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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0<sub>6,1</sub> for its branching Coxeter-Dynkin diagram, shown as . Acronym: rene (Jonathan Bowers)
The rectified 8-simplex is the vertex figure of the 9-demicube, and the edge figure of the uniform 2<sub>61</sub> honeycomb.
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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0<sub>5,2</sub> for its branching Coxeter-Dynkin diagram, shown as . Acronym: brene (Jonathan Bowers)
The birectified 8-simplex is the vertex figure of the 1<sub>52</sub> honeycomb.
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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0<sub>4,3</sub> for its branching Coxeter-Dynkin diagram, shown as . Acronym: trene (Jonathan Bowers)
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.
The three presented polytopes are in the family of 135 uniform 8-polytopes with A<sub>8</sub> symmetry.