# Rectified 8-simplexes

(Redirected from Trirectified 8-simplex)
 Orthogonal projections in A8 Coxeter plane 8-simplex Rectified 8-simplex Birectified 8-simplex Trirectified 8-simplex

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 061
Schläfli symbol t1{37}
r{37} = {36,1}
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 A8, , order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
8
. It is also called 06,1 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.

## Birectified 8-simplex

Birectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 052
Schläfli symbol t2{37}
2r{37} = {35,2} 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 A8, , order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
8
. It is also called 05,2 for its branching Coxeter-Dynkin diagram, shown as            .

The birectified 8-simplex is the vertex figure of the 152 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.

## Trirectified 8-simplex

Trirectified 8-simplex
Type uniform 8-polytope
Coxeter symbol 043
Schläfli symbol t3{37}
3r{37} = {34,3} 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 A7, , order 362880
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S3
8
. It is also called 04,3 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.

## Related polytopes

This polytope is the vertex figure of the 9-demicube, and the edge figure of the uniform 261 honeycomb.

It is also one of 135 uniform 8-polytopes with A8 symmetry.