Cantellated 5-cubes

5-cube

Cantellated 5-cube

Bicantellated 5-cube

Cantellated 5-orthoplex

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5-orthoplex

Cantitruncated 5-cube

Bicantitruncated 5-cube

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

## Cantellated 5-cube

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Cantellated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | rr{4,3,3,3} =  r { 4 3 , 3 , 3 } {\displaystyle r\left\\{{\begin{array}{l}4\\\3,3,3\end{array}}\right\\}}
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 |
Coxeter group | B5 [4,3,3,3]
Properties  | convex, uniform

### Alternate names

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* Small rhombated penteract (Acronym: sirn) (Jonathan Bowers)

### Coordinates

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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 ) ) {\displaystyle \left(\pm 1,\ \pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)}

### Images

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

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

## Bicantellated 5-cube

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Bicantellated 5-cube

Type  | Uniform 5-polytope
Schläfli symbols | 2rr{4,3,3,3} =  r { 3 , 4 3 , 3 } {\displaystyle r\left\\{{\begin{array}{l}3,4\\\3,3\end{array}}\right\\}}
r{32,1,1} =  r { 3 , 3 3 3 } {\displaystyle r\left\\{{\begin{array}{l}3,3\\\3\\\3\end{array}}\right\\}}
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 |
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

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* Bicantellated penteract, bicantellated 5-orthoplex, or bicantellated pentacross
* Small birhombated penteractitriacontiditeron (Acronym: sibrant) (Jonathan Bowers)

### Coordinates

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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

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

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

## Cantitruncated 5-cube

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Cantitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | tr{4,3,3,3} =  t { 4 3 , 3 , 3 } {\displaystyle t\left\\{{\begin{array}{l}4\\\3,3,3\end{array}}\right\\}}
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 |
Coxeter group | B5 [4,3,3,3]
Properties  | convex, uniform

### Alternate names

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

### Coordinates

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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 + 2 2 , 1 + 2 2 , 1 + 2 2 ) {\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}}\right)}

### Images

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

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|

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

### Related polytopes

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It is third in a series of cantitruncated hypercubes:

Petrie polygon projections  |  |  |  |  |

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Truncated cuboctahedron | Cantitruncated tesseract | Cantitruncated 5-cube | Cantitruncated 6-cube | Cantitruncated 7-cube | Cantitruncated 8-cube
|  |  |  |  |

## Bicantitruncated 5-cube

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Bicantitruncated 5-cube

Type  | uniform 5-polytope
Schläfli symbol | 2tr{3,3,3,4} =  t { 3 , 4 3 , 3 } {\displaystyle t\left\\{{\begin{array}{l}3,4\\\3,3\end{array}}\right\\}}
t{32,1,1} =  t { 3 , 3 3 3 } {\displaystyle t\left\\{{\begin{array}{l}3,3\\\3\\\3\end{array}}\right\\}}
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 |
Coxeter groups | B5, [3,3,3,4]
D5, [32,1,1]
Properties  | convex, uniform

### Alternate names

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* Bicantitruncated penteract
* Bicantitruncated pentacross
* Great birhombated penteractitriacontiditeron (Acronym: gibrant) (Jonathan Bowers)

### Coordinates

[edit]

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

[edit]

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

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|
|

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

## Related polytopes

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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

## References

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* 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, 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. "5D uniform polytopes (polytera)". o3o3x3o4x - sirn, o3x3o3x4o - sibrant, o3o3x3x4x - girn, o3x3x3x4o - gibrant

## External links

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* 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 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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