Dissection of Three Rhombic Solids into an Icosahedron, a Dodecahedron, and an Icosidodecahedron
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This Demonstration gives a dissection of the union of a rhombic triacontahedron, a hexecontahedron, and a 120-hedron into the union of an icosahedron, a dodecahedron, and an icosidodecahedron.
Contributed by: Izidor Hafner (March 2011)
Open content licensed under CC BY-NC-SA
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.