Rectified 6-simplexes
6-simplex |
Rectified 6-simplex |
Birectified 6-simplex |
Orthogonal projections in A6 Coxeter plane |
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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
Rectified 6-simplex | |
---|---|
Type | uniform polypeton |
Schläfli symbol | t1{35} |
Coxeter-Dynkin diagrams | |
Elements |
f5 = 14, f4 = 63, C = 140, F = 175, E = 105, V = 21 |
Coxeter group | A6, [35], order 5040 |
Bowers name and (acronym) | Rectified heptapeton (ril) |
Vertex figure | 5-cell prism |
Circumradius | 0.845154 |
Properties | convex, isogonal |
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
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
Dihedral symmetry | [4] | [3] |
Birectified 6-simplex
Birectified 6-simplex | |
---|---|
Type | uniform polypeton |
Schläfli symbol | t2{3,3,3,3,3} |
Coxeter-Dynkin diagrams | |
5-faces | 14 total: 7 t1{3,3,3,3} 7 t2{3,3,3,3} |
4-faces | 84 |
Cells | 245 |
Faces | 350 |
Edges | 210 |
Vertices | 35 |
Vertex figure | {3}x{3,3} |
Petrie polygon | Heptagon |
Coxeter groups | A6, [3,3,3,3,3] |
Properties | convex |
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
Ak Coxeter plane | A6 | A5 | A4 |
---|---|---|---|
Graph | |||
Dihedral symmetry | [7] | [6] | [5] |
Ak Coxeter plane | A3 | A2 | |
Graph | |||
Dihedral symmetry | [4] | [3] |
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.
t0 |
t1 |
t2 |
t0,1 |
t0,2 |
t1,2 |
t0,3 |
t1,3 |
t2,3 |
t0,4 |
t1,4 |
t0,5 |
t0,1,2 |
t0,1,3 |
t0,2,3 |
t1,2,3 |
t0,1,4 |
t0,2,4 |
t1,2,4 |
t0,3,4 |
t0,1,5 |
t0,2,5 |
t0,1,2,3 |
t0,1,2,4 |
t0,1,3,4 |
t0,2,3,4 |
t1,2,3,4 |
t0,1,2,5 |
t0,1,3,5 |
t0,2,3,5 |
t0,1,4,5 |
t0,1,2,3,4 |
t0,1,2,3,5 |
t0,1,2,4,5 |
t0,1,2,3,4,5 |
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, editied 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]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Richard Klitzing, 6D, uniform polytopes (polypeta) o3x3o3o3o3o - ril, o3x3o3o3o3o - bril
External links
- Olshevsky, George, Simplex at Glossary for Hyperspace.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
Fundamental convex regular and uniform polytopes in dimensions 2–10 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Family | An | BCn | 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 | 5-cell | 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 |