Rectified 8 simplexes

Rectified 8-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S1
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.

Birectified 8-simplex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S2
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 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 S3
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.

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.