The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration.
The vertices of a birectification exist at the center of the faces of the original polytope(s).
This intersection is analogous to the 3D stellated octahedron, seen as a compound of two regular tetrahedra and intersected in a central octahedron, while that is a first rectification where vertices are at the center of the original edges.

It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cube's long diagonal orthogonally.
 In this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell.
 This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space: the 20 distinct permutations of (1,1,1,−1,−1,−1).

The vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of (0,0,0,1,1,1).
 This construction can be seen as facets of the birectified 6-orthoplex.
The birectified 5-simplex is the intersection of two regular 5-simplexes in dual configuration. The vertices of a birectification exist at the center of the faces of the original polytope(s). This intersection is analogous to the 3D stellated octahedron, seen as a compound of two regular tetrahedra and intersected in a central octahedron, while that is a first rectification where vertices are at the center of the original edges.
It is also the intersection of a 6-cube with the hyperplane that bisects the 6-cube's long diagonal orthogonally.  In this sense it is the 5-dimensional analog of the regular hexagon, octahedron, and bitruncated 5-cell.  This characterization yields simple coordinates for the vertices of a birectified 5-simplex in 6-space: the 20 distinct permutations of (1,1,1,−1,−1,−1).
The vertices of the birectified 5-simplex can also be positioned on a hyperplane in 6-space as permutations of (0,0,0,1,1,1).  This construction can be seen as facets of the birectified 6-orthoplex.
Rectified 5-simplexes