Gömböc

A gömböc or gomboc (Hungarian: [ˈɡømbøts]) is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. Its existence was conjectured by Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by Hungarian scientists Gábor Domokos and Péter Várkonyi. The gömböc shape is not unique; it has countless varieties, most of which are very close to a sphere and all have very strict shape tolerance (about 0.1 mm per 100 mm).

The most famous solution has a sharpened top, as shown in the photo. Its shape helped to explain the body structure of some tortoises in relation to their ability to return to equilibrium position after being placed upside down. Copies of gömböc have been donated to institutions and museums, and the biggest one was presented at the World Expo 2010 in Shanghai, China.

In geometry, a body with a single stable resting position is called monostatic, and the term mono-monostatic has been coined to describe a body which additionally has only one unstable point of balance. (The previously known monostatic polyhedron does not qualify, as it has three unstable equilibria.) A sphere weighted so that its center of mass is shifted from the geometrical center is a mono-monostatic body. A more common example is the Comeback Kid, Weeble or roly-poly toy (see left figure). Not only does it have a low center of mass, but it also has a specific shape. At equilibrium, the center of mass and the contact point are on the line perpendicular to the ground. When the toy is pushed, its center of mass rises and also shifts away from that line. This produces a righting moment which returns the toy to the equilibrium position.

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