I presume you mean why you can't flip a rectangular solid (a x b x c with a < b < c) around its middle axis. Ignoring both gravity and air resistance. Well, if you look at the flow on the phase space (a scalene ellipsoid) that shows how physics evolves, it has one pair of saddle points (at +-the middle axis) and two pair of centers (at +- the other two axes). If you flip the rectangular solid around its smallest or largest axis, any small perturbation will just shift it to a nearby closed orbit, so it will stay close to rotating around that axis. But if you flip it about the middle axis, the local saddle structure has all its nearby (but unequal) orbits going far away from rotation about that axis. I was not able to access Gene's writeup since I got a 404 error from the URL below, so can't compare what he wrote to the above. Gene? --Dan rwg wrote: << I wrote: << 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?
. . . 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?