Birectified_6-cube
In six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.
 6-cube     |
 Rectified 6-cube |
 Birectified 6-cube | |
 Birectified 6-orthoplex |
 Rectified 6-orthoplex |
 6-orthoplex | |
| Orthogonal projections in A6 Coxeter plane | |||
|---|---|---|---|
There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the edge-centers of the 6-cube. Vertices of the birectified 6-cube are located in the square face centers of the 6-cube.
| Rectified 6-cube | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t1{4,34} or r{4,34} |
| Coxeter-Dynkin diagrams | =       |
| 5-faces | 76 |
| 4-faces | 444 |
| Cells | 1120 |
| Faces | 1520 |
| Edges | 960 |
| Vertices | 192 |
| Vertex figure | 5-cell prism |
| Petrie polygon | Dodecagon |
| Coxeter groups | B6, [3,3,3,3,4] D6, [33,1,1] |
| Properties | convex |
Alternate names
- Rectified hexeract (acronym: rax) (Jonathan Bowers)
Construction
The rectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:
Images
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
...| Birectified 6-cube | |
|---|---|
| Type | uniform 6-polytope |
| Coxeter symbol | 0311 |
| Schläfli symbol | t2{4,34} or 2r{4,34} |
| Coxeter-Dynkin diagrams | =   =    |
| 5-faces | 76 |
| 4-faces | 636 |
| Cells | 2080 |
| Faces | 3200 |
| Edges | 1920 |
| Vertices | 240 |
| Vertex figure | {4}x{3,3} duoprism |
| Coxeter groups | B6, [3,3,3,3,4] D6, [33,1,1] |
| Properties | convex |
Alternate names
- Birectified hexeract (acronym: brox) (Jonathan Bowers)
- Rectified 6-demicube
Construction
The birectified 6-cube may be constructed from the 6-cube by truncating its vertices at the midpoints of its edges.
Coordinates
The Cartesian coordinates of the vertices of the rectified 6-cube with edge length √2 are all permutations of:
Images
| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | |||
| Dihedral symmetry | [6] | [4] |
These polytopes are part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
- 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
- (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)". o3x3o3o3o4o - rax, o3o3x3o3o4o - brox,
- Weisstein, Eric W. "Hypercube". MathWorld.
- 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 | ||||||||||||