Cantellated 5-orthoplexes
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In five-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.
 5-orthoplex     |
 Cantellated 5-orthoplex |
 Bicantellated 5-cube |
Cantellated 5-cube |
 5-cube |
 Cantitruncated 5-orthoplex |
 Bicantitruncated 5-cube |
 Cantitruncated 5-cube |
| Orthogonal projections in B5 Coxeter plane | |||
|---|---|---|---|
There are 6 cantellation for the 5-orthoplex, including truncations. Some of them are more easily constructed from the dual 5-cube.
Cantellated 5-orthoplex
| Cantellated 5-orthoplex | ||
| Type | Uniform 5-polytope | |
| Schläfli symbol | rr{3,3,3,4} rr{3,3,31,1} | |
| Coxeter-Dynkin diagrams |   | |
| 4-faces | 82 | 10  40   32  |
| Cells | 640 | 80  160  320  80  |
| Faces | 1520 | 640  320 480  80 |
| Edges | 1200 | 960 240 |
| Vertices | 240 | |
| Vertex figure | Square pyramidal prism  | |
| Coxeter group | B5, [4,3,3,3], order 3840 D5, [32,1,1], order 1920 | |
| Properties | convex | |
Alternate names
- Cantellated 5-orthoplex
- Bicantellated 5-demicube
- Small rhombated triacontaditeron (Acronym: sart) (Jonathan Bowers)[1]
Coordinates
The vertices of the can be made in 5-space, as permutations and sign combinations of:
- (0,0,1,1,2)
Images
The cantellated 5-orthoplex is constructed by a cantellation operation applied to the 5-orthoplex.
| Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B2 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [4] | [4] |
Cantitruncated 5-orthoplex
| Cantitruncated 5-orthoplex | ||
|---|---|---|
| Type | uniform 5-polytope | |
| Schläfli symbol | tr{3,3,3,4} tr{3,31,1} | |
| Coxeter-Dynkin diagrams |  | |
| 4-faces | 82 | 10  40 32  |
| Cells | 640 | 80 160  320 80  |
| Faces | 1520 | 640 320  480 80 |
| Edges | 1440 | 960 240 240 |
| Vertices | 480 | |
| Vertex figure | Square pyramidal pyramid  | |
| Coxeter groups | B5, [3,3,3,4], order 3840 D5, [32,1,1], order 1920 | |
| Properties | convex | |
Alternate names
- Cantitruncated pentacross
- Cantitruncated triacontaditeron (Acronym: gart) (Jonathan Bowers)[2]
Coordinates
Cartesian coordinates for the vertices of a cantitruncated 5-orthoplex, centered at the origin, are all sign and coordinate permutations of
- (±3,±2,±1,0,0)
Images
| Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [10] | [8] | [6] |
| Coxeter plane | B2 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [4] | [4] |
Related polytopes
These polytopes are from a set of 31 uniform 5-polytopes generated from the regular 5-cube or 5-orthoplex.
