If one starts with a single icosidodecahedron, and then augments its pentagonal faces with dodecahedra, and its trianguar faces with icosahedra, this is the result.

This figure has gaps in it where two pentagons and two triangles meet around a vertex. If one puts icosidodecahedra in those gaps, this is the resulting figure.

Next, once again, the pentagonal faces are augmented with dodecahedra, and the triangular faces with icosahedra.

These virtual polyhedral models were all built using Stella 4d, available at http://www.software3d.com/Stella.php.