# 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*.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S^{1}_{6}. It is also called **0 _{4,1}** 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^{2}_{6}. It is also called **0 _{3,2}** 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_{41} 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_{6} Coxeter plane orthographic projections.