It was proved  that the combination of the icosahedron, the dodecahedron, and the icosidodecahedron has Dehn invariant 0, so by Sydler's theorem it is possible to dissect the combination to form a cube. In a related Demonstration (see Related Links), an example is given of a dissection of this combination to rhombic solids. In this Demonstration the connection of combinations is done using the larger diagonal of the golden rhombus.
 J. H. Conway, C. Radin, and L. Sadun, "On Angles Whose Squared Trigonometric Functions Are Rational," Discrete & Computational Geometry, 22(3), 1999 pp. 321–332.