Pentellated 6-cubes

6-cube

6-orthoplex

Pentellated 6-cube

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Pentitruncated 6-cube

Penticantellated 6-cube

Penticantitruncated 6-cube

Pentiruncitruncated 6-cube

Pentiruncicantellated 6-cube

Pentiruncicantitruncated 6-cube

Pentisteritruncated 6-cube

Pentistericantitruncated 6-cube

Omnitruncated 6-cube

Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.

There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications. The simple pentellated 6-cube is also called an expanded 6-cube, constructed by an expansion operation applied to the regular 6-cube. The highest form, the pentisteriruncicantitruncated 6-cube, is called an omnitruncated 6-cube with all of the nodes ringed. Six of them are better constructed from the 6-orthoplex given at pentellated 6-orthoplex.

## Contents

* 1 Pentellated 6-cube
* 1.1 Alternate names
* 1.2 Images
* 2 Pentitruncated 6-cube
* 2.1 Alternate names
* 2.2 Images
* 3 Penticantellated 6-cube
* 3.1 Alternate names
* 3.2 Images
* 4 Penticantitruncated 6-cube
* 4.1 Alternate names
* 4.2 Images
* 5 Pentiruncitruncated 6-cube
* 5.1 Alternate names
* 5.2 Images
* 6 Pentiruncicantellated 6-cube
* 6.1 Alternate names
* 6.2 Images
* 7 Pentiruncicantitruncated 6-cube
* 7.1 Alternate names
* 7.2 Images
* 8 Pentisteritruncated 6-cube
* 8.1 Alternate names
* 8.2 Images
* 9 Pentistericantitruncated 6-cube
* 9.1 Alternate names
* 9.2 Images
* 10 Omnitruncated 6-cube
* 10.1 Alternate names
* 10.2 Images
* 10.3 Full snub 6-cube
* 11 Related polytopes
* 12 Notes
* 13 References
* 14 External links

## Pentellated 6-cube

Pentellated 6-cube

Type | Uniform 6-polytope
Schläfli symbol | t0,5{4,3,3,3,3}
Coxeter-Dynkin diagram |
5-faces |
4-faces |
Cells |
Faces |
Edges | 1920
Vertices | 384
Vertex figure | 5-cell antiprism
Coxeter group | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Pentellated 6-orthoplex
* Expanded 6-cube, expanded 6-orthoplex
* Small teri-hexeractihexacontitetrapeton (Acronym: stoxog) (Jonathan Bowers)[1]

### Images

## Pentitruncated 6-cube

Pentitruncated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,1,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 8640
Vertices | 1920
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Teritruncated hexeract (Acronym: tacog) (Jonathan Bowers)[2]

### Images

orthographic projections Coxeter plane | B6 | B5 | B4

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Graph  |  |  |
Dihedral symmetry | [12]  | [10]  | [8]
Coxeter plane  | B3 | B2
Graph  |  |
Dihedral symmetry  | [6]  | [4]
Coxeter plane  | A5 | A3
Graph  |  |
Dihedral symmetry  | [6]  | [4]

## Penticantellated 6-cube

Penticantellated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 21120
Vertices | 3840
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Terirhombated hexeract (Acronym: topag) (Jonathan Bowers)[3]

### Images

orthographic projections Coxeter plane | B6 | B5 | B4

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Graph  |  |  |
Dihedral symmetry | [12]  | [10]  | [8]
Coxeter plane  | B3 | B2
Graph  |  |
Dihedral symmetry  | [6]  | [4]
Coxeter plane  | A5 | A3
Graph  |  |
Dihedral symmetry  | [6]  | [4]

## Penticantitruncated 6-cube

Penticantitruncated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,1,2,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 30720
Vertices | 7680
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Terigreatorhombated hexeract (Acronym: togrix) (Jonathan Bowers)[4]

### Images

orthographic projections Coxeter plane | B6 | B5 | B4

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|
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Graph  |  |  |
Dihedral symmetry | [12]  | [10]  | [8]
Coxeter plane  | B3 | B2
Graph  |  |
Dihedral symmetry  | [6]  | [4]
Coxeter plane  | A5 | A3
Graph  |  |
Dihedral symmetry  | [6]  | [4]

## Pentiruncitruncated 6-cube

Pentiruncitruncated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,1,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 151840
Vertices | 11520
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Tericellirhombated hexacontitetrapeton (Acronym: tocrag) (Jonathan Bowers)[5]

### Images

orthographic projections Coxeter plane | B6 | B5 | B4

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|
|

Graph  |  |  |
Dihedral symmetry | [12]  | [10]  | [8]
Coxeter plane  | B3 | B2
Graph  |  |
Dihedral symmetry  | [6]  | [4]
Coxeter plane  | A5 | A3
Graph  |  |
Dihedral symmetry  | [6]  | [4]

## Pentiruncicantellated 6-cube

Pentiruncicantellated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 46080
Vertices | 11520
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Teriprismatorhombi-hexeractihexacontitetrapeton (Acronym: tiprixog) (Jonathan Bowers)[6]

