Runcic 5-cubes

Short description: Concept in geometry

160px
5-cube
| 160px
Runcic 5-cube

|

160px
5-demicube
=  | 160px
Runcicantic 5-cube

Orthogonal projections in B5 Coxeter plane

In five-dimensional geometry, a runcic 5-cube, runcic 5-demicube or runcihalf 5-cube, is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

## Contents

* 1 Runcic 5-cube
* 1.1 Alternate names
* 1.2 Cartesian coordinates
* 1.3 Images
* 1.4 Related polytopes
* 2 Runcicantic 5-cube
* 2.1 Alternate names
* 2.2 Cartesian coordinates
* 2.3 Images
* 2.4 Related polytopes
* 3 Related polytopes
* 4 Notes
* 5 References
* 6 External links

## Runcic 5-cube

Runcic 5-cube

Type | uniform 5-polytope
Schläfli symbol | h3{4,3,3,3}
Coxeter-Dynkin diagram |

4-faces | 42
Cells | 360
Faces | 880
Edges | 720
Vertices | 160
Vertex figure |
Coxeter groups | D5, [32,1,1]
Properties | convex

### Alternate names

* Cantellated 5-demicube/demipenteract
* Small rhombated hemipenteract (sirhin) (Jonathan Bowers)[1]

### Cartesian coordinates

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

(±1,±1,±1,±3,±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

It has half the vertices of the runcinated 5-cube, as compared here in the B5 Coxeter plane projections:

Runcic 5-cube  |
Runcinated 5-cube

|

Runcic n-cubes

n | 4 | 5 | 6 | 7 | 8
[1+,4,3n-2]
= [3,3n-3,1]  | [1+,4,32]
= [3,31,1]  | [1+,4,33]
= [3,32,1]  | [1+,4,34]
= [3,33,1]  | [1+,4,35]
= [3,34,1]  | [1+,4,36]
= [3,35,1]
Runcic
figure  |  |  |  |  |
Coxeter |
=  |
=  |
=  |
=  |

Schläfli | h3{4,32} | h3{4,33} | h3{4,34} | h3{4,35} | h3{4,36}

## Runcicantic 5-cube

Runcicantic 5-cube

Type | uniform 5-polytope
Schläfli symbol | t0,1,2{3,32,1}
h2,3{4,33}
Coxeter-Dynkin diagram |
4-faces | 42
Cells | 360
Faces | 1040
Edges | 1200
Vertices | 480
Vertex figure |
Coxeter groups | D5, [32,1,1]
Properties | convex

### Alternate names

* Cantitruncated 5-demicube/demipenteract
* Great rhombated hemipenteract (girhin) (Jonathan Bowers)[2]

### Cartesian coordinates

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

(±1,±1,±3,±5,±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]

### Related polytopes

It has half the vertices of the runcicantellated 5-cube, as compared here in the B5 Coxeter plane projections:

Runcicantic 5-cube  |
Runcicantellated 5-cube

|

## Related polytopes

These polytopes are based on the 5-demicube, a member of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 uniform 5-polytopes 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}

## Notes

1. ↑ Klitzing, (x3o3o *b3x3o - sirhin).
2. ↑ Klitzing, (x3x3o *b3x3o - girhin).

## 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 Ivić 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. x3o3o *b3x3o - sirhin, x3x3o *b3x3o - girhin

## External links

* Weisstein, Eric W.. "Hypercube". http://mathworld.wolfram.com/Hypercube.html.
* 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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