The Gomboc

Movie source: www.gomboc.eu, Sound source: Gnomusy: ballerina

The Hungarian Mathematicians Peter Varkonyi and Gabor Domokos have constructed a solid which is now called the Gomboc. It has the remarkable property that it has only one stable equilibrium position and one unstable equilibrium. Such bodies are called mono-monostatic. Nature seems have used those shapes for turtles.
The construction of the Gomboc is a problem for multivariable calculus: we can assume that its shape is given in spherical coordinates as rho = f(theta,phi) such that the north pole is the stable and the south pole is the unstable equilibrium point. Let g(theta,phi) be the potential energy of the body when it the surface point (theta,phi,f(theta,phi)) is the contact point with the surface. The function g should have no equilibria in [0,2pi] x (0,pi) and go to a maximum for phi to pi and to a minimum for phi to -phi. The problem is that even if we know a function g with this property, it must be able to be realized by a homogenous body. For example g(theta,phi) = phi is a function with this property. But there is no solid which has this potential energy function g. It is possible to realize it when putting weight into the bottom.

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