# Runcic 5-cubes

| 5-cube | Runcic 5-cube = |
| 5-demicube = | Runcicantic 5-cube = |
| Orthogonal projections in B$_5$ Coxeter plane ||
In six-dimensional geometry, a runcic 5-cube or (runcic 5-demicube, 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.

## Runcic 5-cube

Runcic 5-cube :
| Type | uniform 5-polytope |
| Schläfli symbol | h$_3${4,3,3,3} |
| Coxeter-Dynkin diagram | |
| 4-faces | 42 |
| Cells | 360 |
| Faces | 880 |
| Edges | 720 |
| Vertices | 160 |
| Vertex figure | |
| Coxeter groups | D$_5$, [3$^{2,1,1}$] |
| Properties | convex |

### Alternate names

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

### 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 | B$_5$ |
| Graph | |
| Dihedral symmetry | [10/2] |
| Coxeter plane | D$_5$ | D$_4$ |
| Graph | | |
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D$_3$ | A$_3$ |
| 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$^+$,4,3$^{n-2}$] = [3,3$^{n-3,1}$] | [1$^+$,4,3$^2$] = [3,3$^{1,1}$] | [1$^+$,4,3$^3$] = [3,3$^{2,1}$] | [1$^+$,4,3$^4$] = [3,3$^{3,1}$] | [1$^+$,4,3$^5$] = [3,3$^{4,1}$] | [1$^+$,4,3$^6$] = [3,3$^{5,1}$] |
| Runcic figure | | | | | |
| Coxeter | = | = | = | = | = |
| Schläfli | h$_3${4,3$^2$} | h$_3${4,3$^3$} | h$_3${4,3$^4$} | h$_3${4,3$^5$} | h$_3${4,3$^6$} |

## Runcicantic 5-cube

Runcicantic 5-cube :
| Type | uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2}${3,3$^{2,1}$} h$_3${4,3$^3$} |
| Coxeter-Dynkin diagram | |
| 4-faces | 42 |
| Cells | 360 |
| Faces | 1040 |
| Edges | 1200 |
| Vertices | 480 |
| Vertex figure | |
| Coxeter groups | D$_5$, [3$^{2,1,1}$] |
| Properties | convex |

### Alternate names

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

### 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 | B$_5$ |
| Graph | |
| Dihedral symmetry | [10/2] |
| Coxeter plane | D$_5$ | D$_4$ |
| Graph | | |
| Dihedral symmetry | [8] | [6] |
| Coxeter plane | D$_3$ | A$_3$ |
| 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

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 5-polytopes that can be constructed from the D$_5$ 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} | h$_2${4,3,3,3} | h$_3${4,3,3,3} | h$_4${4,3,3,3} | h$_{2,3}${4,3,3,3} | h$_{2,4}${4,3,3,3} | h$_{3,4}${4,3,3,3} | h$_{2,3,4}${4,3,3,3} |