Rectified 9-orthoplex
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 A9 Coxeter plane | ||||
|---|---|---|---|---|
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.
There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.
These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.
Contents |
Rectified 9-orthoplex
| Rectified 9-orthoplex | |
|---|---|
| Type | uniform 9-polytope |
| Schläfli symbol | t1{37,4} |
| Coxeter-Dynkin diagrams | |
| 7-faces | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2016 |
| Vertices | 144 |
| Vertex figure | 7-orthoplex prism |
| Petrie polygon | octakaidecagon |
| Coxeter groups | C9, [4,37] D9, [36,1,1] |
| Properties | convex |
The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.
- or
Rectified 9-orthoplex
Alternate names
- rectified enneacross (Acronym riv) (Jonathan Bowers)[1]
Construction
There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or [4,37] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or [36,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length 
are all permutations of:
- (±1,±1,0,0,0,0,0,0,0)
Root vectors
Its 144 vertices represent the root vectors of the simple Lie group D9. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups.
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
| [8] | [6] | [4] | |||
Birectified 9-orthoplex
Alternate names
- Rectified 9-demicube
- Birectified enneacross (Acronym brav) (Jonathan Bowers)[2]
Images
| B9 | B8 | B7 | |||
|---|---|---|---|---|---|
| [18] | [16] | [14] | |||
| B6 | B5 | ||||
| [12] | [10] | ||||
| B4 | B3 | B2 | |||
| [8] | [6] | [4] | |||
Trirectified 9-orthoplex
Alternate names
- trirectified enneacross (Acronym tarv) (Jonathan Bowers)[3]
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, editied 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. (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