Truncated 6-orthoplexes
 6-orthoplex    
|
 Truncated 6-orthoplex
|
 Bitruncated 6-orthoplex
|
 Tritruncated 6-cube
|
 6-cube
|
 Truncated 6-cube
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 Bitruncated 6-cube
| |
| Orthogonal projections in B6 Coxeter plane | |||
|---|---|---|---|
In six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.
There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the bitruncated 6-orthoplex are located on the triangular faces of the 6-orthoplex. Vertices of the tritruncated 6-orthoplex are located inside the tetrahedral cells of the 6-orthoplex.
Truncated 6-orthoplex
[edit]| Truncated 6-orthoplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t{3,3,3,3,4} |
| Coxeter-Dynkin diagrams |
|
| 5-faces | 76 |
| 4-faces | 576 |
| Cells | 1200 |
| Faces | 1120 |
| Edges | 540 |
| Vertices | 120 |
| Vertex figure |  ( )v{3,4} |
| Coxeter groups | B6, [3,3,3,3,4] D6, [33,1,1] |
| Properties | convex |
Alternate names
[edit]- Truncated hexacross
- Truncated hexacontatetrapeton (Acronym: tag) (Jonathan Bowers)[1]
Construction
[edit]There are two Coxeter groups associated with the truncated hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with the D6 or [33,1,1] Coxeter group.
Coordinates
[edit]Cartesian coordinates for the vertices of a truncated 6-orthoplex, centered at the origin, are all 120 vertices are sign (4) and coordinate (30) permutations of
- (±2,±1,0,0,0,0)
Images
[edit]| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph |
|
|
|
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | 
|
| |
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | 
|
| |
| Dihedral symmetry | [6] | [4] |
Bitruncated 6-orthoplex
[edit]| Bitruncated 6-orthoplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | 2t{3,3,3,3,4} |
| Coxeter-Dynkin diagrams |
|
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure |  { }v{3,4} |
| Coxeter groups | B6, [3,3,3,3,4] D6, [33,1,1] |
| Properties | convex |
Alternate names
[edit]- Bitruncated hexacross
- Bitruncated hexacontatetrapeton (Acronym: botag) (Jonathan Bowers)[2]
Images
[edit]| Coxeter plane | B6 | B5 | B4 |
|---|---|---|---|
| Graph |
|
|
|
| Dihedral symmetry | [12] | [10] | [8] |
| Coxeter plane | B3 | B2 | |
| Graph | 
|
| |
| Dihedral symmetry | [6] | [4] | |
| Coxeter plane | A5 | A3 | |
| Graph | 
|
| |
| Dihedral symmetry | [6] | [4] |
Related polytopes
[edit]These polytopes are a part of a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.
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. "6D uniform polytopes (polypeta)". x3x3o3o3o4o - tag, o3x3x3o3o4o - botag