This is the compound of five dodecahedra, a shape which is included in the built-in polyhedral library of Stella 4d, a program you can try for yourself, free, right here.
I wanted to see what I could make, starting from this compound. My first modification to it was to create its convex hull, which is shown below.
The next move was to use Stella‘s “Try to Make Faces Regular” function, which produced this:
Next, I augmented this figure’s thirty yellow rhombi with prisms.
I then created the convex hull of this augmented polyhedron.
Next, I used the “Try to Make Faces Regular” function again, producing a solid that looks, to me, like a hybrid of the rhombicosidodecahedron and the rhombic triacontahedron.
This polyhedron has yellow faces that are almost squares. Careful inspection reveals that they are actually isosceles trapezoids. The next thing I did was to augment each of these trapezoids with a tall prism.
The next step was to, again, create the convex hull.
That was the end of this polyhedral journey, but I am confident there will be others.