Sure, make it "up to diffeomorphism". The shapes you get fit in a larger class of piecewise-smooth closed surfaces, but all the interesting information is encod

Sure, make it "up to diffeomorphism". The shapes you get fit in a larger class of piecewise-smooth closed surfaces, but all the interesting information is encoded in the polyhedron. Note that the other shapes they're dealing with - e.g. prisms - also don't have planar faces (but are defined as having such on Wikipedia). The point is that mathematical definitions are quite malleable, and depend on the context. "Lop a vertex off a prism and deform a bit" is hardly a radical notion in itself. As for combining these shapes - the term for that is tiling or tessellation. You can tesselate the space with all kinds of shapes. The "faces match" is a property of objects a tessellation, not of any given object in itself.