Stability of Polygons Inscribed in an Ellipse
This Demonstration concerns polygons with sides inscribed in an ellipse with semimajor axis 1 and semiminor axis . If there exists a perpendicular line from a side that intersects the center of gravity, then the side is stable. The stable sides are shown in green.[more]
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.[less]
 G. Domokos, Z. Lángi, and T. Szabó, "On the Equilibria of Finely Discretized Curves and Surfaces," Monatshefte für Mathematik, 168(3–4), 2012 pp. 321–345.
 "The Gömböc." (Jun 25, 2015) www.gomboc.eu.