Stericated 5-cubes

5-cube

Stericated 5-cube

Steritruncated 5-cube

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Stericantellated 5-cube

Steritruncated 5-orthoplex

Stericantitruncated 5-cube

Steriruncitruncated 5-cube

Stericantitruncated 5-orthoplex

Omnitruncated 5-cube

Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-cube.

There are eight degrees of sterication for the 5-cube, including permutations of runcination, cantellation, and truncation. The simple stericated 5-cube is also called an expanded 5-cube, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-cube. The highest form, the steriruncicantitruncated 5-cube, is more simply called an omnitruncated 5-cube with all of the nodes ringed.

## Stericated 5-cube

[edit]

Stericated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | 2r2r{4,3,3,3}
Coxeter-Dynkin diagram |

4-faces  | 242
Cells  | 800
Faces  | 1040
Edges  | 640
Vertices  | 160
Vertex figure |
Coxeter group | B5 [4,3,3,3]
Properties  | convex

### Alternate names

[edit]

* Stericated penteract / Stericated 5-orthoplex / Stericated pentacross
* Expanded penteract / Expanded 5-orthoplex / Expanded pentacross
* Small cellated penteractitriacontaditeron (Acronym: scant) (Jonathan Bowers)[1]

### Coordinates

[edit]

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

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

### Images

[edit]

The stericated 5-cube is constructed by a sterication operation applied to the 5-cube.

### Dissections

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The stericated 5-cube can be dissected into two tesseractic cupolae and a runcinated tesseract between them. This dissection can be seen as analogous to the 4D runcinated tesseract being dissected into two cubic cupolae and a central rhombicuboctahedral prism between them, and also the 3D rhombicuboctahedron being dissected into two square cupolae with a central octagonal prism between them.

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]

## Steritruncated 5-cube

[edit]

Steritruncated 5-cube

Type | uniform 5-polytope
Schläfli symbol | t0,1,4{4,3,3,3}
Coxeter-Dynkin diagrams |
4-faces | 242
Cells | 1600
Faces | 2960
Edges | 2240
Vertices | 640
Vertex figure |
Coxeter groups | B5, [3,3,3,4]
Properties | convex

### Alternate names

[edit]

* Steritruncated penteract
* Celliprismated triacontaditeron (Acronym: capt) (Jonathan Bowers)[2]

### Construction and coordinates

[edit]

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

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

### 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]

## Stericantellated 5-cube

[edit]

Stericantellated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | t0,2,4{4,3,3,3}
Coxeter-Dynkin diagram |

4-faces | 242
Cells | 2080
Faces | 4720
Edges | 3840
Vertices | 960
Vertex figure |
Coxeter group | B5 [4,3,3,3]
Properties  | convex

### Alternate names

[edit]

* Stericantellated penteract
* Stericantellated 5-orthoplex, stericantellated pentacross
* Cellirhombated penteractitriacontiditeron (Acronym: carnit) (Jonathan Bowers)[3]

### Coordinates

[edit]

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

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

### 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]

## Stericantitruncated 5-cube

[edit]

Stericantitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | t0,1,2,4{4,3,3,3}
Coxeter-Dynkin
diagram |
4-faces | 242
Cells | 2400
Faces | 6000
Edges | 5760
Vertices | 1920
Vertex figure |
Coxeter group | B5 [4,3,3,3]
Properties  | convex, isogonal

### Alternate names

[edit]

* Stericantitruncated penteract
* Steriruncicantellated triacontiditeron / Biruncicantitruncated pentacross
* Celligreatorhombated penteract (cogrin) (Jonathan Bowers)[4]

### Coordinates

[edit]

The Cartesian coordinates of the vertices of a stericantitruncated 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 + 3 2 ) {\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}

### 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]

## Steriruncitruncated 5-cube

[edit]

Steriruncitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | 2t2r{4,3,3,3}
Coxeter-Dynkin
diagram |

4-faces | 242
Cells | 2160
Faces | 5760
Edges | 5760
Vertices | 1920
Vertex figure |
Coxeter group | B5 [4,3,3,3]
Properties  | convex, isogonal

### Alternate names

[edit]

* Steriruncitruncated penteract / Steriruncitruncated 5-orthoplex / Steriruncitruncated pentacross
* Celliprismatotruncated penteractitriacontiditeron (captint) (Jonathan Bowers)[5]

### Coordinates

[edit]

