The Rhombic Triacontahedron
The rhombic triacontahedron is a 3D Catalan solid bounded by 30 rhombuses, 60 edges, and 32 vertices. It is the dual of the icosidodecahedron.
The faces are transitive, each a rhombus with a diagonal ratio of 1 : φ, where φ=(1+√5)/2 is the Golden Ratio. Unlike most of the other Catalan solids, the edges are also transitive, due to the icosidodecahedron being quasiregular.
The 32 vertices are of two kinds: 20 vertices where 5 edges meet, corresponding to an inscribed icosahedron, and 12 vertices where 3 edges meet, corresponding to an inscribed dodecahedron.
The rhombic triacontahedron is one of the projection envelopes of the castellated rhombicosidodecahedral prism, a CRF polychoron that has bilunabirotunda cells. In this projection, each rhombus is the projection image of a bilunabirotunda, each order 3 vertex corresponds with a tetrahedron, and each order 5 vertex corresponds with a pentagonal prism.
Projections
The following are images of the rhombic triacontahedron from various viewpoints:
Projection  Description 

Projection centered on an order5 vertex. 

Centered on an order3 vertex. 

Centered on a face. 

Centered on an edge. 
Animation
Here's an animation of a rhombic triacontahedron rotating around the vertical axis:
Coordinates
The Cartesian coordinates for the rhombic triacontahedron are all permutations of coordinate and all changes of sign of:
 (φ, φ, φ)
along with even permutations of coordinate and all changes of sign of:
 (0, φ^{2}, 1)
 (0, φ, φ^{2})
where φ=(1+√5)/2 is the Golden Ratio.
These coordinates are obtained by inverting an icosidodecahedron of edge length 1/φ^{3}.