Rhombic hexecontahedron

The rhombic hexecontahedron is a polyhedron with 60 golden rhombus faces.

Golden rhombohedra
The rhombic hexecontahedron can be constructed by joining 20 acute golden rhombohedra around a single point. This gives an easy way to determine the volume of rhombic hexecontahedron given the volume of the acute golden rhombohedron. The volume of a single acute golden rhombohedron of edge length 1 is $$\frac{5+\sqrt{5}}{10}$$ thus the volume of the rhombic hexecontahedron of edge length 1 is 20 times that making it $$10+2\sqrt{5}$$. The surface area can also be calculated by similar means.

Stellation
The rhombic hexecontahedron is a stellation of the rhombic triacontahedron.

Its stellation is given by the following stellation diagram: Yellow represents the original face, red represents the new faces, green represents space internal to the new polyhedron and dark blue represents space external to the new polyhedron. Since the faces form coplanar pairs two faces appear in the stellation diagram.

Related polytopes
The rhombic hexecontahedron is topologically equivalent to the deltoidal hexecontahedron which has kites for faces instead of rhombi.

Grünbaum's new rhombic hexecontahedron
Grünbaum constructs an additional rhombic hexecontahedron, which he calls the new rhombic hexecontahedron, by swapping the acute angles ($$\arctan(2)$$) of the rhombic faces with their obtuse angles ($$\pi-\arctan(2)$$). The result is another 60 sided polyhedron with golden rhombi for faces, however the faces of the new rhombic hexecontahedron intersect.

This stellation is given by the following stellation diagram: The left shows the new faces in red and the old face in yellow, and the right shows the interior of the polyhedron with an even-odd-filling in green and exterior in dark blue.