Rectified 9-cubes
 9-orthoplex     |
 Rectified 9-orthoplex |
 Birectified 9-orthoplex | |
 Trirectified 9-orthoplex |
 Quadrirectified 9-cube |
 Trirectified 9-cube | |
 Birectified 9-cube |
 Rectified 9-cube |
 9-cube | |
| Orthogonal projections in BC9 Coxeter plane | |||
|---|---|---|---|
In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube.
There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-cube are located in the square face centers of the 9-cube. Vertices of the trirectified 9-orthoplex are located in the cube cell centers of the 9-cube. Vertices of the quadrirectified 9-cube are located in the tesseract centers of the 9-cube.
These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.
Rectified 9-cube
Alternate names
- Rectified enneract (Acronym ren) (Jonathan Bowers)[1]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
Birectified 9-cube
Alternate names
- Birectified enneract (Acronym barn) (Jonathan Bowers)[2]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
Trirectified 9-cube
Alternate names
- Trirectified enneract (Acronym tarn) (Jonathan Bowers)[3]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
 |
 | |||
| [8] | [6] | [4] | |||
Quadrirectified 9-cube
Alternate names
- Quadrirectified enneract (Acronym nav) (Jonathan Bowers)[4]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
|
 |
 | |||
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
 |
 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
 |
|
 | |||
| [8] | [6] | [4] | |||
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
- (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. (1966)
- Richard Klitzing, 9D, uniform polytopes (polyyotta) x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne
External links
- Olshevsky, George, Cross polytope at Glossary for Hyperspace.
- 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 4-polytope | 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 and compounds | ||||||||||||
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