# Cantellated 5-simplexes

| 5-simplex | Cantellated 5-simplex | Bicantellated 5-simplex |
| Birectified 5-simplex | Cantitruncated 5-simplex | Bicantitruncated 5-simplex |
| Orthogonal projections in A$_5$ Coxeter plane |||
In five-dimensional geometry, a cantellated 5-simplex is a convex uniform 5-polytope, being a cantellation of the regular 5-simplex.
There are unique 4 degrees of cantellation for the 5-simplex, including truncations.

## Cantellated 5-simplex

Cantellated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | rr{3,3,3,3} = $r\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ | rr{3,3,3,3} = $r\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 27 | 6 r{3,3,3} 6 rr{3,3,3} 15x{3,3} |
| Cells | 135 | 30 {3,3} 30 r{3,3} 15 rr{3,3} 60x{3} |
| Faces | 290 | 200 {3} 90 {4} |
| Edges | 240 | 240 |
| Vertices | 60 | 60 |
| Vertex figure | Tetrahedral prism | Tetrahedral prism |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex | convex |
The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).

### Alternate names

* Cantellated hexateron
* Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)

### Coordinates

The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract 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] |

## Bicantellated 5-simplex

Bicantellated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | 2rr{3,3,3,3} = $r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | 2rr{3,3,3,3} = $r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 32 | 12 t02{3,3,3} 20 {3}x{3} |
| Cells | 180 | 30 t1{3,3} 120x{3} 30 t02{3,3} |
| Faces | 420 | 240 {3} 180 {4} |
| Edges | 360 | 360 |
| Vertices | 90 | 90 |
| Vertex figure | | |
| Coxeter group | A$_5$×2, [[3,3,3,3]], order 1440 | A$_5$×2, [[3,3,3,3]], order 1440 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Bicantellated hexateron
* Small birhombated dodecateron (Acronym: sibrid) (Jonathan Bowers)

### Coordinates

The coordinates can be made in 6-space, as 90 permutations of:
    (0,0,1,1,2,2)
This construction exists as one of 64 orthant facets of the bicantellated 6-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |

## Cantitruncated 5-simplex

cantitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | tr{3,3,3,3} = $t\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ | tr{3,3,3,3} = $t\left\{{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 27 | 6 t012{3,3,3} 6 t{3,3,3} 15x{3,3} |
| Cells | 135 | 15 t012{3,3} 30 t{3,3} 60x{3} 30 {3,3} |
| Faces | 290 | 120 {3} 80 {6} 90x |
| Edges | 300 | 300 |
| Vertices | 120 | 120 |
| Vertex figure | Irr. 5-cell | Irr. 5-cell |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex | convex |

### Alternate names

* Cantitruncated hexateron
* Great rhombated hexateron (Acronym: garx) (Jonathan Bowers)

### Coordinates

The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 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] |

## Bicantitruncated 5-simplex

Bicantitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | 2tr{3,3,3,3} = $t\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | 2tr{3,3,3,3} = $t\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagram | or | or |
| 4-faces | 32 | 12 tr{3,3,3} 20 {3}x{3} |
| Cells | 180 | 30 t{3,3} 120x{3} 30 t{3,4} |
| Faces | 420 | 240 {3} 180 {4} |
| Edges | 450 | 450 |
| Vertices | 180 | 180 |
| Vertex figure | | |
| Coxeter group | A$_5$×2, [[3,3,3,3]], order 1440 | A$_5$×2, [[3,3,3,3]], order 1440 |
| Properties | convex, isogonal | convex, isogonal |

### Alternate names

* Bicantitruncated hexateron
* Great birhombated dodecateron(Acronym: gibrid) (Jonathan Bowers)

### Coordinates

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

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_5$ | A$_4$ |
| Graph | | |
| Dihedral symmetry | [6] | [[5]]=[10] |
| A$_k$ Coxeter plane | A$_3$ | A$_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [[3]]=[6] |

## Related uniform 5-polytopes

The cantellated 5-simplex is one 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$_{0,1,2,3,4}$ |