Monostatic body

A monostatic body is a shape that, when resting on a flat surface in uniform gravity, has exactly one stable equilibrium. A physically realized monostatic body will always right itself on a flat surface regardless of how it is placed. Most research has concentrated on monostatic shapes that are convex and made of a homogeneous material.

It has been shown that no such shape exists in 2D, but in 3D an example is a slightly bloated cylinder whose ends are cut at angles. Conway and Guy found such a polyhedron with 19 faces, which is the least known face count for such a polyhedron. It is known that no homogeneous monostatic tetrahedron exists, but inhomogeneous monostatic tetrahedra have been discovered, and homogeneous monostatic simplices exist in 8D and higher.

Monostatic bodies also possess unstable and saddle-type equilibria. Let S, U, and D be the number of stable, unstable, and saddle-type equilibria. The Poincaré-Hopf theorem implies that in 3D $$S + U - D = 2$$. A mono-monostatic body is a special class of 3D monostatic bodies with only one unstable equilibrium and no saddle-type equilibria (S = U = 1, D = 0). The existence of convex homogeneous mono-monostatic bodies was conjectured by Arnold in 2005, and confirmed in 2006 by Domokos and Várkonyi, who showed that a sphere can be slightly perturbed to produce such a shape, but the perturbations from a perfect sphere are so slight that physical realization is not practical.

However, in the same year, the same authors discovered a convex homogeneous mono-monostatic body that is not a close approximation of the sphere. This shape is known as the Gömböc, which is sold commercially as a toy and has received much attention in the popular press. Physical realizations of the Gömböc are however very sensitive to imperfections.

Kovács and Domokos found a 21-face convex polyhedron which is mono-monostatic if realized as equal point masses at each of its vertices.