This is the tetrahedrally-diminished dodecahedron, which can be formed from the Platonic dodecahedron by using faceting. In this case, I used Zometools. If you’d like to look into buying some Zome of your own, the website to visit is http://www.zometool.com.

Stellating polyhedra is one of the things Zome was invented for. The images below are two different rotations of a stellated tetrahedrally-diminished dodecahedron.