
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 monomonostatic. 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.

