Quiz

I have a feeling that it doesn't exist; there should be a state of equilibrium, otherwise such a constantly rotating polyhedron would already be exploited as a perpetual motion machine. A regular sphere can essentially be represented as a polyhedron if the faces are made 2-3 atoms long, but I think we know that if the plane is truly flat, then gravity + friction will prevent the sphere from rolling anywhere. And the less the polyhedron resembles a sphere, the less likely it is to be able to roll anywhere at all – and essentially, that's what's needed: it shouldn't be in a state of equilibrium = it should roll. Without the application of an external force, the sphere will not roll. In some way, for example, fill the sphere (okay, the polyhedron) with some volatile substance like iodine and heat the horizontal surface. Then, iodine vapor from the bottom evaporates and settles on top – the center of mass shifts, the ball rolls, and the iodine crystals end up near the horizontal surface and re-sublimate... In short, here, heating is the external force. But I don't have any proper proof.