What is the inverse of a cube?

Schatz discovered in 1929 that the Platonic solids could be inverted, and one of the products of the inversion of the cube was the oloid.

Link: Oloid at Mathematica mathworld

The mysteries of kinematic inversion: the old man and the oloid

( 30 min. sort of documentary follows, where Don Cruse talks about Paul Schatz, in three parts, more interesting than I expected! )