Truncated 6-simplexes
# Truncated 6-simplexes

| 6-simplex | Truncated 6-simplex |
| - | - |
| Bitruncated 6-simplex | Tritruncated 6-simplex |
| Orthogonal projections in $A_7$ Coxeter plane ||

In six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.

There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are located on the triangular faces of the 6-simplex. Vertices of the tritruncated 6-simplex are located inside the tetrahedral cells of the 6-simplex.

## Truncated 6-simplex

Truncated 6-simplex :

| Type | uniform 6-polytope |
| - | - |
| Class | A6 polytope |
| Schläfli symbol | t{3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | 14: 7 {3,3,3,3} 7 t{3,3,3,3} |
| 4-faces | 63: 42 {3,3,3} 21 t{3,3,3} |
| Cells | 140: 105 {3,3} 35 t{3,3} |
| Faces | 175: 140 {3} 35 {6} |
| Edges | 126 |
| Vertices | 42 |
| Vertex figure | ( )v{3,3,3} |
| Coxeter group | $A_6,$ $[3^5],$ order 5040 |
| Dual | ? |
| Properties | convex |

### Alternate names

* Truncated heptapeton (Acronym: til) (Jonathan Bowers)

### Coordinates

The vertices of the truncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,2). This construction is based on facets of the truncated 7-orthoplex.

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_6$ | $A_5$ | $A_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [7] | [6] | [5] |
| $A_k$ Coxeter plane | $A_3$ | $A_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Bitruncated 6-simplex

Bitruncated 6-simplex :

| Type | uniform 6-polytope |
| - | - |
| Class | A6 polytope |
| Schläfli symbol | 2t{3,3,3,3,3} |
| Coxeter-Dynkin diagram | |
| 5-faces | 14 |
| 4-faces | 84 |
| Cells | 245 |
| Faces | 385 |
| Edges | 315 |
| Vertices | 105 |
| Vertex figure | { }v{3,3} |
| Coxeter group | $A_6,$ $[3^5],$ order 5040 |
| Properties | convex |

### Alternate names

* Bitruncated heptapeton (Acronym: batal) (Jonathan Bowers)

### Coordinates

The vertices of the bitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,2,2). This construction is based on facets of the bitruncated 7-orthoplex.

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_6$ | $A_5$ | $A_4$ |
| - | - | - | - |
| Graph | | | |
| Dihedral symmetry | [7] | [6] | [5] |
| $A_k$ Coxeter plane | $A_3$ | $A_2$ |
| Graph | | |
| Dihedral symmetry | [4] | [3] |

## Tritruncated 6-simplex

Tritruncated 6-simplex :

| Type | uniform 6-polytope |
| - | - |
| Class | A6 polytope |
| Schläfli symbol | 3t{3,3,3,3,3} |
| Coxeter-Dynkin diagram | or |
| 5-faces | 14 2t{3,3,3,3} |
| 4-faces | 84 |
| Cells | 280 |
| Faces | 490 |
| Edges | 420 |
| Vertices | 140 |
| Vertex figure | {3}v{3} |
| Coxeter group | $A_6,$ $[[3^5]],$ order 10080 |
| Properties | convex, isotopic |

The tritruncated 6-simplex is an isotopic uniform polytope, with 14 identical bitruncated 5-simplex facets.

The tritruncated 6-simplex is the intersection of two 6-simplexes in dual configuration: and .

### Alternate names

* Tetradecapeton (as a 14-facetted 6-polytope) (Acronym: fe) (Jonathan Bowers)

### Coordinates

The vertices of the tritruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,2). This construction is based on facets of the bitruncated 7-orthoplex. Alternately it can be centered on the origin as permutations of (-1,-1,-1,0,1,1,1).

### Images

orthographic projections :

| $A_k$ Coxeter plane | $A_6$ | $A_5$ | $A_4$ |
| - | - | - | - |
| Graph | | | |
| Symmetry | $[[7]]^{(*)}=[14]$ | [6] | $[[5]]^{(*)}=[10]$ |
| $A_k$ Coxeter plane | $A_3$ | $A_2$ |
| Graph | | |
| Symmetry | [4] | $[[3]]^{(*)}=[6]$ |

    Note: (*) Symmetry doubled for $A_k$ graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.

### Related polytopes

Isotopic uniform truncated simplices :

| Dim. | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| - | - | - | - | - | - | - | - |
| Name Coxeter | Hexagon = t{3} = {6} | Octahedron = r{3,3} = ${3^{1,1}}$ = {3,4} $\left\{{\begin{array}{l}3\\3\end{array}}\right\}$ | Decachoron $2t{3^3}$ | Dodecateron $2r{3^4}$ = ${3^{2,2}}$ $\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}$ | Tetradecapeton $3t{3^5}$ | Hexadecaexon $3r{3^6}$ = ${3^{3,3}}$ $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ | Octadecazetton $4t{3^7}$ |
| Images | | | | | | | |
| Vertex figure | ( )∨( ) | { }×{ } | { }∨{ } | {3}×{3} | {3}∨{3} | {3,3}×{3,3} | {3,3}∨{3,3} |
| Facets | | {3} | t{3,3} | r{3,3,3} | 2t{3,3,3,3} | 2r{3,3,3,3,3} | 3t{3,3,3,3,3,3} |
| As intersecting dual simplexes | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ | ∩ |

## Related uniform 6-polytopes

The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in $A_6$ Coxeter plane orthographic projections.

A6 polytopes :

| $t_0$ | $t_1$ | $t_2$ | $t_{0,1}$ | $t_{0,2}$ | $t_{1,2}$ | $t_{0,3}$ | $t_{1,3}$ | $t_{2,3}$ |
| - | - | - | - | - | - | - | - | - |
| $t_{0,4}$ | $t_{1,4}$ | $t_{0,5}$ | $t_{0,1,2}$ | $t_{0,1,3}$ | $t_{0,2,3}$ | $t_{1,2,3}$ | $t_{0,1,4}$ | $t_{0,2,4}$ |
| $t_{1,2,4}$ | $t_{0,3,4}$ | $t_{0,1,5}$ | $t_{0,2,5}$ | $t_{0,1,2,3}$ | $t_{0,1,2,4}$ | $t_{0,1,3,4}$ | $t_{0,2,3,4}$ | $t_{1,2,3,4}$ |
| $t_{0,1,2,5}$ | $t_{0,1,3,5}$ | $t_{0,2,3,5}$ | $t_{0,1,4,5}$ | $t_{0,1,2,3,4}$ | $t_{0,1,2,3,5}$ | $t_{0,1,2,4,5}$ | $t_{0,1,2,3,4,5}$ |
