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.[1][2]
Coordinates
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.
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.
Images
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.
A8 polytopes | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
t0 | t1 | t2 | t3 | t01 | t02 | t12 | t03 | t13 | t23 | t04 | t14 | t24 | t34 | t05 |
t15 | t25 | t06 | t16 | t07 | t012 | t013 | t023 | t123 | t014 | t024 | t124 | t034 | t134 | t234 |
t015 | t025 | t125 | t035 | t135 | t235 | t045 | t145 | t016 | t026 | t126 | t036 | t136 | t046 | t056 |
t017 | t027 | t037 | t0123 | t0124 | t0134 | t0234 | t1234 | t0125 | t0135 | t0235 | t1235 | t0145 | t0245 | t1245 |
t0345 | t1345 | t2345 | t0126 | t0136 | t0236 | t1236 | t0146 | t0246 | t1246 | t0346 | t1346 | t0156 | t0256 | t1256 |
t0356 | t0456 | t0127 | t0137 | t0237 | t0147 | t0247 | t0347 | t0157 | t0257 | t0167 | t01234 | t01235 | t01245 | t01345 |
t02345 | t12345 | t01236 | t01246 | t01346 | t02346 | t12346 | t01256 | t01356 | t02356 | t12356 | t01456 | t02456 | t03456 | t01237 |
t01247 | t01347 | t02347 | t01257 | t01357 | t02357 | t01457 | t01267 | t01367 | t012345 | t012346 | t012356 | t012456 | t013456 | t023456 |
t123456 | t012347 | t012357 | t012457 | t013457 | t023457 | t012367 | t012467 | t013467 | t012567 | t0123456 | t0123457 | t0123467 | t0123567 | t01234567 |
References
- ^ Coxeter 1973, §1.8 Configurations
- ^ Coxeter, H.S.M. (1991). Regular Complex Polytopes (2nd ed.). Cambridge University Press. p. 117. ISBN 9780521394901.
- Coxeter, H.S.M.:
- — (1973). "Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)". Regular Polytopes (3rd ed.). Dover. pp. 296. ISBN 0-486-61480-8.
- Sherk, F. Arthur; McMullen, Peter; Thompson, Anthony C.; Weiss, Asia Ivic, eds. (1995). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Wiley. ISBN 978-0-471-01003-6.
- (Paper 22) — (1940). "Regular and Semi Regular Polytopes I". Math. Zeit. 46: 380–407. doi:10.1007/BF01181449. S2CID 186237114.
- (Paper 23) — (1985). "Regular and Semi-Regular Polytopes II". Math. Zeit. 188 (4): 559–591. doi:10.1007/BF01161657. S2CID 120429557.
- (Paper 24) — (1988). "Regular and Semi-Regular Polytopes III". Math. Zeit. 200: 3–45. doi:10.1007/BF01161745. S2CID 186237142.
- Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "26. Hemicubes: 1n1". The Symmetries of Things. p. 409. ISBN 978-1-56881-220-5.
- Johnson, Norman (1991). "Uniform Polytopes" (Manuscript). Norman Johnson (mathematician).
- Johnson, N.W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD). University of Toronto. OCLC 258527038.
- Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o3o — ene".
External links
- Glossary for hyperspace, George Olshevsky.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn | |||||||
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
Uniform polychoron | Pentachoron | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell | |||||||
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube | |||||||||
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 | ||||||||
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 | ||||||||
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 | ||||||||
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube | |||||||||
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube | |||||||||
Uniform n-polytope | 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 |