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.

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
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. 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
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. 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
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. 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.