I think you need to distinguish between no stable position vs. no stable face. The perpetual motion argument says there must be a stable position, but it does not say that the position is one with a face flush against the surface the polyhedron is resting on. Tom Dan Asimov writes:
Which brings up the old problem: Does there exist a polyhedron with no stable face on a tabletop? (I.e., for which the center-of-gravity's projection to the plane of any face lies outside that face.)
The standard argument for why no such polyhedron exists is that it would keep rolling forever, so be a perpetual motion machine.
Some time ago there was no known proof purely by geometry. Does anyone know if that's still the case?
--Dan
On 2012-09-20, at 9:32 PM, Brent wrote:
And whether the projection of the CG onto the plane of the face falls within the face. :-)
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