### Images

## Pentiruncicantitruncated 6-cube

Pentiruncicantitruncated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,1,2,3,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 80640
Vertices | 23040
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Terigreatoprismated hexeract (Acronym: tagpox) (Jonathan Bowers)[7]

### Images

orthographic projections Coxeter plane | B6 | B5 | B4

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|
|

Graph  |  |  |
Dihedral symmetry | [12]  | [10]  | [8]
Coxeter plane  | B3 | B2
Graph  |  |
Dihedral symmetry  | [6]  | [4]
Coxeter plane  | A5 | A3
Graph  |  |
Dihedral symmetry  | [6]  | [4]

## Pentisteritruncated 6-cube

Pentisteritruncated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,1,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 30720
Vertices | 7680
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Tericellitrunki-hexeractihexacontitetrapeton (Acronym: tactaxog) (Jonathan Bowers)[8]

### Images

## Pentistericantitruncated 6-cube

Pentistericantitruncated 6-cube

Type | uniform 6-polytope
Schläfli symbol | t0,1,2,4,5{4,3,3,3,3}
Coxeter-Dynkin diagrams |
5-faces |
4-faces |
Cells |
Faces |
Edges | 80640
Vertices | 23040
Vertex figure |
Coxeter groups | B6, [4,3,3,3,3]
Properties | convex

### Alternate names

* Tericelligreatorhombated hexeract (Acronym: tocagrax) (Jonathan Bowers)[9]

### Images

orthographic projections Coxeter plane | B6 | B5 | B4

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|
|

Graph  |  |  |
Dihedral symmetry | [12]  | [10]  | [8]
Coxeter plane  | B3 | B2
Graph  |  |
Dihedral symmetry  | [6]  | [4]
Coxeter plane  | A5 | A3
Graph  |  |
Dihedral symmetry  | [6]  | [4]

## Omnitruncated 6-cube

Omnitruncated 6-cube

Type  | Uniform 6-polytope
Schläfli symbol | t0,1,2,3,4,5{35}
Coxeter-Dynkin diagrams |
5-faces | 728:
12 t0,1,2,3,4{3,3,3,4}
60 {}×t0,1,2,3{3,3,4} 40px×40px
160 {6}×t0,1,2{3,4} 40px×40px
240 {8}×t0,1,2{3,3} 40px×40px
192 {}×t0,1,2,3{33} 40px×40px
64 t0,1,2,3,4{34}

4-faces | 14168
Cells | 72960
Faces | 151680
Edges | 138240
Vertices | 46080
Vertex figure | irregular 5-simplex
Coxeter group | B6, [4,3,3,3,3]
Properties | convex, isogonal

The omnitruncated 6-cube has 5040 vertices, 15120 edges, 16800 faces (4200 hexagons and 1260 squares), 8400 cells, 1806 4-faces, and 126 5-faces. With 5040 vertices, it is the largest of 35 uniform 6-polytopes generated from the regular 6-cube.

### Alternate names

* Pentisteriruncicantituncated 6-cube or 6-orthoplex (omnitruncation for 6-polytopes)
* Omnitruncated hexeract
* Great teri-hexeractihexacontitetrapeton (Acronym: gotaxog) (Jonathan Bowers)[10]

### Images

### Full snub 6-cube

The full snub 6-cube or omnisnub 6-cube, defined as an alternation of the omnitruncated 6-cube is not uniform, but it can be given Coxeter diagram  and symmetry [4,3,3,3,3]+, and constructed from 12 snub 5-cubes, 64 snub 5-simplexes, 60 snub tesseract antiprisms, 192 snub 5-cell antiprisms, 160 3-sr{4,3} duoantiprisms, 240 4-s{3,4} duoantiprisms, and 23040 irregular 5-simplexes filling the gaps at the deleted vertices.

## Related polytopes

These polytopes are from a set of 63 uniform 6-polytopes generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

## Notes

1. ↑ Klitzing, (x4o3o3o3o3x - stoxog)
2. ↑ Klitzing, (x4x3o3o3o3x - tacog)
3. ↑ Klitzing, (x4o3x3o3o3x - topag)
4. ↑ Klitzing, (x4x3x3o3o3x - togrix)
5. ↑ Klitzing, (x4x3o3x3o3x - tocrag)
6. ↑ Klitzing, (x4o3x3x3o3x - tiprixog)
7. ↑ Klitzing, (x4x3x3o3x3x - tagpox)
8. ↑ Klitzing, (x4x3o3o3x3x - tactaxog)
9. ↑ Klitzing, (x4x3x3o3x3x - tocagrax)
10. ↑ Klitzing, (x4x3x3x3x3x - gotaxog)

## 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. "6D uniform polytopes (polypeta)". https://bendwavy.org/klitzing/dimensions/polypeta.htm. x4o3o3o3o3x - stoxog, x4x3o3o3o3x - tacog, x4o3x3o3o3x - topag, x4x3x3o3o3x - togrix, x4x3o3x3o3x - tocrag, x4o3x3x3o3x - tiprixog, x4x3x3o3x3x - tagpox, x4x3o3o3x3x - tactaxog, x4x3x3o3x3x - tocagrax, x4x3x3x3x3x - gotaxog

## External links

* Glossary for hyperspace, George Olshevsky.
* 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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