In eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.
There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the edge-centers of the 8-cube. Vertices of the birectified 8-cube are located in the square face centers of the 8-cube. Vertices of the trirectified 8-cube are located in the 7-cube cell centers of the 8-cube.
Rectified 8-cube
| Rectified 8-cube |
|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t1{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 256 + 16 |
| 6-faces | 2048 + 112 |
| 5-faces | 7168 + 448 |
| 4-faces | 14336 + 1120 |
| Cells | 17920 +* 1792 |
| Faces | 4336 + 1792 |
| Edges | 7168 |
| Vertices | 1024 |
| Vertex figure | 6-simplex prism {3,3,3,3,3}×{} |
| Coxeter groups | B8, [36,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
- Rectified octeract
- Acronym: recto (Jonathan Bowers)
Images
orthographic projections| B8 | B7 |
|---|
| |
| [16] | [14] |
| B6 | B5 |
| |
| [12] | [10] |
| B4 | B3 | B2 |
| | |
| [8] | [6] | [4] |
| A7 | A5 | A3 |
| | |
| [8] | [6] | [4] |
Birectified 8-cube
| Birectified 8-cube |
|---|
| Type | uniform 8-polytope |
| Coxeter symbol | 0511 |
| Schläfli symbol | t2{4,3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 7-faces | 256 + 16 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 7168 + 7168 + 448 |
| 4-faces | 21504 + 14336 + 1120 |
| Cells | 35840 + 17920 + 1792 |
| Faces | 35840 + 14336 |
| Edges | 21504 |
| Vertices | 1792 |
| Vertex figure | {3,3,3,3}x{4} |
| Coxeter groups | B8, [36,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
- Birectified octeract
- Rectified 8-demicube
- Acronym: bro (Jonathan Bowers)
Images
orthographic projections| B8 | B7 |
|---|
| |
| [16] | [14] |
| B6 | B5 |
| |
| [12] | [10] |
| B4 | B3 | B2 |
| | |
| [8] | [6] | [4] |
| A7 | A5 | A3 |
| | |
| [8] | [6] | [4] |
Trirectified 8-cube
| Triectified 8-cube |
|---|
| Type | uniform 8-polytope |
| Schläfli symbol | t3{4,3,3,3,3,3,3} |
| Coxeter diagrams | |
| 7-faces | 16+256 |
| 6-faces | 1024 + 2048 + 112 |
| 5-faces | 1792 + 7168 + 7168 + 448 |
| 4-faces | 1792 + 10752 + 21504 +14336 |
| Cells | 8960 + 26880 + 35840 |
| Faces | 17920+35840 |
| Edges | 17920 |
| Vertices | 1152 |
| Vertex figure | {3,3,3}x{3,4} |
| Coxeter groups | B8, [36,4] D8, [35,1,1] |
| Properties | convex |
Alternate names
- Trirectified octeract
- Acronym: tro (Jonathan Bowers)
Images
orthographic projections| B8 | B7 |
|---|
| |
| [16] | [14] |
| B6 | B5 |
| |
| [12] | [10] |
| B4 | B3 | B2 |
| | |
| [8] | [6] | [4] |
| A7 | A5 | A3 |
| | |
| [8] | [6] | [4] |
Notes
- 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. . o3o3o3o3o3o3x4o - recto, o3o3o3o3o3x3o4o - bro, o3o3o3o3x3o3o4o - tro
External links