Every convex polygon can be defined by a function in a polar coordinate system with origin at the center of gravity of an object with cross section . On horizontal surfaces, all objects start rolling in a way that sends the center of gravity lower such that decreases at the point of contact with the underlying surface. Equilibria occur if at this point. A balance point is stable at the minima of , where .

The number of vertices because in these cases, the center of gravity of the polygon and the ellipse are equal.