To make the polyhedron shown above, I started with the great icosahedron, then applied the “morph duals by truncation” function in Stella 4d (a computer program available, with a free trial download, here). The faces of this polyhedron are twenty equilateral triangles, in red, and twelve regular star pentagons, colored yellow. [Later edit: the solid above is called the great icosidodecahedron, and is one of the uniform solids, although I did not realize this immediately.]

Next, I started with the polyhedron above, and applied the same dual-morph operation to it. The resulting polyhedron (shown below) has twenty equilateral triangles and twelve star pentagons (like its “parent”), plus thirty rectangles, which are shown in blue. Here’s the result, with 62 faces, total.