# Runcinated 5-simplexes

| 5-simplex | Runcinated 5-simplex | Runcitruncated 5-simplex |
| Birectified 5-simplex | Runcicantellated 5-simplex | Runcicantitruncated 5-simplex |
| Orthogonal projections in A$_5$ Coxeter plane |||
In six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations (Runcination) of the regular 5-simplex.
There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations.

## Runcinated 5-simplex

Runcinated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,3}${3,3,3,3} | t$_{0,3}${3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | 6 t$_{0,3}${3,3,3} 20 {3}×{3} 15 { }×r{3,3} 6 r{3,3,3} |
| Cells | 255 | 45 {3,3} 180 { }×{3} 30 r{3,3} |
| Faces | 420 | 240 {3} 180 {4} |
| Edges | 270 | 270 |
| Vertices | 60 | 60 |
| Vertex figure | | |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex | convex |

### Alternate names

* Runcinated hexateron
* Small prismated hexateron (Acronym: spix) (Jonathan Bowers)

### Coordinates

The vertices of the runcinated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,1,1,1,2) or of (0,1,1,1,2,2), seen as facets of a runcinated 6-orthoplex, or a biruncinated 6-cube respectively.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Runcitruncated 5-simplex

Runcitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,3}${3,3,3,3} | t$_{0,1,3}${3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | 6 t$_{0,1,3}${3,3,3} 20 {3}×{6} 15 { }×r{3,3} 6 rr{3,3,3} |
| Cells | 315 | |
| Faces | 720 | |
| Edges | 630 | 630 |
| Vertices | 180 | 180 |
| Vertex figure | | |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Runcitruncated hexateron
* Prismatotruncated hexateron (Acronym: pattix) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 180 permutations of:
    (0,0,1,1,2,3)
This construction exists as one of 64 orthant facets of the runcitruncated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Runcicantellated 5-simplex

Runcicantellated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,2,3}${3,3,3,3} | t$_{0,2,3}${3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | |
| Cells | 255 | |
| Faces | 570 | |
| Edges | 540 | 540 |
| Vertices | 180 | 180 |
| Vertex figure | | |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Runcicantellated hexateron
* Biruncitruncated 5-simplex/hexateron
* Prismatorhombated hexateron (Acronym: pirx) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 180 permutations of:
    (0,0,1,2,2,3)
This construction exists as one of 64 orthant facets of the runcicantellated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Runcicantitruncated 5-simplex

Runcicantitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t$_{0,1,2,3}${3,3,3,3} | t$_{0,1,2,3}${3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 47 | 6 t$_{0,1,2,3}${3,3,3} 20 {3}×{6} 15×t{3,3} 6 tr{3,3,3} |
| Cells | 315 | 45 t$_{0,1,2}${3,3} 120 { }×{3} 120 { }×{6} 30 t{3,3} |
| Faces | 810 | 120 {3} 450 {4} 240 {6} |
| Edges | 900 | 900 |
| Vertices | 360 | 360 |
| Vertex figure | Irregular 5-cell | Irregular 5-cell |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Runcicantitruncated hexateron
* Great prismated hexateron (Acronym: gippix) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 360 permutations of:
    (0,0,1,2,3,4)
This construction exists as one of 64 orthant facets of the runcicantitruncated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [5] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Related uniform 5-polytopes

These polytopes are in a set of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A$_5$ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)
A5 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{0,4}$ | t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ |
| t$_{0,2,4}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,1,2,3,4}$ |