On equilibria of tetrahedra

Abstract: The monostatic property of polyhedra (i.e. the property of having just one stable or unstable static equilibrium point) has been in a focus of research ever since Conway and Guy \cite{Conway} published the proof of the existence of the first such object. In the same article they also proved that a homogeneous tetrahedron has at least two stable equilibrium points. By using polar duality, the same idea has been used \cite{balancing} to prove that a homogeneous tetrahedron has at least two unstable equilibria. Conway \cite{Dawson} also claimed that among inhomogeneous tetrahedra one can find monostable ones. Here we not only give a formal proof of this statement and show that monostatic tetrahedra have exactly 4 equilibria, but also demonstrate a startling new aspect of this problem: being monostatic implies certain \emph{visible} features of the shape and vice versa. Our results also imply that mono-monostatic tetrahedra (having just one stable and just one unstable equilibrium point) do not exist. In contrast, we show that for any other legal number of faces, edges, and vertices there is a mono-monostatic polyhedron with that face vector.