This is a modified stellated icosahedron.

This is basically a stellated truncated icosahedron.

its a dodecahedron - 12 sides, but each side is 5 more sides, so it turns out to be a star with 60 sides, 12 points

Here is your plain, simple tetrahedon, dressed up with a couple of rounds of double stellations as well as two doo sabin remeshes.

One ofCoxeter's 59 stellations of the icosahedron.SeeCoxeter's book: e1, f1, g1, (e1 and g1 in bold!), plate XIII Number 39 inWenninger's polyhedro...

One ofCoxeter's 59 stellations of the icosahedron.SeeCoxeter's book: e1, f1, g1, (e1 and g1 in bold!), plate XIII Number 39 inWenninger's polyhedro...

A Kepler-Poinsot solid

Small stellated dodecahedron. A ball-and-stick version

A Kepler-Poinsot polyhedron