Cantic 5-cube
| Truncated 5-demicube Cantic 5-cube | |
|---|---|
| File:Truncated 5-demicube D5.svg D5 Coxeter plane projection | |
| Type | uniform 5-polytope |
| Schläfli symbol | h2{4,3,3,3} t{3,32,1} |
| Coxeter-Dynkin diagram | File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = File:CDel node h1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png |
| 4-faces | 42 total: 16 r{3,3,3} 16 t{3,3,3} 10 t{3,3,4} |
| Cells | 280 total: 80 {3,3} 120 t{3,3} 80 {3,4} |
| Faces | 640 total: 480 {3} 160 {6} |
| Edges | 560 |
| Vertices | 160 |
| Vertex figure | File:Truncated 5-demicube verf.png ( )v{ }×{3} |
| Coxeter groups | D5, [32,1,1] |
| Properties | convex |
In geometry of five dimensions or higher, a cantic 5-cube, cantihalf 5-cube, truncated 5-demicube is a uniform 5-polytope, being a truncation of the 5-demicube. It has half the vertices of a cantellated 5-cube.
Cartesian coordinates
[edit | edit source]The Cartesian coordinates for the 160 vertices of a cantic 5-cube centered at the origin and edge length 6√2 are coordinate permutations:
- (±1,±1,±3,±3,±3)
with an odd number of plus signs.
Alternate names
[edit | edit source]- Cantic penteract, truncated demipenteract
- Truncated hemipenteract (thin) (Jonathan Bowers)[1]
Images
[edit | edit source]| Coxeter plane | B5 | |
|---|---|---|
| Graph | File:5-demicube t01 B5.svg | |
| Dihedral symmetry | [10/2] | |
| Coxeter plane | D5 | D4 |
| Graph | File:5-demicube t01 D5.svg | File:5-demicube t01 D4.svg |
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D3 | A3 |
| Graph | File:5-demicube t01 D3.svg | File:5-demicube t01 A3.svg |
| Dihedral symmetry | [4] | [4] |
Related polytopes
[edit | edit source]It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections:
| File:5-demicube t01 B5.svg Cantic 5-cube |
File:5-cube t02.svg Cantellated 5-cube |
This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.
There are 23 uniform 5-polytope that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.
Notes
[edit | edit source]- ^ Klitzing, (x3x3o *b3o3o - thin)
References
[edit | edit source]- 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, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). [1] Archived 2016-07-11 at the Wayback Machine
- (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.
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
External links
[edit | edit source]- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Polytopes of Various Dimensions
- Multi-dimensional Glossary