If a zonish dodecahedron is created with zones based on the dodecahedron’s vertices, here is the result.
If the same thing is done with edges, this is the result — an edge-distorted version of the great rhombicosidodecahedron.
Another option is faces-only. Although I haven’t checked the bond-lengths, this one does have the general shape of the most-symmetrical 80-carbon-atom fullerene molecule. Also, this shape is sometimes called the “pseudo-truncated-icosahedron.”
The next zonish dodecahedron has had zones added based on the dodecahedron’s faces and edges, both.
Here’s the one for vertices and edges.
Here’s the one for faces and vertices.
Finally, the last of this set of seven has had zones added based on all three: faces, vertices, and edges.
All seven of these were made with Stella 4d, which is available at http://www.software3d.com/Stella.php.