- Source: Rectified 6-simplexes
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.
Rectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S16. It is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as .
= Alternate names
=Rectified heptapeton (Acronym: ril) (Jonathan Bowers)
= Coordinates
=The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.
= Images
=Birectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S26. It is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as .
= Alternate names
=Birectified heptapeton (Acronym: bril) (Jonathan Bowers)
= Coordinates
=The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.
= Images
=Related uniform 6-polytopes
The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 241 polytope.
These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
Notes
References
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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]
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
Klitzing, Richard. "6D uniform polytopes (polypeta)". o3x3o3o3o3o - ril, o3x3o3o3o3o - bril
External links
Polytopes of Various Dimensions
Multi-dimensional Glossary
Kata Kunci Pencarian:
- Rectified 6-simplexes
- Pentellated 6-simplexes
- Rectified 9-simplexes
- Rectified 5-simplexes
- Rectified 7-simplexes
- Rectified 10-simplexes
- Rectified 8-simplexes
- 4 21 polytope
- 2 31 polytope
- Rectified 5-cell
