There are two dodecahedra I know of whose faces are identical rhombuses. One has rhombuses whose diagonals have a ratio of $\sqrt{2}$ -- this one is often simply called the "rhombic dodecahedron". The other has rhombuses whose diagonals have a ratio of $\phi$ (the golden ratio) -- this is the so-called "Bilinski dodecahedron".
Are there any other rhombuses which can be the faces of a dodecahedron like this, such that all the faces are congruent rhombuses? If not, why not?