# Decomposition of three bilunabirotundas and dodecahedron to rhombic solids

Izidor Hafner

Faculty of Electrical Engineering, University of Ljubljana

Trzaska 25, 1000 Ljubljana, Slovenia

e-mail: izidor.hafner@fe.uni-lj.si

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].

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