# Rectified 8-simplexes

| 8-simplex | Rectified 8-simplex |
| Birectified 8-simplex | Trirectified 8-simplex |
| Orthogonal projections in A$_8$ Coxeter plane ||
In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.
There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified 8-simplex are located in the triangular face centers of the 8-simplex. Vertices of the trirectified 8-simplex are located in the tetrahedral cell centers of the 8-simplex.

## Rectified 8-simplex

Rectified 8-simplex :
| Type | uniform 8-polytope |
| Coxeter symbol | 0$_{61}$ |
| Schläfli symbol | t$_1${3$^7$} r{3$^7$} = {3$^{6,1}$} or $\left\{{\begin{array}{l}3,3,3,3,3,3\\3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 7-faces | 18 |
| 6-faces | 108 |
| 5-faces | 336 |
| 4-faces | 630 |
| Cells | 756 |
| Faces | 588 |
| Edges | 252 |
| Vertices | 36 |
| Vertex figure | 7-simplex prism,×{3,3,3,3,3} |
| Petrie polygon | enneagon |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^1$ $_8$. It is also called 0$_{6,1}$ for its branching Coxeter-Dynkin diagram, shown as .

### Coordinates

The Cartesian coordinates of the vertices of the rectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 9-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_8$ | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Birectified 8-simplex

Birectified 8-simplex :
| Type | uniform 8-polytope |
| Coxeter symbol | 0$_{52}$ |
| Schläfli symbol | t$_2${3$^7$} 2r{3$^7$} = {3$^{5,2}$} or $\left\{{\begin{array}{l}3,3,3,3,3\\3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 7-faces | 18 |
| 6-faces | 144 |
| 5-faces | 588 |
| 4-faces | 1386 |
| Cells | 2016 |
| Faces | 1764 |
| Edges | 756 |
| Vertices | 84 |
| Vertex figure | {3}×{3,3,3,3} |
| Coxeter group | A$_8$, [3$^7$], order 362880 |
| Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^2$ $_8$. It is also called 0$_{5,2}$ for its branching Coxeter-Dynkin diagram, shown as .
The birectified 8-simplex is the vertex figure of the 1$_{52}$ honeycomb.

### Coordinates

The Cartesian coordinates of the vertices of the birectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 9-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_8$ | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Trirectified 8-simplex

Trirectified 8-simplex :
| Type | uniform 8-polytope |
| Coxeter symbol | 0$_{43}$ |
| Schläfli symbol | t$_3${3$^7$} 3r{3$^7$} = {3$^{4,3}$} or $\left\{{\begin{array}{l}3,3,3,3\\3,3,3\end{array}}\right\}$ |
| Coxeter-Dynkin diagrams | or |
| 7-faces | 9 + 9 |
| 6-faces | 36 + 72 + 36 |
| 5-faces | 84 + 252 + 252 + 84 |
| 4-faces | 126 + 504 + 756 + 504 |
| Cells | 630 + 1260 + 1260 |
| Faces | 1260 + 1680 |
| Edges | 1260 |
| Vertices | 126 |
| Vertex figure | {3,3}×{3,3,3} |
| Petrie polygon | enneagon |
| Coxeter group | A$_7$, [3$^7$], order 362880 |
| Properties | convex |
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S$^3$ $_8$. It is also called 0$_{4,3}$ for its branching Coxeter-Dynkin diagram, shown as .

### Coordinates

The Cartesian coordinates of the vertices of the trirectified 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 9-orthoplex.

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_8$ | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | | |
| Dihedral symmetry | [9] | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Related polytopes

This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 2$_{61}$ honeycomb.
It is also one of 135 uniform 8-polytopes with A$_8$ symmetry.
A8 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_3$ | t$_{01}$ | t$_{02}$ | t$_{12}$ | t$_{03}$ | t$_{13}$ | t$_{23}$ | t$_{04}$ | t$_{14}$ | t$_{24}$ | t$_{34}$ | t$_{05}$ |
| t$_{15}$ | t$_{25}$ | t$_{06}$ | t$_{16}$ | t$_{07}$ | t$_{012}$ | t$_{013}$ | t$_{023}$ | t$_{123}$ | t$_{014}$ | t$_{024}$ | t$_{124}$ | t$_{034}$ | t$_{134}$ | t$_{234}$ |
| t$_{015}$ | t$_{025}$ | t$_{125}$ | t$_{035}$ | t$_{135}$ | t$_{235}$ | t$_{045}$ | t$_{145}$ | t$_{016}$ | t$_{026}$ | t$_{126}$ | t$_{036}$ | t$_{136}$ | t$_{046}$ | t$_{056}$ |
| t$_{017}$ | t$_{027}$ | t$_{037}$ | t$_{0123}$ | t$_{0124}$ | t$_{0134}$ | t$_{0234}$ | t$_{1234}$ | t$_{0125}$ | t$_{0135}$ | t$_{0235}$ | t$_{1235}$ | t$_{0145}$ | t$_{0245}$ | t$_{1245}$ |
| t$_{0345}$ | t$_{1345}$ | t$_{2345}$ | t$_{0126}$ | t$_{0136}$ | t$_{0236}$ | t$_{1236}$ | t$_{0146}$ | t$_{0246}$ | t$_{1246}$ | t$_{0346}$ | t$_{1346}$ | t$_{0156}$ | t$_{0256}$ | t$_{1256}$ |
| t$_{0356}$ | t$_{0456}$ | t$_{0127}$ | t$_{0137}$ | t$_{0237}$ | t$_{0147}$ | t$_{0247}$ | t$_{0347}$ | t$_{0157}$ | t$_{0257}$ | t$_{0167}$ | t$_{01234}$ | t$_{01235}$ | t$_{01245}$ | t$_{01345}$ |
| t$_{02345}$ | t$_{12345}$ | t$_{01236}$ | t$_{01246}$ | t$_{01346}$ | t$_{02346}$ | t$_{12346}$ | t$_{01256}$ | t$_{01356}$ | t$_{02356}$ | t$_{12356}$ | t$_{01456}$ | t$_{02456}$ | t$_{03456}$ | t$_{01237}$ |
| t$_{01247}$ | t$_{01347}$ | t$_{02347}$ | t$_{01257}$ | t$_{01357}$ | t$_{02357}$ | t$_{01457}$ | t$_{01267}$ | t$_{01367}$ | t$_{012345}$ | t$_{012346}$ | t$_{012356}$ | t$_{012456}$ | t$_{013456}$ | t$_{023456}$ |
| t$_{123456}$ | t$_{012347}$ | t$_{012357}$ | t$_{012457}$ | t$_{013457}$ | t$_{023457}$ | t$_{012367}$ | t$_{012467}$ | t$_{013467}$ | t$_{012567}$ | t$_{0123456}$ | t$_{0123457}$ | t$_{0123467}$ | t$_{0123567}$ | t$_{01234567}$ |