Rectified 8-orthoplexes
![]() 8-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Rectified 8-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Birectified 8-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Trirectified 8-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Trirectified 8-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Birectified 8-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() Rectified 8-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() 8-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Orthogonal projections in A8 Coxeter plane |
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In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.
Rectified 8-orthoplex
[edit]Rectified 8-orthoplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t1{3,3,3,3,3,3,4} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | 272 |
6-faces | 3072 |
5-faces | 8960 |
4-faces | 12544 |
Cells | 10080 |
Faces | 4928 |
Edges | 1344 |
Vertices | 112 |
Vertex figure | 6-orthoplex prism |
Petrie polygon | hexakaidecagon |
Coxeter groups | C8, [4,36] D8, [35,1,1] |
Properties | convex |
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.
Related polytopes
[edit]The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.
or
Alternate names
[edit]- rectified octacross
- rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)[1]
Construction
[edit]There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.
Cartesian coordinates
[edit]Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length are all permutations of:
- (±1,±1,0,0,0,0,0,0)
Images
[edit]B8 | B7 | ||||
---|---|---|---|---|---|
![]() | ![]() | ||||
[16] | [14] | ||||
B6 | B5 | ||||
![]() | ![]() | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] |
Birectified 8-orthoplex
[edit]Birectified 8-orthoplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t2{3,3,3,3,3,3,4} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | 272 |
6-faces | 3184 |
5-faces | 16128 |
4-faces | 34048 |
Cells | 36960 |
Faces | 22400 |
Edges | 6720 |
Vertices | 448 |
Vertex figure | {3,3,3,4}x{3} |
Coxeter groups | C8, [3,3,3,3,3,3,4] D8, [35,1,1] |
Properties | convex |
Alternate names
[edit]- birectified octacross
- birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]
Cartesian coordinates
[edit]Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,0,0,0,0,0)
Images
[edit]B8 | B7 | ||||
---|---|---|---|---|---|
![]() | ![]() | ||||
[16] | [14] | ||||
B6 | B5 | ||||
![]() | ![]() | ||||
[12] | [10] | ||||
B4 | B3 | B2 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
![]() | ![]() | ![]() | |||
[8] | [6] | [4] |
Trirectified 8-orthoplex
[edit]Trirectified 8-orthoplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t3{3,3,3,3,3,3,4} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | 16+256 |
6-faces | 1024 + 2048 + 112 |
5-faces | 1792 + 7168 + 7168 + 448 |
4-faces | 1792 + 10752 + 21504 + 14336 |
Cells | 8960 + 126880 + 35840 |
Faces | 17920 + 35840 |
Edges | 17920 |
Vertices | 1120 |
Vertex figure | {3,3,4}x{3,3} |
Coxeter groups | C8, [3,3,3,3,3,3,4] D8, [35,1,1] |
Properties | convex |
The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.
Alternate names
[edit]- trirectified octacross
- trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)[3]
Cartesian coordinates
[edit]Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length are all permutations of:
- (±1,±1,±1,±1,0,0,0,0)
Images
[edit]B8 | B7 | ||||
---|---|---|---|---|---|
![]() | ![]() | ||||
[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.
- Klitzing, Richard. "8D uniform polytopes (polyzetta)". o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark