Steric 5-cubes

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* 5-cube

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* Steric 5-cube
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* Stericantic 5-cube
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* Half 5-cube
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* Steriruncic 5-cube
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* Steriruncicantic 5-cube
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Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a steric 5-cube or (steric 5-demicube or sterihalf 5-cube) is a convex uniform 5-polytope. There are unique 4 steric forms of the 5-cube. Steric 5-cubes have half the vertices of stericated 5-cubes.

## Contents

* 1 Steric 5-cube
* 1.1 Alternate names
* 1.2 Cartesian coordinates
* 1.3 Images
* 1.4 Related polytopes
* 2 Stericantic 5-cube
* 2.1 Alternate names
* 2.2 Cartesian coordinates
* 2.3 Images
* 3 Steriruncic 5-cube
* 3.1 Alternate names
* 3.2 Cartesian coordinates
* 3.3 Images
* 4 Steriruncicantic 5-cube
* 4.1 Alternate names
* 4.2 Cartesian coordinates
* 4.3 Images
* 5 Related polytopes
* 6 References
* 7 Further reading
* 8 External links

## Steric 5-cube

Steric 5-cube

Type | uniform polyteron
Schläfli symbol |

* t0,3{3,32,1}
* h4{4,3,3,3

Coxeter-Dynkin diagram |

*   *

4-faces | 82
Cells | 480
Faces | 720
Edges | 400
Vertices | 80
Vertex figure | {3,3}-t1{3,3} antiprism
Coxeter groups | D5, [32,1,1]
Properties | convex

### Alternate names

* Steric penteract, runcinated demipenteract
* Small prismated hemipenteract (siphin) (Jonathan Bowers)[1]: (x3o3o *b3o3x - siphin)

### Cartesian coordinates

The Cartesian coordinates for the 80 vertices of a steric 5-cube centered at the origin are the permutations of

(±1,±1,±1,±1,±3)

with an odd number of plus signs.

### Images

orthographic projections Coxeter plane | B5

|

Graph  |
Dihedral symmetry | [10/2]
Coxeter plane  | D5 | D4
Graph  |  |
Dihedral symmetry  | [8]  | [6]
Coxeter plane  | D3 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

### Related polytopes

Template:Steric cube table

## Stericantic 5-cube

Stericantic 5-cube

Type | uniform polyteron
Schläfli symbol |

* t0,1,3{3,32,1}
* h2,4{4,3,3,3

Coxeter-Dynkin diagram |

*   *

4-faces | 82
Cells | 720
Faces | 1840
Edges | 1680
Vertices | 480
Vertex figure |
Coxeter groups | D5, [32,1,1]
Properties | convex

### Alternate names

* Prismatotruncated hemipenteract (pithin) (Jonathan Bowers)[1]: (x3x3o *b3o3x - pithin)

### Cartesian coordinates

The Cartesian coordinates for the 480 vertices of a stericantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±3,±5)

with an odd number of plus signs.

### Images

orthographic projections Coxeter plane | B5

|

Graph  |
Dihedral symmetry | [10/2]
Coxeter plane  | D5 | D4
Graph  |  |
Dihedral symmetry  | [8]  | [6]
Coxeter plane  | D3 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Steriruncic 5-cube

Steriruncic 5-cube

Type | uniform polyteron
Schläfli symbol |

* t0,2,3{3,32,1}
* h3,4{4,3,3,3

Coxeter-Dynkin diagram |

*   *

4-faces | 82
Cells | 560
Faces | 1280
Edges | 1120
Vertices | 320
Vertex figure |
Coxeter groups | D5, [32,1,1]
Properties | convex

### Alternate names

* Prismatorhombated hemipenteract (pirhin) (Jonathan Bowers)[1]: (x3o3o *b3x3x - pirhin)

### Cartesian coordinates

The Cartesian coordinates for the 320 vertices of a steriruncic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±1,±3,±5)

with an odd number of plus signs.

