In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
The rows and columns correspond to vertices, edges, faces, cells and 4-faces.
Each progressive uniform polytope is constructed from the previous as its vertex figure.
It has 20 vertices, 90 edges, 120 triangular faces, 60 cells (30 tetrahedral, and 30 octahedral).
Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors).
The vertices of a birectification exist at the center of the faces of the original polytope(s).
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).
Each progressive uniform polytope is constructed from the previous as its vertex figure.