Think about a solid uniform polyhedron. How many faces are stable? A face is stable if it does not tip over when we place the object on a table.

An appeal to the impossibility of perpetual motion should convince you that at least one face is always stable. But, is there always more than one stable face? If there is only one stable face, then this implies that no matter how the object is placed on a table it always rolls around onto its only stable face!

The two-tip tetrahedron

A two-tip tetrahedron

This tetrahedron has only two stable faces.

The uni-stable polyhedron

The uni-stable polyhedron

Conway and Guy proved that there are uni-stable polyhedra. That is to say, polyhedra which are stable on only one face.