Dissection of a Cube into Three Bilunabirotundas, a Dodecahedron, and a Smaller Cube
This Demonstration shows a dissection of a cube into three bilunabirotundas, a dodecahedron, and a smaller cube.
The existence of such a decomposition follows from  or from the Demonstration referenced in the Related Links and . This dissection as well as the dissection of a dodecahedron into eight parts (if original) could be made into a commercial puzzle.  V. G. Boltyanskii, Tretja Problema Hilberta, Moscow: Nauka, 1977. Translated by R. A. Silverman as Hilbert's Third Problem (New York: John Wiley & Sons, Inc., 1978).  I. Hafner, "Decomposition of Three Bilunabirotundas and Dodecahedron to Rhombic Solids," Visual Mathematics, 9(2), 2007 article 6.