Regular enneazetton (8simplex)

Orthogonal projection inside Petrie polygon

Type 
Regular 8polytope

Family 
simplex

Schläfli symbol 
{3,3,3,3,3,3,3}

CoxeterDynkin diagram 

7faces 
9 7simplex

6faces 
36 6simplex

5faces 
84 5simplex

4faces 
126 5cell

Cells 
126 tetrahedron

Faces 
84 triangle

Edges 
36

Vertices 
9

Vertex figure 
7simplex

Petrie polygon 
enneagon

Coxeter group 
A_{8} [3,3,3,3,3,3,3]

Dual 
Selfdual

Properties 
convex

In geometry, an 8simplex is a selfdual regular 8polytope. It has 9 vertices, 36 edges, 84 triangle faces, 126 tetrahedral cells, 126 5cell 4faces, 84 5simplex 5faces, 36 6simplex 6faces, and 9 7simplex 7faces. Its dihedral angle is cos^{−1}(1/8), or approximately 82.82°.
It can also be called an enneazetton, or ennea8tope, as a 9facetted polytope in eightdimensions. The name enneazetton is derived from ennea for nine facets in Greek and zetta for having sevendimensional facets, and on.
As a configuration[]
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, etc, with diagonal element their counts (fvectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^{[1]}^{[2]}
${\begin{bmatrix}{\begin{matrix}9&8&28&56&70&56&28&8\\2&36&7&21&35&35&21&7\\3&3&84&6&15&20&15&6\\4&6&4&126&5&10&10&5\\5&10&10&5&126&4&6&4\\6&15&20&15&6&84&3&3\\7&21&35&35&21&7&36&2\\8&28&56&70&56&28&8&9\end{matrix}}\end{bmatrix}}$
Coordinates[]
The Cartesian coordinates of the vertices of an origincentered 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 8simplex can be positioned in 9space as permutations of (0,0,0,0,0,0,0,0,1). This construction is based on facets of the 9orthoplex.
Images[]
Related polytopes and honeycombs[]
This polytope is a facet in the uniform tessellations: 2_{51}, and 5_{21} with respective CoxeterDynkin diagrams:
 ,
This polytope is one of 135 uniform 8polytopes with A_{8} symmetry.
References[]
 ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
 ^ Coxeter, Complex Regular Polytopes, p.117
 H.S.M. Coxeter:
 Coxeter, Regular Polytopes, (3rd ion, 1973), Dover ion, ISBN 0486614808, p.296, Table I (iii): Regular Polytopes, three regular polytopes in ndimensions (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 ndimensions (n≥5)
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, ed by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036 [1]
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 (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[]