The Gömböc

Consider an object placed on a horizontal surface. It is in a state of equlibrium if two conditions are satisfied. First, the total force on the object must vanish,

secondly, the total torque around any point must vanish

The first condition says, that the objects CM does not accelerate, the second condition that it does not begin to rotate. As an example, a usual dice has 6 stable equilibria.

In 1995 it was conjecture by the mathematician V.I. Arnold that there exists a homogeneous, three-dimensional object with just two balance points: one stable and one unstable. The conjecture sounds simple, but was hard to prove. Only in 2006 a first proof appeared. Later it was demonstrated how one can actually build such an object, now called a Gömböc. (This appears to have been known by turtles for millions of years, by the way).

See also: Wikipedia entry