Rhombicosidodecahedron

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 mean.

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 sqare 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.

It has 20 regular triangular faces, 30 regular square faces, 12 regular pentagonal faces, 60 vertices and 120 edges.

See dodecahedron, icosahedron, icosidodecahedron, truncated icosidodecahedron (great rhombicosidodecahedron).