he name rhombicosidodecahedron refers to the fact that the 30 square faces lie in the same planes as the 30 faces of the rhombic triacontahedron which is dual to the icosidodecahedron.
Canonical coordinates for the vertices of a rhombicosidodecahedron centered at the origin are (±1, ±1, ±τ3), (±τ3, ±1, ±1), (±1, ±τ3, ±1), and (±τ2, ±τ, ±2τ), (±2τ, ±τ2, ±τ), (±τ, ±2τ, ±τ2), and (±(2+τ), 0, ±τ2), (±τ2, ±(2+τ), 0), (0, ±τ2, ±(2+τ)), where τ = (1+√5)/2 is the golden ratio.
If you blow up an icosahedron by moving the faces away from the origin the right amount, without changing the orientation or size of the faces, and do the same to its dual dodecahedron, and patch the square holes in the result, you get a rhombicosadodecahedron. Therefore, it has the same number of triangles as an icosahedron and the same number of pentagons as a dodecahedron.
The Zometool kits for making geodesic domes and other polyhedra use slotted balls as connectors. The balls are "expanded" small rhombicosidodecahedra, with the squares replaced by rectangles. The expansion is chosen so that the resulting rectangles are golden rectangles.
