Polyhedra

While studying mesh-refinement schemes in the context of so-called
"multigrid" methods, I stumbled upon the following issue.  Can anyone
give me hints?

Start from a simplex  S0 in n-dimensional space.  Take the mid-edges,
take their convex hull, S1 say.  Take the barycenters of the
2-dimensional faces, take their convex hull, S2.  And so on.  It happens
that each polyhedron  Si  is a tessellation of pieces homothetic to  Si 
and other  Sj's, with the reduction factor 2.  Have these polyhedra been
studied in other contexts?  What is known about them?

Thanks in advance for any pointers.