These two polyhedra are the icosidodecahedron (left), and its dual, the rhombic triacontahedron (right).
One nice thing about these two polyhedra is that one of them, the rhombic triacontahedron, can be used repeatedly, as a building-block, to build the other one, the icosidodecahedron. To get this started, I first constructed one edge of the icosidodecahedron, simply by lining up four rhombic triacontahedra.
Three of these lines of rhombic triacontahedra make one of the icosidodecahedron’s triangular faces.
Next, a pentagon is attached to this triangle.
Next, the pentagonal ring is surrounded by triangles.
More triangles and pentagons bring this process to the half-way point. If we were building a pentagonal rotunda (one of the Johnson solids), this would be the finished product.
Adding the other half completes the icosidodecahedron.
All of these images were created using Stella 4d: Polyhedron Navigator. You may try this program yourself, for free, at http://www.software3d.com/Stella.php. The last thing I did with Stella, for this post, was to put the finished model into rainbow color mode.