inside Petrie polygon
|Coxeter group||A8 [3,3,3,3,3,3,3]|
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
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 (f-vectors). 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.
The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:
|Ak Coxeter plane||A8||A7||A6||A5|
|Ak Coxeter plane||A4||A3||A2|
Related polytopes and honeycombs
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
- Coxeter, Regular Polytopes, sec 1.8 Configurations
- Coxeter, Complex Regular Polytopes, p.117
- 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 
- (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: 1n1)
- 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".
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary