# Truncated 5-simplexes

| 5-simplex | Truncated 5-simplex | Bitruncated 5-simplex |
| Orthogonal projections in A$_5$ Coxeter plane |||
In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.

## Truncated 5-simplex

Truncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | t{3,3,3,3} | t{3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 12 | 6 {3,3,3} 6 t{3,3,3} |
| Cells | 45 | 30 {3,3} 15 t{3,3} |
| Faces | 80 | 60 {3} 20 {6} |
| Edges | 75 | 75 |
| Vertices | 30 | 30 |
| Vertex figure | ( )v{3,3} | ( )v{3,3} |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex | convex |
The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).

### Alternate names

* Truncated hexateron (Acronym: tix) (Jonathan Bowers)

### Coordinates

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 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] |

## Bitruncated 5-simplex

bitruncated 5-simplex :
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Schläfli symbol | 2t{3,3,3,3} | 2t{3,3,3,3} |
| Coxeter-Dynkin diagram | | |
| 4-faces | 12 | 6 2t{3,3,3} 6 t{3,3,3} |
| Cells | 60 | 45 {3,3} 15 t{3,3} |
| Faces | 140 | 80 {3} 60 {6} |
| Edges | 150 | 150 |
| Vertices | 60 | 60 |
| Vertex figure | { }v{3} | { }v{3} |
| Coxeter group | A$_5$ [3,3,3,3], order 720 | A$_5$ [3,3,3,3], order 720 |
| Properties | convex | convex |

### Alternate names

* Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)

### Coordinates

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

## Related uniform 5-polytopes

The truncated 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$_{0,1,2,3,4}$ |