The dodecahedron and the icosahedron are dual to each other, and can be combined to make this well-known compound.
In hyperspace, the analog to the dodecahedron is the hyperdodecahedron, also known as the 120-cell, as well as the hecatonicosachoron. Its dual is the 600-cell, or hexacosichoron, made of 600 tetrahedral cells. The image at the top is the compound of these two polychora, rotating in hyperspace.
These images were made using Stella 4d, available at http://www.software3d.com/Stella.php.