Quiz

Now, let's try this approach. For a polyhedron to rotate from one face to another, it must have potential energy, meaning the potential energy on some neighboring face must be lower (the center of gravity's position when rotating onto it must become lower). Since the number of faces is finite, the polyhedron can roll from face to face, and its potential energy can decrease. Eventually, it will reach a state where, in the position on all neighboring faces, its potential energy will be greater than in the position on the face it is resting on, and it will no longer be able to roll onto a neighboring face.

Added 2 minutes later
The center of gravity can only descend, and for a polyhedron, it can only descend in discrete steps when rolling, and it cannot descend indefinitely.