4& 126& 5& 10& 10& 5\\ 5& 10& 10& 5& 126& 4& 6& 4\\ 6& 15& 20& 15& 6& 84& 3& 3\\ 7& 21& 35& 35& 21& 7& 36& 2\\ 8& 28& 56& 70& 56& 28& 8& 9\end{array}\end{array}\right]$

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

$(1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},\sqrt{1/10},\sqrt{1/6},\sqrt{1/3},\pm 1)$

$(1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},\sqrt{1/10},\sqrt{1/6},-2\sqrt{1/3},0)$

$(1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},\sqrt{1/10},-\sqrt{3/2},0,0)$

$(1/6,\sqrt{1/28},\sqrt{1/21},\sqrt{1/15},-2\sqrt{2/5},0,0,0)$

$(1/6,\sqrt{1/28},\sqrt{1/21},-\sqrt{5/3},0,0,0,0)$

$(1/6,\sqrt{1/28},-\sqrt{12/7},0,0,0,0,0)$

$(1/6,-\sqrt{7/4},0,0,0,0,0,0)$

$(-4/3,0,0,0,0,0,0,0)$

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.

Related polytopes and honeycombs

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

Template:CDD, Template:CDD

This polytope is one of 135 uniform 8-polytopes with A₈ symmetry. Template:Enneazetton family

References

• Coxeter, H.S.M.:
  • Template:Cite book
  • Template:Cite book
    • (Paper 22) Template:Cite journal
    • (Paper 23) Template:Cite journal
    • (Paper 24) Template:Cite journal
• Template:Cite book
• Template:Cite document
  • Template:Cite thesis
• Template:KlitzingPolytopes

External links

• Template:PolyCell
• Polytopes of Various Dimensions
• Multi-dimensional Glossary