Cantellated 7-cubes
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In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.
 7-cube     |
 Cantellated 7-cube |
 Bicantellated 7-cube |
 Tricantellated 7-cube |
 Birectified 7-cube |
 Cantitruncated 7-cube |
 Bicantitruncated 7-cube |
 Tricantitruncated 7-cube |
 Cantellated 7-orthoplex |
 Bicantellated 7-orthoplex |
 Cantitruncated 7-orthoplex |
 Bicantitruncated 7-orthoplex |
| Orthogonal projections in B6 Coxeter plane | |||
|---|---|---|---|
There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.
Cantellated 7-cube
| Cantellated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | rr{4,3,3,3,3,3} |
| Coxeter diagram | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16128 |
| Vertices | 2688 |
| Vertex figure | |
| Coxeter groups | B7, [4,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Small rhombated hepteract (acronym: sersa) (Jonathan Bowers)[1]
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
Bicantellated 7-cube
| Bicantellated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | r2r{4,3,3,3,3,3} |
| Coxeter diagrams |   |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 40320 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | B7, [4,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Small birhombated hepteract (acronym: sibrosa) (Jonathan Bowers)[2]
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
Tricantellated 7-cube
| Tricantellated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | r3r{4,3,3,3,3,3} |
| Coxeter diagrams |  |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 47040 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter groups | B7, [4,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Small trirhombihepteractihecatonicosaoctaexon (acronym: strasaz) (Jonathan Bowers)[3]
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
Cantitruncated 7-cube
| Cantitruncated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | tr{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 18816 |
| Vertices | 5376 |
| Vertex figure | |
| Coxeter groups | B7, [4,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Great rhombated hepteract (acronym: gersa) (Jonathan Bowers)[4]
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
It is fifth in a series of cantitruncated hypercubes:
  |
  |
  |
  |
|
  |
| Truncated cuboctahedron | Cantitruncated tesseract | Cantitruncated 5-cube | Cantitruncated 6-cube | Cantitruncated 7-cube | Cantitruncated 8-cube |
|
|
|
|
|
|
Bicantitruncated 7-cube
| Bicantitruncated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | r2r{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 47040 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B7, [4,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Great birhombated hepteract (acronym: gibrosa) (Jonathan Bowers)[5]
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph |  |
|
 |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
Tricantitruncated 7-cube
| Tricantitruncated 7-cube | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t3r{4,3,3,3,3,3} |
| Coxeter diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 53760 |
| Vertices | 13440 |
| Vertex figure | |
| Coxeter groups | B7, [4,3,3,3,3,3] |
| Properties | convex |
Alternate names
- Great trirhombihepteractihecatonicosaoctaexon (acronym: gotrasaz) (Jonathan Bowers)[6]
Images
| Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
|---|---|---|---|
| Graph | too complex | |
 |
| Dihedral symmetry | [14] | [12] | [10] |
| Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
| Graph |  |
 |
 |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A5 | A3 | |
| Graph |  |
 | |
| Dihedral symmetry | [6] | [4] |
Related polytopes
These polytopes are from a family of 127 uniform 7-polytopes with B7 symmetry.