It turns out that it is possible to use multiple icosidodecahedra as building blocks to build that polyhedron’s dual, the rhombic triacontahedron. This construction appears below, in four different coloring-schemes.
These rotating virtual models were constructed using Stella 4d, a program available at http://www.software3d.com/Stella.php.
RobertLovesPi.net 
I think this is now at the top of my favorites list. You can see through it! Wow!
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