f5 = 7, f4 = 21, C = 35, F = 35, E = 21, V = 7
|Coxeter group||A6, , order 5040|
|Properties||convex, isogonal self-dual|
In geometry, a 6-simplex is a self-dual regular 6-polytope. It has 7 vertices, 21 edges, 35 triangle faces, 35 tetrahedral cells, 21 5-cell 4-faces, and 7 5-simplex 5-faces. Its dihedral angle is cos−1(1/6), or approximately 80.41°.
It can also be called a heptapeton, or hepta-6-tope, as a 7-facetted polytope in 6-dimensions. The name heptapeton is derived from hepta for seven facets in Greek and -peta for having five-dimensional facets, and -on. Jonathan Bowers gives a heptapeton the acronym hop.
As a configurationEdit
This configuration matrix represents the 6-simplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-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.
The Cartesian coordinates for an origin-centered regular heptapeton having edge length 2 are:
The vertices of the 6-simplex can be more simply positioned in 7-space as permutations of:
|Ak Coxeter plane||A6||A5||A4|
|Ak Coxeter plane||A3||A2|
Related uniform 6-polytopesEdit
- Klitzing, (x3o3o3o3o3o - hop)
- 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. "6D uniform polytopes (polypeta) x3o3o3o3o - hix".
- Olshevsky, George. "Simplex". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
|An||Bn||I2(p) / Dn||E6 / E7 / E8 / F4 / G2||Hn|
|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|