Rectified 9-orthoplexes
![]() 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 |
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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.
Rectified 9-orthoplex
[edit]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
Alternate names
[edit]- rectified enneacross (Acronym riv) (Jonathan Bowers)[1]
Construction
[edit]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
[edit]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
[edit]Its 144 vertices represent the root vectors of the simple Lie group D9. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. 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
[edit]B9 | B8 | B7 | |||
---|---|---|---|---|---|
![]() | ![]() | ![]() | |||
[18] | [16] | [14] | |||
B6 | B5 | ||||
![]() | ![]() | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
— | — | — | |||
[8] | [6] | [4] |
Birectified 9-orthoplex
[edit]Alternate names
[edit]- Rectified 9-demicube
- Birectified enneacross (Acronym brav) (Jonathan Bowers)[2]
Images
[edit]B9 | B8 | B7 | |||
---|---|---|---|---|---|
![]() | ![]() | ![]() | |||
[18] | [16] | [14] | |||
B6 | B5 | ||||
![]() | ![]() | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
— | — | — | |||
[8] | [6] | [4] |
Trirectified 9-orthoplex
[edit]Alternate names
[edit]- trirectified enneacross (Acronym tarv) (Jonathan Bowers)[3]
Images
[edit]B9 | B8 | B7 | |||
---|---|---|---|---|---|
![]() | ![]() | ![]() | |||
[18] | [16] | [14] | |||
B6 | B5 | ||||
![]() | ![]() | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
— | — | — | |||
[8] | [6] | [4] |
Notes
[edit]References
[edit]- 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 [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)
- Klitzing, Richard. "9D uniform polytopes (polyyotta)". x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne