# Rectified 7-simplexes

| 7-simplex | Rectified 7-simplex |
| Birectified 7-simplex | Trirectified 7-simplex |
| Orthogonal projections in A$_7$ Coxeter plane ||
In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.
There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the 7-simplex. Vertices of the birectified 7-simplex are located in the triangular face centers of the 7-simplex. Vertices of the trirectified 7-simplex are located in the tetrahedral cell centers of the 7-simplex.

## Rectified 7-simplex

Rectified 7-simplex :
| Type | uniform 7-polytope |
| Coxeter symbol | 0$_{51}$ |
| Schläfli symbol | r{3$^6$} = {3$^{5,1}$} or $\left\{{\begin{array}{l}3,3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter diagrams | Or |
| 6-faces | 16 |
| 5-faces | 84 |
| 4-faces | 224 |
| Cells | 350 |
| Faces | 336 |
| Edges | 168 |
| Vertices | 28 |
| Vertex figure | 6-simplex prism |
| Petrie polygon | Octagon |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |
The rectified 7-simplex is the edge figure of the 2$_{51}$ honeycomb. It is called 0$_{5,1}$ for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^1$ $_7$.

### Alternate names

* Rectified octaexon (Acronym: roc) (Jonathan Bowers)

### Coordinates

The vertices of the rectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Birectified 7-simplex

Birectified 7-simplex :
| Type | uniform 7-polytope |
| Coxeter symbol | 0$_{42}$ |
| Schläfli symbol | 2r{3,3,3,3,3,3} = {3$^{4,2}$} or $\left\{{\begin{array}{l}3,3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter diagrams | Or |
| 6-faces | 16: 8 r{3$^5$} 8 2r{3$^5$} |
| 5-faces | 112: 28 {3$^4$} 56 r{3$^4$} 28 2r{3$^4$} |
| 4-faces | 392: 168 {3$^3$} (56+168) r{3$^3$} |
| Cells | 770: (420+70) {3,3} 280 {3,4} |
| Faces | 840: (280+560) {3} |
| Edges | 420 |
| Vertices | 56 |
| Vertex figure | {3}x{3,3,3} |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^2$ $_7$. It is also called 0$_{4,2}$ for its branching Coxeter-Dynkin diagram, shown as .

### Alternate names

* Birectified octaexon (Acronym: broc) (Jonathan Bowers)

### Coordinates

The vertices of the birectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 8-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Trirectified 7-simplex

Trirectified 7-simplex :
| Type | uniform 7-polytope |
| Coxeter symbol | 0$_{33}$ |
| Schläfli symbol | 3r{3$^6$} = {3$^{3,3}$} or $\left\{{\begin{array}{l}3,3,3\\3,3,3\end{array}}\right\}$ |
| Coxeter diagrams | Or |
| 6-faces | 16 2r{3$^5$} |
| 5-faces | 112 |
| 4-faces | 448 |
| Cells | 980 |
| Faces | 1120 |
| Edges | 560 |
| Vertices | 70 |
| Vertex figure | {3,3}x{3,3} |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex, isotopic |
The trirectified 7-simplex is the intersection of two regular 7-simplexes in dual configuration.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^3$ $_7$.
This polytope is the vertex figure of the 1$_{33}$ honeycomb. It is called 0$_{3,3}$ for its branching Coxeter-Dynkin diagram, shown as .

### Alternate names

* Hexadecaexon (Acronym: he) (Jonathan Bowers)

### Coordinates

The vertices of the trirectified 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 8-orthoplex.
The trirectified 7-simplex is the intersection of two regular 7-simplices in dual configuration. This characterization yields simple coordinates for the vertices of a trirectified 7-simplex in 8-space: the 70 distinct permutations of (1,1,1,1,−1,−1,−1,-1).

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

### 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 polytopes

These polytopes are three of 71 uniform 7-polytopes with A$_7$ symmetry.
A7 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_3$ | 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$_{2,4}$ | t$_{0,5}$ | t$_{1,5}$ | t$_{0,6}$ |
| 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$_{1,3,4}$ | t$_{2,3,4}$ | t$_{0,1,5}$ | t$_{0,2,5}$ | t$_{1,2,5}$ | t$_{0,3,5}$ | t$_{1,3,5}$ | t$_{0,4,5}$ |
| t$_{0,1,6}$ | t$_{0,2,6}$ | t$_{0,3,6}$ | 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$_{1,2,3,5}$ | t$_{0,1,4,5}$ | t$_{0,2,4,5}$ | t$_{1,2,4,5}$ | t$_{0,3,4,5}$ |
| t$_{0,1,2,6}$ | t$_{0,1,3,6}$ | t$_{0,2,3,6}$ | t$_{0,1,4,6}$ | t$_{0,2,4,6}$ | t$_{0,1,5,6}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ |
| t$_{0,1,2,4,5}$ | t$_{0,1,3,4,5}$ | t$_{0,2,3,4,5}$ | t$_{1,2,3,4,5}$ | t$_{0,1,2,3,6}$ | t$_{0,1,2,4,6}$ | t$_{0,1,3,4,6}$ | t$_{0,2,3,4,6}$ |
| t$_{0,1,2,5,6}$ | t$_{0,1,3,5,6}$ | t$_{0,1,2,3,4,5}$ | t$_{0,1,2,3,4,6}$ | t$_{0,1,2,3,5,6}$ | t$_{0,1,2,4,5,6}$ | t$_{0,1,2,3,4,5,6}$ |