The Cartesian coordinates of the vertices of a steriruncitruncated penteract having an edge length of 2 are given by all permutations of coordinates and sign of:

( 1 , 1 + 2 , 1 + 1 2 , 1 + 2 2 , 1 + 3 2 ) {\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+1{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}

### 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]

## Steritruncated 5-orthoplex

[edit]

Steritruncated 5-orthoplex

Type | uniform 5-polytope
Schläfli symbol | t0,1,4{3,3,3,4}
Coxeter-Dynkin diagrams |
4-faces | 242
Cells | 1520
Faces | 2880
Edges | 2240
Vertices | 640
Vertex figure |
Coxeter group | B5, [3,3,3,4]
Properties | convex

### Alternate names

[edit]

* Steritruncated pentacross
* Celliprismated penteract (Acronym: cappin) (Jonathan Bowers)[6]

### Coordinates

[edit]

Cartesian coordinates for the vertices of a steritruncated 5-orthoplex, centered at the origin, are all permutations of

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

### Images

[edit]

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]

## Stericantitruncated 5-orthoplex

[edit]

Stericantitruncated 5-orthoplex

Type  | Uniform 5-polytope
Schläfli symbol | t0,2,3,4{4,3,3,3}
Coxeter-Dynkin
diagram |
4-faces | 242
Cells | 2320
Faces | 5920
Edges | 5760
Vertices | 1920
Vertex figure |
Coxeter group | B5 [4,3,3,3]
Properties  | convex, isogonal

### Alternate names

[edit]

* Stericantitruncated pentacross
* Celligreatorhombated triacontaditeron (cogart) (Jonathan Bowers)[7]

### Coordinates

[edit]

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

( 1 , 1 , 1 + 2 , 1 + 2 2 , 1 + 3 2 ) {\displaystyle \left(1,\ 1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}}\right)}

### Images

[edit]

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]

## Omnitruncated 5-cube

[edit]

Omnitruncated 5-cube

Type  | Uniform 5-polytope
Schläfli symbol | tr2r{4,3,3,3}
Coxeter-Dynkin
diagram |

4-faces | 242
Cells | 2640
Faces | 8160
Edges | 9600
Vertices | 3840
Vertex figure |
irr. {3,3,3}
Coxeter group | B5 [4,3,3,3]
Properties  | convex, isogonal

### Alternate names

[edit]

* Steriruncicantitruncated 5-cube (Full expansion of omnitruncation for 5-polytopes by Johnson)
* Omnitruncated penteract
* Omnitruncated triacontiditeron / omnitruncated pentacross
* Great cellated penteractitriacontiditeron (Jonathan Bowers)[8]

### Coordinates

[edit]

The Cartesian coordinates of the vertices of an omnitruncated 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 + 3 2 , 1 + 4 2 ) {\displaystyle \left(1,\ 1+{\sqrt {2}},\ 1+2{\sqrt {2}},\ 1+3{\sqrt {2}},\ 1+4{\sqrt {2}}\right)}

### 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]

### Full snub 5-cube

[edit]

The full snub 5-cube or omnisnub 5-cube, defined as an alternation of the omnitruncated 5-cube is not uniform, but it can be given Coxeter diagram  and symmetry [4,3,3,3]+, and constructed from 10 snub tesseracts, 32 snub 5-cells, 40 snub cubic antiprisms, 80 snub tetrahedral antiprisms, 80 3-4 duoantiprisms, and 1920 irregular 5-cells filling the gaps at the deleted vertices.

## Related polytopes

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This polytope is one 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

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1. ^ Klitzing, (x3o3o3o4x - scant)
2. ^ Klitzing, (x3o3o3x4x - capt)
3. ^ Klitzing, (x3o3x3o4x - carnit)
4. ^ Klitzing, (x3o3x3x4x - cogrin)
5. ^ Klitzing, (x3x3o3x4x - captint)
6. ^ Klitzing, (x3x3o3o4x - cappin)
7. ^ Klitzing, (x3x3x3o4x - cogart)
8. ^ Klitzing, (x3x3x3x4x - gacnet)

## 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, 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)". x3o3o3o4x - scan, x3o3o3x4x - capt, x3o3x3o4x - carnit, x3o3x3x4x - cogrin, x3x3o3x4x - captint, x3x3x3x4x - gacnet, x3x3x3o4x - cogart

## External links

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* Glossary for hyperspace, George Olshevsky.
* Polytopes of Various Dimensions, 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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