I've been thinking about polyhedrons, when placed on a table on a certain face, will tip over and keep tipping over infinitely. I'm trying to prove mathematically that such a polyhedron doesn't exist.
Rules: the polyhedron does not need to be uniform nor convex, but at any point the mass there is non-negative. this means you can have holes and caves and points and fun stuff.
Right now I'm stuck because I'm not sure what determines if the polyhedron tips over. Either it's 1. The COM of the polyhedron, when projected onto the table, lies outside of the polygon face touching the table, and 2. The gravitational potential energy, mgh, is lowered by tipping over.
I understand with some basic geometry that 1 implies 2, but condition 2 does not necessarily imply that it will tip over.
How do I proceed to prove that such a perpetual polyhedron doesn't exist?
Edit: Thought about it some more and solved my question. 1 and 2 are equivalent in the real world because if the COM was above the base face, then tipping over would require that mgh first increase before finally adopting its final value after tipping over. If the COM is not above the face then there is no necessary activation energy. Therefore, the polyhedron tips over if and only if the energy strictly decreases, which happens if and only if the COM lies outside the face. After this process the potential energy decreases.
Now assume that the polyhedron spontaneously remains in perpetual motion. Then the energy is strictly decreasing; but the potential energy clearly has a minimum value, so it can't go on forever.