DanA> Yeah, I forgot. There's the Jacbobi sn, cn, dn, giving the solutions to the ideal rotations on the surface of an ellipsoid. (Showing why you can't flip a book in midair, halfway around its middle axis.) Hmm, but how would you assign probabilities to the initial conditions? --Dan On 2012-09-21, at 6:13 AM, Veit Elser wrote:

On Sep 21, 2012, at 12:32 AM, meekerdb <meekerdb@verizon.net <http://gosper.org/webmail/src/compose.php?send_to=meekerdb%40verizon.net>>> wrote:> >> On 9/20/2012 9:02 PM, Dan Asimov wrote:>>> Back in H.S. I wondered about this for any polyhedron, and guessed a face's probability of landing down is proportional to the solid angle it subtends from the center of gravity.>> >> And whether the projection of the CG onto the plane of the face falls within the face. :-)>> >> Brent> That's close to my physical model. It came up in connection with estimating the probability of a coin landing on its side. I had my students> design a cylindrical dice that has equal odds for heads, tails and "side". But because this was a physics course we did not make the> ad hoc assumption that the rotation group is sampled uniformly in a toss. Instead we used the coin tossing model where the cylinder axis rotates rapidly about a horizontal axis. Non-uniform sampling is especially valid for symmetric objects, where the three moments of inertia are equal and there is no free precession. So I don't find Michael Kleber's "fairness by symmetry" very compelling. I bet if one "rolled" a cubic dice more like the standard coin toss one could strongly bias the outcome (against the faces intersected by the rotation axis).

I've always wanted an ISS astronaut to demonstrate this with some sort of brick with a little dimple in the center of each face, held between pencil points, then spun (with breath?) and released. Unfortunately, the "breath" part reminds us that the experiment is tainted by aerodynamics. Dan, it sounds like you have a fairly simple geometric argument. Several years ago, Gene wrote up a fairly complicated physics argument <http://gosper.org/Rigid Body (1).pdf>. Does this mean you can simplify it? --rwg