Stericated 7-cubes

Orthogonal projections in B6 Coxeter plane

7-cube

Stericated 7-cube

Bistericated 7-cube

Steritruncated 7-cube

Bisteritruncated 7-cube

Stericantellated 7-cube

Bistericantellated 7-cube

Stericantitruncated 7-cube

Bistericantitruncated 7-cube

Steriruncinated 7-cube

Steriruncitruncated 7-cube

Steriruncicantellated 7-cube

Bisteriruncitruncated 7-cube

Steriruncicantitruncated 7-cube

Bisteriruncicantitruncated 7-cube

In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th-order truncations (sterication) of the regular 7-cube.

There are 24 unique sterication for the 7-cube with permutations of truncations, cantellations, and runcinations. 10 are more simply constructed from the 7-orthoplex.

This polytope is one of 127 uniform 7-polytopes with B7 symmetry.

## Stericated 7-cube

[edit]

Stericated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Small cellated hepteract (acronym: ) (Jonathan Bowers)[1]

### Images

[edit]

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]

## Bistericated 7-cube

[edit]

bistericated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t1,5{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Small bicellated hepteractihecatonicosoctaexon (acronym: ) (Jonathan Bowers)[2]

### Images

[edit]

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]

## Steritruncated 7-cube

[edit]

steritruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,1,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Cellitruncated hepteract (acronym: ) (Jonathan Bowers)[3]

### Images

[edit]

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]

## Bisteritruncated 7-cube

[edit]

bisteritruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t1,2,5{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Bicellitruncated hepteract (acronym: ) (Jonathan Bowers)[4]

### Images

[edit]

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]

## Stericantellated 7-cube

[edit]

Stericantellated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,2,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Cellirhombated hepteract (acronym: ) (Jonathan Bowers)[5]

### Images

[edit]

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]

## Bistericantellated 7-cube

[edit]

Bistericantellated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t1,3,5{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Bicellirhombihepteract (acronym: ) (Jonathan Bowers)[6]

### Images

[edit]

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]

## Stericantitruncated 7-cube

[edit]

stericantitruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,1,2,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Celligreatorhombated hepteract (acronym: ) (Jonathan Bowers)[7]

### Images

[edit]

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]

## Bistericantitruncated 7-cube

[edit]

bistericantitruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t1,2,3,5{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Bicelligreatorhombated hepteract (acronym: ) (Jonathan Bowers)[8]

### Images

[edit]

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]

## Steriruncinated 7-cube

[edit]

Steriruncinated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,3,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Celliprismated hepteract (acronym: ) (Jonathan Bowers)[9]

### Images

[edit]

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]

## Steriruncitruncated 7-cube

[edit]

steriruncitruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,1,3,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Celliprismatotruncated hepteract (acronym: ) (Jonathan Bowers)[10]

### Images

[edit]

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]

## Steriruncicantellated 7-cube

[edit]

steriruncicantellated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,2,3,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Celliprismatorhombated hepteract (acronym: ) (Jonathan Bowers)[11]

### Images

[edit]

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]

## Bisteriruncitruncated 7-cube

[edit]

bisteriruncitruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t1,2,4,5{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Bicelliprismatotruncated hepteractihecatonicosoctaexon (acronym: ) (Jonathan Bowers)[12]

### Images

[edit]

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]

## Steriruncicantitruncated 7-cube

[edit]

steriruncicantitruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t0,1,2,3,4{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Great cellated hepteract (acronym: ) (Jonathan Bowers)[13]

### Images

[edit]

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  | too complex | too complex
Dihedral symmetry  | [6]  | [4]

## Bisteriruncicantitruncated 7-cube

[edit]

bisteriruncicantitruncated 7-cube

Type | uniform 7-polytope
Schläfli symbol | t1,2,3,4,5{4,35}
Coxeter-Dynkin diagrams |
6-faces |
5-faces |
4-faces |
Cells |
Faces |
Edges |
Vertices |
Vertex figure |
Coxeter groups | B7, [4,35]
Properties | convex

### Alternate names

[edit]

* Great bicellated hepteractihecatonicosoctaexon (Acronym ) (Jonathan Bowers) [14]

### Images

[edit]

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]

## Notes

[edit]

1. ^ Klitizing, (x3o3o3o3x3o4o - )
2. ^ Klitizing, (x3o3x3o3x3o4o - )
3. ^ Klitizing, (x3x3o3o3x3o4o - )
4. ^ Klitizing, (o3x3x3o3o3x4o - )
5. ^ Klitizing, (x3o3x3o3x3o4o - )
6. ^ Klitizing, (o3x3o3x3o3x4o - )
7. ^ Klitizing, (x3x3x3o3x3o4o - )
8. ^ Klitizing, (o3x3x3x3o3x4o - )
9. ^ Klitizing, (x3o3o3x3x3o4o - )
10. ^ Klitizing, (x3x3x3o3x3o4o - )
11. ^ Klitizing, (x3o3x3x3x3o4o - )
12. ^ Klitizing, (o3x3x3o3x3x4o - )
13. ^ Klitizing, (x3x3x3x3x3o4o - )
14. ^ Klitizing, (o3x3x3x3x3x4o - )

## References

[edit]

* 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)". x3o3o3o3x3o4o - , x3o3x3o3x3o4o - , x3x3o3o3x3o4o - , o3x3x3o3o3x4o - , x3o3x3o3x3o4o - , o3x3o3x3o3x4o - , x3x3x3o3x3o4o - , o3x3x3x3o3x4o - , x3o3o3x3x3o4o - , x3x3x3o3x3o4o - , x3o3x3x3x3o4o - , o3x3x3o3x3x4o - , x3x3x3x3x3o4o - , o3x3x3x3x3x4o -

## External links

[edit]

* 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 polychoron | Pentachoron | 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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