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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0<sub>4,1</sub> for its branching Coxeter-Dynkin diagram, shown as .
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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S. It is also called 0<sub>3,2</sub> for its branching Coxeter-Dynkin diagram, shown as .
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.
The rectified 6-simplex polytope is the vertex figure of the 7-demicube, and the edge figure of the uniform 2<sub>41</sub> 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 A<sub>6</sub> Coxeter plane orthographic projections.