Arnold's conjecture that convex, homogeneous bodies with less than 4 equilibria points (mono-monostatic bodies) exist was shown to be correct. In other words, it's possible to make a "come-back kid" {"ванька-встанька"} from a block of homogeneous material. So no matter how you drop it, it will straighten itself out and get on all four paws :-). Here is a page with a neat video demonstrating how "GÖMBÖC" rolls itself into equilibrium. And this Wikipedia article does a much better job of concisely explaining what this thing is (and what it does).
This picture is taken from a full paper: P. L. Varkonyi, G. Domokos Mathematical Intelligencer 2006, 28(4), 1.P.S. Why less than four equilibria points, not just one? Because both stable and unstable equilibria count, hence gomboc has two equilibria points, not one.
5 comments:
Still, why "less than four equilibria points" and not "just two"?
I wonder if you just "stumbled upon" this site or if you are an avid reader of Mathematical Intelligencer...
Good job by Arnold, BTW (and by the Hungarians too, of course)!
I think having four equilibria points is a fairly common situation, but less than that is tricky. And it's a "stumbled up" kind of thing (I'm not such a fan of Math. Intelligencer ;-)).
Oh, and one more thing: for 2D body you can prove that it cannot have less than four equilibria points. Arnold came up with his conjecture that for 3D body it is not true. I think somebody proved for 6(or 7)D body, and than Hungarians showed the existence of such body for 3D case.
Where can I buy one?
No idea :-). Although it should make a good toy, it's surprising nobody makes these things. Here is a good (and free) idea for a start-up!
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