Polyhedra which can be perfectly split into self-similar pieces

Question

A cube can be perfectly split into smaller equally sized cubes. Similarly, a triangular prism can be perfectly split into smaller equally sized triangular prisms.

Is there a name for or list of the set of polyhedra which can be split into smaller self-similar pieces?

Answer 1

You can find some non-trivial examples on my page Polyhedron rep-tiles (rep-8) Actually, there are thousands of such polyhedrons even for 8 equal pieces.