Mathieu equation
$$\ddot{x} + (a + bp(t))x = 0,\ \ p(t) \equiv p(t + 2\pi),$$
where $a$ and $b$ are real parameters is a ubiquitous system in mechanics; it arises as a linearization around periodic solutions in many Hamiltonian systems with two degrees of freedom, in quantum mechanics, in the study of complete integrability of the KdV equation and other settings.
This problem has been studied for nearly a century by van der Pol, Gelfand, Keller, Arnold and others, but the complete understanding of stability diagram is still lacking. This paper describes a new {\it geometrical} picture which uncovers a new property of the stability diagaram and gives a new insight into some results of Arnold, Keller and van der Pol.