Cantellated 5-cubes

120px
5-cube
| 120px
Cantellated 5-cube
| 120px
Bicantellated 5-cube
| 120px
Cantellated 5-orthoplex

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120px
5-orthoplex
| 120px
Cantitruncated 5-cube
| 120px
Bicantitruncated 5-cube
| 120px
Cantitruncated 5-orthoplex

Orthogonal projections in B5 Coxeter plane

In six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.

There are 6 unique cantellation for the 5-cube, including truncations. Half of them are more easily constructed from the dual 5-orthoplex

## Contents

* 1 Cantellated 5-cube
* 1.1 Alternate names
* 1.2 Coordinates
* 1.3 Images
* 2 Bicantellated 5-cube
* 2.1 Alternate names
* 2.2 Coordinates
* 2.3 Images
* 3 Cantitruncated 5-cube
* 3.1 Alternate names
* 3.2 Coordinates
* 3.3 Images
* 3.4 Related polytopes
* 4 Bicantitruncated 5-cube
* 4.1 Alternate names
* 4.2 Coordinates
* 4.3 Images
* 5 Related polytopes
* 6 Notes
* 7 References
* 8 External links

## Cantellated 5-cube

Cantellated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | rr{4,3,3,3} = r{43,3,3}
Coxeter-Dynkin diagram  |  =
4-faces  | 122  | 10

Cells  | 680  | 40
320
160
160
Faces  | 1520  | 80
480
320
640
Edges  | 1280  | 320+960
Vertices  | 320
Vertex figure | 60px
Coxeter group | B5 [4,3,3,3]
Properties  | convex, uniform

### Alternate names

* Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)[1]

### Coordinates

The Cartesian coordinates of the vertices of a cantellated 5-cube having edge length 2 are all permutations of:

(±1, ±1, ±(1+2), ±(1+2), ±(1+2))

### Images

orthographic projections Coxeter plane | B5 | B4 / D5 | B3 / D4 / A2

|
|
|

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

## Bicantellated 5-cube

Bicantellated 5-cube

Type  | Uniform 5-polytope
Schläfli symbols | 2rr{4,3,3,3} = r{3,43,3}
r{32,1,1} = r{3,333}
Coxeter-Dynkin diagrams  |  =

4-faces  | 122  | 10

Cells  | 840  | 40
240
160
320

Faces  | 2160  | 240
320
960
320
320
Edges  | 1920  | 960+960
Vertices  | 480
Vertex figure | 60px
Coxeter groups | B5, [3,3,3,4]
D5, [32,1,1]
Properties  | convex, uniform

In five-dimensional geometry, a bicantellated 5-cube is a uniform 5-polytope.

### Alternate names

* Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
* Small birhombated penteractitriacontaditeron (Acronym: sibrant) (Jonathan Bowers)[2]

### Coordinates

The Cartesian coordinates of the vertices of a bicantellated 5-cube having edge length 2 are all permutations of:

(0,1,1,2,2)

### Images

orthographic projections 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-cube

Cantitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | tr{4,3,3,3} = t{43,3,3}
Coxeter-Dynkin
diagram  |  =
4-faces  | 122  | 10

Cells  | 680  | 40
320
160
160
Faces  | 1520  | 80
480
320
640
Edges  | 1600  | 320+320+960
Vertices  | 640
Vertex figure | 80px
Coxeter group | B5 [4,3,3,3]
Properties  | convex, uniform

### Alternate names

* Tricantitruncated 5-orthoplex / tricantitruncated pentacross
* Great rhombated penteract (girn) (Jonathan Bowers)[3]

### Coordinates

The Cartesian coordinates of the vertices of a cantitruncated 5-cube having an edge length of 2 are given by all permutations of coordinates and sign of:

(1, 1+2, 1+22, 1+22, 1+22)

### Images

orthographic projections 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

It is third in a series of cantitruncated hypercubes: Template:Cantitruncated hypercube polytopes

## Bicantitruncated 5-cube

Bicantitruncated 5-cube

Type  | uniform 5-polytope
Schläfli symbol | 2tr{3,3,3,4} = t{3,43,3}
t{32,1,1} = t{3,333}
Coxeter-Dynkin diagrams  |  =

4-faces  | 122  | 10

Cells  | 840  | 40
240
160
320

Faces  | 2160  | 240
320
960
320
320
Edges  | 2400  | 960+480+960
Vertices  | 960
Vertex figure | 60px
Coxeter groups | B5, [3,3,3,4]
D5, [32,1,1]
Properties  | convex, uniform

### Alternate names

* Bicantitruncated penteract
* Bicantitruncated pentacross
* Great birhombated penteractitriacontaditeron (Acronym: gibrant) (Jonathan Bowers)[4]

### Coordinates

Cartesian coordinates for the vertices of a bicantitruncated 5-cube, centered at the origin, are all sign and coordinate permutations of

(±3,±3,±2,±1,0)

### Images

orthographic projections 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.

B5 polytopes

β5 |
t1β5 |
t2γ5 |
t1γ5 |
γ5 |
t0,1β5 |
t0,2β5 |
t1,2β5

t0,3β5 |
t1,3γ5 |
t1,2γ5 |
t0,4γ5 |
t0,3γ5 |
t0,2γ5 |
t0,1γ5 |
t0,1,2β5

t0,1,3β5 |
t0,2,3β5 |
t1,2,3γ5 |
t0,1,4β5 |
t0,2,4γ5 |
t0,2,3γ5 |
t0,1,4γ5 |
t0,1,3γ5

t0,1,2γ5 |
t0,1,2,3β5 |
t0,1,2,4β5 |
t0,1,3,4γ5 |
t0,1,2,4γ5 |
t0,1,2,3γ5 |
t0,1,2,3,4γ5

## Notes

1. ↑ Klitzing, (o3o3x3o4x - sirn).
2. ↑ Klitzing, (o3x3o3x4o - sibrant).
3. ↑ Klitzing, (o3o3x3x4x - girn).
4. ↑ Klitzing, (o3x3x3x4o - gibrant).

## 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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
* (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. "5D uniform polytopes (polytera) with acronyms". https://bendwavy.org/klitzing/dimensions/polytera.htm. o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant

## External links

* Glossary for hyperspace, George Olshevsky.
* Polytopes of Various Dimensions, Jonathan Bowers
* Runcinated uniform polytera (spid), Jonathan Bowers
* 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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