Cantic 6-cube
| Cantic 6-cube Truncated 6-demicube | |
|---|---|
 D6 Coxeter plane projection | |
| Type | uniform polypeton |
| Schläfli symbol | t0,1{3,33,1} h2{4,34} |
| Coxeter-Dynkin diagram |      =   |
| 5-faces | 76 |
| 4-faces | 636 |
| Cells | 2080 |
| Faces | 3200 |
| Edges | 2160 |
| Vertices | 480 |
| Vertex figure | ( )v[{ }x{3,3}] |
| Coxeter groups | D6, [33,1,1] |
| Properties | convex |
In six-dimensional geometry, a cantic 6-cube (or a truncated 6-demicube) is a uniform 6-polytope.
Alternate names
[edit]- Truncated 6-demicube
- Truncaced demihexeract
- Truncated hemihexeract (Acronym: thax) (Jonathan Bowers)[1]
Cartesian coordinates
[edit]The Cartesian coordinates for the 480 vertices of a cantic 6-cube centered at the origin and edge length 6√2 are coordinate permutations:
- (±1,±1,±3,±3,±3,±3)
with an odd number of plus signs.
Images
[edit]| Coxeter plane | B6 | |
|---|---|---|
| Graph |  | |
| Dihedral symmetry | [12/2] | |
| Coxeter plane | D6 | D5 |
| Graph |  |
 |
| Dihedral symmetry | [10] | [8] |
| Coxeter plane | D4 | D3 |
| Graph |  |
 |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A5 | A3 |
| Graph |  |
 |
| Dihedral symmetry | [6] | [4] |
Related polytopes
[edit]| n | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|
| Symmetry [1+,4,3n-2] |
[1+,4,3] = [3,3] |
[1+,4,32] = [3,31,1] |
[1+,4,33] = [3,32,1] |
[1+,4,34] = [3,33,1] |
[1+,4,35] = [3,34,1] |
[1+,4,36] = [3,35,1] |
| Cantic figure |
 |
 |
 |
|
 |
 |
| Coxeter | = |
= |
= |
= |
= |
= |
| Schläfli | h2{4,3} | h2{4,32} | h2{4,33} | h2{4,34} | h2{4,35} | h2{4,36} |
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
| D6 polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
 h{4,34} |
h2{4,34} |
 h3{4,34} |
 h4{4,34} |
 h5{4,34} |
 h2,3{4,34} |
 h2,4{4,34} |
 h2,5{4,34} | ||||
 h3,4{4,34} |
 h3,5{4,34} |
 h4,5{4,34} |
 h2,3,4{4,34} |
 h2,3,5{4,34} |
 h2,4,5{4,34} |
 h3,4,5{4,34} |
 h2,3,4,5{4,34} | ||||
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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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.
- Klitzing, Richard. "6D uniform polytopes (polypeta) with acronyms". x3x3o *b3o3o3o – thax