Yes, the rhombic hexecontahedron is a polyhedron such that the projection of the centroid to any face lies outside that face: http://cp4space.wordpress.com/2012/09/01/activities-with-golden-rhombi/ Sincerely, Adam P. Goucher http://cp4space.wordpress.com/ ----- Original Message ----- From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Friday, September 21, 2012 7:39 PM Subject: Re: [math-fun] Polyhedral center-of-gravity problem
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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