Decomposition of a combination of solids to a combination of rhombohedra means that former can be decomposed to the cube as well [1]. In the paper we give visual evidence that 3 bilunabirotundas (Johnson solid 91) and the dodecahedron can be decomposed to three rhombic solids and rhombic hexecontahedron. The former rhombic solid can be constructed from triacontahedron, doubled in two direction, decreased of 4 oblate rhombohedra and two halves of rhombic dodecahedron of the second kind. Observe that the bilunabirotunda inscribed in the "rhombic" solid has a deficit of 4 pentagonal cups, while triangular pyramids annihilate one another. On the other hand the dodecahedron has surplus of 12 pentagonal cups, which annihilate the deficit of 12 cups of bilunabirotundas. Click the picture to get the animation [2].
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References
[1] V.G. Boltjanskii, Tretja problema Hilberta, Nauka, Moskva 1977.
[2] Martin Kraus' Live3D applet http://www.vis.uni-stuttgart.de/~kraus/index.html