# Rectified 6-simplexes

| 6-simplex | Rectified 6-simplex | Birectified 6-simplex |
| Orthogonal projections in A$_6$ Coxeter plane |||
In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.
There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.

## Rectified 6-simplex

Rectified 6-simplex :
| Type | uniform polypeton |
| Schläfli symbol | t$_1${3$^5$} r{3$^5$} = {3$^{4,1}$} or $\left\{{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter diagrams | |
| Elements | f$_5$ = 14, f$_4$ = 63, C = 140, F = 175, E = 105, V = 21 (χ=0) |
| Coxeter group | A$_6$, [3$^5$], order 5040 |
| Bowers name and (acronym) | Rectified heptapeton (ril) |
| Vertex figure | 5-cell prism |
| Circumradius | 0.845154 |
| Properties | convex, isogonal |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^1$ $_6$. It is also called 0$_{4,1}$ for its branching Coxeter-Dynkin diagram, shown as .

### Alternate names

* Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

### Coordinates

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

## Birectified 6-simplex

Birectified 6-simplex :
| Type | uniform 6-polytope |
| Class | A6 polytope |
| Schläfli symbol | t$_2${3,3,3,3,3} 2r{3$^5$} = {3$^{3,2}$} or $\left\{{\begin{array}{l}3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter symbol | 0$_{32}$ |
| Coxeter diagrams | |
| 5-faces | 14 total: 7 t$_1${3,3,3,3} 7 t$_2${3,3,3,3} |
| 4-faces | 84 |
| Cells | 245 |
| Faces | 350 |
| Edges | 210 |
| Vertices | 35 |
| Vertex figure | {3}x{3,3} |
| Petrie polygon | Heptagon |
| Coxeter groups | A$_6$, [3,3,3,3,3] |
| Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^2$ $_6$. It is also called 0$_{3,2}$ for its branching Coxeter-Dynkin diagram, shown as .

### Alternate names

* Birectified heptapeton (Acronym: bril) (Jonathan Bowers)

### Coordinates

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

## Related uniform 6-polytopes

The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 2$_{41}$ polytope.
These polytopes are a part 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$_{0,1,2,3,4,5}$ |