### Images

orthographic projections Coxeter plane | B5

|

Graph  |
Dihedral symmetry | [10/2]
Coxeter plane  | D5 | D4
Graph  |  |
Dihedral symmetry  | [8]  | [6]
Coxeter plane  | D3 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Steriruncicantic 5-cube

Steriruncicantic 5-cube

Type | uniform polyteron
Schläfli symbol |

* t0,1,2,3{3,32,1}
* h2,3,4{4,3,3,3

Coxeter-Dynkin diagram |

*   *

4-faces | 82
Cells | 720
Faces | 2080
Edges | 2400
Vertices | 960
Vertex figure |
Coxeter groups | D5, [32,1,1]
Properties | convex

### Alternate names

* Great prismated hemipenteract (giphin) (Jonathan Bowers)[1]: (x3x3o *b3x3x - giphin)

### Cartesian coordinates

The Cartesian coordinates for the 960 vertices of a steriruncicantic 5-cube centered at the origin are coordinate permutations:

(±1,±1,±3,±5,±7)

with an odd number of plus signs.

### Images

orthographic projections Coxeter plane | B5

|

Graph  |
Dihedral symmetry | [10/2]
Coxeter plane  | D5 | D4
Graph  |  |
Dihedral symmetry  | [8]  | [6]
Coxeter plane  | D3 | A3
Graph  |  |
Dihedral symmetry  | [4]  | [4]

## Related polytopes

This polytope is based on the 5-demicube, a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform polytera (uniform 5-polytope) that can be constructed from the D5 symmetry of the 5-demicube, of which are unique to this family, and 15 are shared within the 5-cube family.

D5 polytopes

h{4,3,3,3} |
h2{4,3,3,3} |
h3{4,3,3,3} |
h4{4,3,3,3} |
h2,3{4,3,3,3} |
h2,4{4,3,3,3} |
h3,4{4,3,3,3} |
h2,3,4{4,3,3,3}

## References

1. ↑ 1.0 1.1 1.2 1.3 Klitzing, Richard. "5D uniform polytopes (polytera) with acronyms". https://bendwavy.org/klitzing/dimensions/polytera.htm.

## Further reading

* Coxeter, H. S. M. (1973). Regular Polytopes (3rd ed.). New York City: Dover. https://books.google.com/books?id=iWvXsVInpgMC. Retrieved 2022-05-19.
* Coxeter, H. S. M. (1995-05-17) (in en-CA). Kaleidoscopes: Selected Writings of H.S.M. Coxeter. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons. ISBN 978-0-471-01003-6. OCLC 632987525. http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html. Retrieved 2022-05-19.
* Coxeter, H. S. M. (1940-12-01). "Regular and Semi Regular Polytopes I" (in en-CA). Mathematische Zeitschrift (Springer Nature) 46: 380–407. doi:10.1007/BF01181449. ISSN 1432-1823. https://link.springer.com/article/10.1007/BF01181449. Retrieved 2022-05-19.
* Coxeter, H. S. M. (1985-12-01). "Regular and Semi-Regular Polytopes II" (in en-CA). Mathematische Zeitschrift (Springer Nature) 188 (4): 559–591. doi:10.1007/BF01161657. ISSN 1432-1823. https://link.springer.com/article/10.1007/BF01161657. Retrieved 2022-05-19.
* Coxeter, H. S. M. (1988-03-01). "Regular and Semi-Regular Polytopes III" (in en-CA). Mathematische Zeitschrift (Springer Nature) 200 (1): 3–45. doi:10.1007/BF01161745. ISSN 1432-1823. https://link.springer.com/article/10.1007/BF01161745. Retrieved 2022-05-19.
* Johnson, Norman W. (1991). Uniform Polytopes (Unfinished manuscript thesis). `{{cite thesis}}`: Cite has empty unknown parameter: `|1=` (help)
* Johnson, Norman W. (1966). The Theory of Uniform Polytopes and Honeycombs (PhD thesis). University of Toronto. Retrieved 2022-05-19.

## External links

* Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html.
* Polytopes of Various Dimensions
* Multi-dimensional Glossary

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* 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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