# Rectified 8-simplexes

In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.

 Orthogonal projections in A8 Coxeter plane 8-simplex Rectified 8-simplex Birectified 8-simplex Trirectified 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
Typeuniform 8-polytope
Coxeter symbol061
Schläfli symbolt1{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-faces18
6-faces108
5-faces336
4-faces630
Cells756
Faces588
Edges252
Vertices36
Vertex figure7-simplex prism, {}×{3,3,3,3,3}
Petrie polygonenneagon
Coxeter groupA8, , order 362880
Propertiesconvex

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.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry    
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry   

## Birectified 8-simplex

Birectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol052
Schläfli symbolt2{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-faces18
6-faces144
5-faces588
4-faces1386
Cells2016
Faces1764
Edges756
Vertices84
Vertex figure{3}×{3,3,3,3}
Coxeter groupA8, , order 362880
Propertiesconvex

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.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry    
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry   

## Trirectified 8-simplex

Trirectified 8-simplex
Typeuniform 8-polytope
Coxeter symbol043
Schläfli symbolt3{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-faces9 + 9
6-faces36 + 72 + 36
5-faces84 + 252 + 252 + 84
4-faces126 + 504 + 756 + 504
Cells630 + 1260 + 1260
Faces1260 + 1680
Edges1260
Vertices126
Vertex figure{3,3}×{3,3,3}
Petrie polygonenneagon
Coxeter groupA7, , order 362880
Propertiesconvex

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.

### Images

orthographic projections
Ak Coxeter plane A8 A7 A6 A5
Graph
Dihedral symmetry    
Ak Coxeter plane A4 A3 A2
Graph
Dihedral symmetry   

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.

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