Cantellated 7-cubes

150px
7-cube
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Cantellated 7-cube
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Bicantellated 7-cube
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Tricantellated 7-cube

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150px
Birectified 7-cube
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Cantitruncated 7-cube
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Bicantitruncated 7-cube
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Tricantitruncated 7-cube

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Cantellated 7-orthoplex
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Bicantellated 7-orthoplex
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Cantitruncated 7-orthoplex
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Bicantitruncated 7-orthoplex

Orthogonal projections in B6 Coxeter plane

In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.

There are 10 degrees of cantellation for the 7-cube, including truncations. 4 are most simply constructible from the dual 7-orthoplex.

## Contents

* 1 Cantellated 7-cube
* 1.1 Alternate names
* 1.2 Images
* 2 Bicantellated 7-cube
* 2.1 Alternate names
* 2.2 Images
* 3 Tricantellated 7-cube
* 3.1 Alternate names
* 3.2 Images
* 4 Cantitruncated 7-cube
* 4.1 Alternate names
* 4.2 Images
* 5 Bicantitruncated 7-cube
* 5.1 Alternate names
* 5.2 Images
* 6 Tricantitruncated 7-cube
* 6.1 Alternate names
* 6.2 Images
* 7 Related polytopes
* 8 See also
* 9 Notes
* 10 References
* 11 External links

## 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

orthographic projections 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

orthographic projections 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 trirhombihepteractihecatonicosoctaexon (acronym: strasaz) (Jonathan Bowers)[3]

### Images

orthographic projections 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

orthographic projections 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: Template:Cantitruncated hypercube polytopes

## 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

orthographic projections 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 trirhombihepteractihecatonicosoctaexon (acronym: gotrasaz) (Jonathan Bowers)[6]

### Images

orthographic projections 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.

## See also

* List of B7 polytopes

## Notes

1. ↑ Klitizing, (x3o3x3o3o3o4o - sersa)
2. ↑ Klitizing, (o3x3o3x3o3o4o - sibrosa)
3. ↑ Klitizing, (o3o3x3o3x3o4o - strasaz)
4. ↑ Klitizing, (x3x3x3o3o3o4o - gersa)
5. ↑ Klitizing, (o3x3x3x3o3o4o - gibrosa)
6. ↑ Klitizing, (o3o3x3x3x3o4o - gotrasaz)

## References

* 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, ISBN 978-0-471-01003-6 [1]
* (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. "7D uniform polytopes (polyexa)". https://bendwavy.org/klitzing/dimensions/polyexa.htm. x3o3x3o3o3o4o- sersa, o3x3o3x3o3o4o - sibrosa, o3o3x3o3x3o4o - strasaz, x3x3x3o3o3o4o - gersa, o3x3x3x3o3o4o - gibrosa, o3o3x3x3x3o4o - gotrasaz

## External links

* Polytopes of Various Dimensions
* Multi-dimensional Glossary

* v
* t
* e

Fundamental convex regular and uniform polytopes in dimensions 2–10

Family | An | Bn | I2(p) / Dn | E6 / E7 / E8 / F4 / G2 | Hn
Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon
Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube |  | Dodecahedron • Icosahedron
Uniform 4-polytope | 5-cell | 16-cell • Tesseract | Demitesseract | 24-cell | 120-cell • 600-cell
Uniform 5-polytope | 5-simplex | 5-orthoplex • 5-cube | 5-demicube |  |
Uniform 6-polytope | 6-simplex | 6-orthoplex • 6-cube | 6-demicube | 122 • 221 |
Uniform 7-polytope | 7-simplex | 7-orthoplex • 7-cube | 7-demicube | 132 • 231 • 321 |
Uniform 8-polytope | 8-simplex | 8-orthoplex • 8-cube | 8-demicube | 142 • 241 • 421 |
Uniform 9-polytope | 9-simplex | 9-orthoplex • 9-cube | 9-demicube |  |
Uniform 10-polytope | 10-simplex | 10-orthoplex • 10-cube | 10-demicube |  |
Uniform n-polytope | n-simplex | n-orthoplex • n-cube | n-demicube | 1k2 • 2k1 • k21 | n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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