Discovering The Mathematics-Based Gömböc Shape

A Gömböc is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. 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.

 

When the existence of the Gömböc shape was discovered in 2007 by Hungarian scientists Gábor Domokos and Péter Várkonyi, it solved a long standing mystery. For years mathematicians had discussed, debated and tried to prove its existence using mathematical equations. Vladimir Arnold, a Russian scientist, had conjectured its existence, but it took a decade to prove it conclusively and create the shape. The New York Times called the discovery “one of the best ideas of the year.” Like in many other mathematical developments, Maple also played a role in creating the Gömböc.

A Gömböc is a convex three-dimensional homogeneous body which, when resting on a flat surface, has just one stable and one unstable point of equilibrium. 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 (less than 0.1 mm per 10 cm). The most famous solution has a sharpened top as shown in the figure.

If you put a Gömböc down on a flat surface, resting on its stable equilibrium point, it will stay in the same position. "Even if you kick it a little, it will come back to its resting position at the stable equilibrium point," says Domokos, one of the inventors of Gömböc. If it is put down at a non-equilibrium point it will start rolling around in a systematic way until it has reached the stable equilibrium position. In other words, the Gömböc is self-righting¹. In fact, Wired Magazine calls it the “world’s first self-righting object.”

Gömböc has found an entry in the Guinness Book of Records as “the first homogenous self-righting shape.” The Natural History Magazine illustrates that “the secret is in the mathematics of its shape.”

The invention of Gömböc is the culmination of a long process of mathematical research and Maple, the mathematical computation engine from Maplesoft, played an important role in its discovery. The yet-undiscovered shape was known to be a convex mono-monostatic object — a three-dimensional object, which because of its geometry had only one possible way to balance upright. Domokos and Várkonyi identified a two-parameter family of objects, all of which had the desired mono-monostatic property. However, not all of them were convex. Maple was used to identify the convex shapes and thus prove the existence of the shape. The process involved a large amount of complex, precise mathematical computation, and Maple’s symbolic computation power made it possible.

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