The Gömböc: Physical Evidence of a Mathematical Theorem

The Gömböc is a round piece of clear synthetic material with gently peaked, organic curves. It looks like a piece of modern art. But if you tip it over, something unusual happens: it rights itself.

It leans off to one side, rocks to and fro as if gathering strength and then, presto, tips itself back into a “standing” position as if by magic. It doesn’t have a hidden counterweight inside that helps it perform this trick, like an inflatable punching-bag doll that uses a central weight to stand upright after you strike it.
Rather, the Gömböc is something new: it is the world’s first self-righting object with just one stable and one unstable point of equilibrium.                                                                          

The Gömböc is a result of a long mathematical quest. In 1995, the great Russian mathematician Vladimir Arnold mused that it would be possible to create a “mono-monostatic” object — a three-dimensional entity that purely by its geometric form had only one possible way to balance upright.

The challenge intrigued two scientists — Gabor Domokos and Peter Varkonyi, both of the Budapest University of Technology and Economics. Rigorous research spanning over numerous years soon followed before both professors understood that such a mono-monostatic object could exist. Presenting it as a physical demonstration proved yet a further challenge. It soon became clear that any technology used to build the Gömböc required high precision. Such an object with a radius of approximately 10cm required a tolerance of less than 0.1mm. Furthermore, both mathematicians shared the belief that the Gömböc should be manufactured from transparent material; otherwise, the homogeneity of the object would be hard to demonstrate.

Long search and many trials led to the 3-D Printing Technology pioneered by Objet Geometries. The PolyJet Technology was the only technology providing both the required extreme precision, transparent and homogeneous material and high reliability. “PolyJet Technology allowed for the ideal homogenous properties during print that we were seeking. It became clear to us this Objet was the key in successfully creating a physical mono-monostatic object” – says Domokos. Objet also allowed for individually marking and numerating the Gömböc. In fact, both the serial number and the Gömböc trademark appear inside the object as thin three-dimensional tubes, filled with support material (which has almost identical density). “Objet was a perfect fit for us!” remarked Domokos. 

The Gömböc is a big hit in the media: a beautiful, but rather abstract mathematical theorem was physically demonstrated in 2007 by using cutting-edge, state-of-art technology thanks to Objet’s PolyJet Technolgy. This discovery generated curiosity for millions of people worldwide. In 2007, New York Times Magazine selected the Gömböc as one of the most interesting seventy inventions of the year; the Gömböc homepage (www.gomboc.eu) received so far over 4.5 million hits from 143 countries. Articles about the Gömböc appeared in 22 languages. The Gömböc stands as a symbol of extreme precision, sensitivity, perfection, harmony and abstract beauty.

Yet the scientists now say that Mother Nature may have beaten them in the race after all. They have noticed that the Gömböc closely resembles the shell of some terrestrial tortoise species, creatures whose round-shelled backs help them right themselves when flipped over. “We discovered it through mathematics and built it with the help of Objet,” Domokos notes, “but evolution got there first.”