Rectified 6-simplexes

In six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.

There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the 6-simplex. Vertices of the birectified 6-simplex are located in the triangular face centers of the 6-simplex.

Rectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 04,1 for its branching Coxeter-Dynkin diagram, shown as.

Alternate names

 * Rectified heptapeton (Acronym: ril) (Jonathan Bowers)

Coordinates
The vertices of the rectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,0,1,1). This construction is based on facets of the rectified 7-orthoplex.

Birectified 6-simplex
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 03,2 for its branching Coxeter-Dynkin diagram, shown as.

Alternate names

 * Birectified heptapeton (Acronym: bril) (Jonathan Bowers)

Coordinates
The vertices of the birectified 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,0,1,1,1). This construction is based on facets of the birectified 7-orthoplex.

Related uniform 6-polytopes
The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 241 polytope.

These polytopes are a part of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.