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
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.

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

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

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

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

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

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

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

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

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

$\left(-4/3,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0,\ 0\right)$

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.

Images

orthographic projections

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

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

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o3o - ene".

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary