This image by Greg Egan shows 5 ways to inscribe a regular tetrahedron in a regular dodecahedron. The union of all these is a nonconvex polyhedron called the compound of 5 tetrahedra, first described by Edmund Hess in 1876.

Here Greg Egan has drawn two regular dodecahedra, in red and blue. They share 8 corners—and these are the corners of a cube, shown in green. Adrian Ocneanu calls these twin dodecahedra, and has proved some fascinating results about them.