  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.
