# Dissection of Three Bilunabirotundas and a Dodecahedron into Rhombic Solids

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This Demonstration shows a dissection of three bilunabirotundas and a dodecahedron into rhombic solids.

Contributed by: Izidor Hafner (March 2011)

Open content licensed under CC BY-NC-SA

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The decomposition of a combination of solids to a combination of rhombohedra means that the former can be decomposed to the cube as well [1]. In [2], visual evidence is given that three bilunabirotundas (Johnson solid 91) and the dodecahedron can be decomposed into three rhombic solids and a rhombic hexecontahedron.

A bilunabirotunda inscribed in the "rhombic" solid has a deficit of four pentagonal cups, while triangular pyramids annihilate one another. On the other hand, the dodecahedron has a surplus of 12 pentagonal cups that annihilate the deficit of 12 cups of the bilunabirotundas. That such a dissection is possible also follows from the Demonstration referenced in Related Links.

[1] 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).
[2] I. Hafner, "Decomposition of Three Bilunabirotundas and Dodecahedron to Rhombic Solids," *Visual Mathematics*, 9(2), 2007 article 6.