This is the truncated dodecahedron. It is one of the Archimedean solids.

This polyhedron has a long stellation-series, from which I selected several on aesthetic grounds. The figure immediately below is the truncated dodecahedron’s 16th stellation.

Here is the 21st stellation.

It’s easy to stellate polyhedra rapidly, and make many other changes to them, with Stella 4d: Polyhedron Navigator. You can try it for free at http://www.software3d.com/Stella.php.

The stellation shown immediately above is the 25th, and the one shown immediately below is the 27th.

Here is the next stellation: the 28th. Unlike the ones shown above, it is chiral.

This is the truncated dodecahedron’s 31st stellation.

This one is the 38th stellation.

This one is the 44th.

The last one shown here is called the truncated dodecahedron’s final stellation because, if it is stellated once more, it returns to the original truncated dodecahedron.