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 configurationEdit
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.[1][2]
CoordinatesEdit
The Cartesian coordinates of the vertices of an origin-centered regular enneazetton having edge length 2 are:
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.
ImagesEdit
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 honeycombsEdit
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.
ReferencesEdit
- 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: 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".
External linksEdit
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10
| ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | ||||||||
Triangle | Square | p-gon | Hexagon | Pentagon | ||||||||
Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | |||||||||
5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | ||||||||
5-simplex | 5-orthoplex • 5-cube | 5-demicube | ||||||||||
6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | |||||||||
7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | |||||||||
8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | |||||||||
9-simplex | 9-orthoplex • 9-cube | 9-demicube | ||||||||||
10-simplex | 10-orthoplex • 10-cube | 10-demicube | ||||||||||
n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope | ||||||||
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds |