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The basic problem comes from V. I. Arnold, one of the greatest living mathematicians, who has made important contributions to topology, chaos theory, mechanics and catastrophe theory—subdisciplines in mathematics which are quite distant from one another. Arnold was the first to conjecture the existence of a homogeneous, convex body with one stable and one unstable point of equilibrium (more precisely, he reckoned there were complex, homogeneous bodies with less than four points of equilibrium).
This is, in fact, a topological problem and a very interesting one at that. Topology is a fairly recent research area of mathematics, a geometry concerned with the mapping of sections, curves, circles and discs. It is also referred to ironically as "rubber sheet geometry", and topologists are described as mathematicians who are unable to tell a tyre from a coffee cup—since the two objects can be transformed into each other by means of topological geometry. However, the relevance of the "Gömböc", the 1.1 mono-monostatic body, will probably extend beyond topology.
Domokos and Várkonyi set about confirming Arnold's conjecture in an astute way. They first envisaged a flat shape rolling on a horizontal plane with exactly two positions of equilibrium (one stable, one unstable) in the plane of rolling, and they promptly managed to prove that no such shape exists. The refutation, however, did not hold up when it comes to threedimensional shapes. They therefore had to assume that such a body exists, but was yet to be constructed. They had to allow for several considerations. | They supposed, and proved mathematically, that such a threedimensional figure had to be of minimal flatness and thinness. Proceeding along these lines, Domokos and Várkonyi got closer and closer to resolving the problem, the essence of which is this: a homogeneous mono-monostatic body in class 1.1 (i.e. with two equilibria altogether) must be envisaged as consisting of two three-dimensional elements conjoined, the one delineated from the other by a spatial pattern something like that found on a tennis ball. Their Mathematical Intelligencer article does not provide exact and detailed parameters for the shape, as their research is still at too early a stage. They have, however, put up a prize of $10,000 for the mathematician who can identify a selfrighting, mono-monostatic polyhedral object (an object with flat sides) similar to the Gömböc with the least number of sides. (The actual prize money will be $10,000 divided by the number of the sides of this object. It is unlikely that anyone will strike it rich, however, as they believe that any such three-dimensional figure would have several thousand sides.) Will there be any practical application for the "Gömböc"? In laymen's eyes, spinoffs from mathematical discoveries often seem disappointingly slow to arrive. However, mathematical discoveries, especially those where several branches of this or other disciplines overlap, may yield answers to questions that their discoverers may never even have dreamed of. Such answers might seem abstract to those less versed in mathematics but, sooner or later, most of them will affect our everyday lives.
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