The polyhedron above is a zonish icosahedron, with zones added to that Platonic solid based on its faces and vertices. Its faces are twenty equilateral triangles, thirty equilateral decagons, and sixty rhombi. After making it, I used faceting to truncate the vertices where sets of five rhombi met, creating the polyhedron below. It has twelve regular pentagons as faces, with the sixty rhombi of the polyhedron above turned into sixty isosceles triangles, along with the thirty decagons and twenty triangles from the first of these two polyhedra. This second one could be called either a faceted zonish icosahedron, or a truncated zonish icosahedron.
Both of these polyhedra were created using Stella 4d: Polyhedron Navigator, software you can try for free at http://www.software3d.com/Stella.php.