Cantic 5-cube
Uniform 5-polytope
From Wikipedia, the free encyclopedia
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.
| Truncated 5-demicube Cantic 5-cube | |
|---|---|
 D5 Coxeter plane projection | |
| Type | uniform 5-polytope |
| Schläfli symbol | h2{4,3,3,3} t{3,32,1} |
| Coxeter-Dynkin diagram |      =   |
| 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 |  ( )v{ }×{3} |
| Coxeter groups | D5, [32,1,1] |
| Properties | convex |
Cartesian coordinates
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
- Cantic penteract, truncated demipenteract
- Truncated hemipenteract (thin) (Jonathan Bowers)[1]
Images
| Coxeter plane | B5 | |
|---|---|---|
| Graph |  | |
| Dihedral symmetry | [10/2] | |
| Coxeter plane | D5 | D4 |
| Graph |  |
 |
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D3 | A3 |
| Graph |  |
 |
| Dihedral symmetry | [4] | [4] |
Related polytopes
It has half the vertices of the cantellated 5-cube, as compared here in the B5 Coxeter plane projections:
Cantic 5-cube |
 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.
