The rigorous proof of the solution can be found in references of their work. In a personal discussion, however, Arnold questioned whether four is a requirement for mono-monostatic bodies and encouraged Domokos to seek examples with fewer equilibria. Domokos met Arnold in 1995 at a major mathematics conference in Hamburg, where Arnold presented a plenary talk illustrating that most geometrical problems have four solutions or extremal points. The problem was solved in 2006 by Gábor Domokos and Péter Várkonyi. Whereas a common anticipation was that a three-dimensional body should also have at least four extrema, Arnold conjectured that this number could be smaller. It was already well known, from a geometrical and topological generalization of the classical four-vertex theorem, that a plane curve has at least four extrema of curvature, specifically, at least two local maxima and at least two local minima (see right figure), meaning that a (convex) mono-monostatic object does not exist in two dimensions. Convex means that a straight line between any two points on a body lies inside the body, or, in other words, that the surface has no sunken regions but instead bulges outward (or is at least flat) at every point. The requirement of being convex is essential as it is trivial to construct a mono-monostatic non-convex body (an example would be a ball with a cavity inside it). The question of whether it is possible to construct a three-dimensional body which is mono-monostatic but also homogeneous and convex was raised by Russian mathematician Vladimir Arnold in 1995. The above examples of mono-monostatic objects are necessarily inhomogeneous, that is, the density of their material varies across their body. This produces a righting moment which returns the toy to the equilibrium position. When the toy is pushed, its center of mass rises and also shifts away from that line. At equilibrium, the center of mass and the contact point are on the line perpendicular to the ground. Not only does it have a low center of mass, but it also has a specific shape. A more common example is the Comeback Kid, Weeble or roly-poly toy (see left figure). (The previously known monostatic polyhedron does not qualify, as it has several unstable equilibria.) A sphere weighted so that its center of mass is shifted from the geometrical center is a mono-monostatic body however, it is not homogeneous. 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. When a roly-poly toy is pushed, the height of the center of mass rises from the green line to the orange line, and the center of mass is no longer over the point of contact with the ground. Copies of the Gömböc have been donated to institutions and museums, and the largest one was presented at the World Expo 2010 in Shanghai, China. 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.
![stable equilibrium 3d stable equilibrium 3d](https://thumbs.dreamstime.com/b/equilibrium-3493129.jpg)
It has a sharpened top, as shown in the photo. Gömböc is the first mono-monostatic shape which has been constructed physically. Mono-monostatic shapes exist in countless varieties, most of which are close to a sphere and all with a very strict shape tolerance (about one part in a thousand). The existence of this class was conjectured by the Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by the Hungarian scientists Gábor Domokos and Péter Várkonyi by constructing at first a mathematical example and subsequently a physical example.
![stable equilibrium 3d stable equilibrium 3d](https://thumbs.dreamstime.com/b/blue-spheres-equilibrium-balance-concept-image-d-illustration-41615060.jpg)
The Gömböc ( Hungarian: ) is the first known physical example of a class of convex three-dimensional homogeneous bodies, called mono-monostatic, which, when resting on a flat surface have just one stable and one unstable point of equilibrium. 4.5 m (15 ft) gömböc statue in the Corvin Quarter in Budapest 2017