Interview with Professor Gábor Domokos about the creation of the ground-breaking gömböc
“How can a mortal have such a thought?” Professor Gábor Domokos is still amazed when he thinks back to how he was given the idea for the now world-famous gömböc.
At a conference in 1995 in Hamburg Domokos approached the Russian mathematician Vladimir Igorevich Arnold, whom he describes as “one of the greatest mathematicians of the last century” and exchanged a few words with him. The conversation was about a topic of common interest: bodies and their stable points of equilibrium. “It can’t have been for more than five minutes that we were standing and talking. We spoke about a few things related to Arnold’s lecture.” And then like a bolt from out of the blue: “Almost in passing Arnold made the remark that in his opinion there must be bodies with just one stable and one unstable point of equilibrium.”
To comprehend the enormity of that thought it is necessary to realise that until then only bodies with at least four points of equilibrium had been conceived of. “Not even the ancient Greeks, who are still among the greatest thinkers of all time and whose knowledge is still amazingly up-to-date today, considered other bodies as far as we know,” says Domokos.
Domokos still has no explanation for why Arnold chose to share his idea with him. “Perhaps he had sensed the seriousness with which I had approached similar problems. Possibly his remark was even linked to the vague hope that he would spur me on to solve a problem that was not in fact of indifference to him,” speculates Domokos. Arnold made it clear to him that he was firmly convinced of the existence of such a body, but had no interest in proving it.
“If somebody else had voiced such a conjecture, I would have dismissed it as an amusing joke and paid it no further attention. But to hear something like that from the mouth of Arnold…”
Where to begin
From then on alongside his duties as a lecturer at the Budapest University of Technology and Economics Domokos had just one goal: “I was determined to prove Arnold’s conjecture and to discover the unknown body!” The question, however, was how to approach the problem. “At first I didn’t have the faintest idea what the body would look like,” recalls Domokos.
He soon realised that he would not even be able to rely on a computer for help. “Since it wasn’t clear what direction we should head in, it was not possible to programme an algorithm. Computers in themselves are blind and have no mind of their own. They only go where they are led.” Solving the problem simply by experimenting was also out of the question: “Since the creation of this body possibly depended on hundredths of millimetres, as in fact proved to be the case, the probability of finding it simply by experimenting was close to zero.”
The impossibility of using such methods to discover its form, combined with the certainty, thanks to Arnold’s words, that it must exist, spurred Domokos on even more and had some strange effects. During a holiday on Rhodes while walking by a pebble beach Domokos suddenly had the thought that the unknown body could be among the pebbles. Together with his untiringly supportive wife he began carrying pebbles back to his hotel room in a rucksack to study them there. “It was an absurd idea that I might solve the problem of its form as a result of a stroll on a beach.” That, however, was not how it turned out: “We gave up after 2,000 pebbles!”
Learning from failure
“On the flight back to Hungary I was ready to abandon the project.” However that feeling of having been defeated did not last for long. Domokos managed to turn that failure into a partial victory. “There were only two possibilities: either Arnold was wrong or there was a reason why the body could not exist among the pebbles”. That train of thought led Domokos to the idea that the body was highly likely to be sphere-like in shape, because it must have a form that is not tolerated by the sea and the stones on the beach. “Approached from the right angle even failures have their positive side.”
The unknown body slowly began to take shape and was even given a name: during conversations with Péter Várkonyi, a scientist who by then was working together with him on the project, they took to referring to the mono-monostatic body (to give it its scientific name) as the gömböc, derived from the Hungarian word “gömb” meaning sphere. However, neither researcher knew what it would look like in more detail. “We sat with a pencil and a sheet of paper and thought about the problem. That is hard for people to understand, even here at the university, but scientific research consists to a significant extent of calmly sitting and thinking.”
From paper to plaster
The results bore out that theory: a year and a half after the holiday on Rhodes the two scientists had managed to refine their idea about the shape of the gömböc sufficiently to be ready to create a proto-gömböc from plaster. And amazingly the very first gömböc worked. Like a weeble it righted itself when turned over (although unlike the weeble the gömböc is homogeneous and relies upon its shape to return to the same position, rather than being weighted). “We were incredibly lucky. The body had a lot of errors. It shouldn’t really have worked.” By chance the accumulation of various errors led to a body with gömböc properties. “If it hadn’t worked, we probably would have thrown in the towel at that point.” By that stage they had long been using their own private funds, rather than public research money.
Whether by luck or not, the two researchers had made enormous progress. The first gömböc, which was not really a gömböc, pointed the way clearly to the first proper design. With more intensive thinking and a lot of stamina the two scientists finally achieved their goal. In 2007 just a few days ahead of a conference in Moscow to celebrate the 70th birthday of Arnold the researchers produced gömböc 001, which was in every way worthy of its name as the first representative of this species. The two scientists decided to travel to Moscow to present the world’s first gömböc to the person who first conceived of the idea.
Honour where it is due
Coincidentally it was on 20 August, Hungary’s most important national holiday, that they met with Arnold. “We went up to him during the break at a conference,” Domokos says. “Arnold immediately recalled our five-minute conversation of twelve years earlier. We immediately got to the point and tried to explain to him that we had brought him a gift, but he insisted that we shouldn’t give him any presents!” Finally the two Hungarians unwrapped the gömböc and pressed it into the professor’s hands.
“The moment Arnold caught sight of the object, he immediately realised its significance. ‘You managed it’ he exclaimed. He hurried into his room, cleared the desk and enthusiastically tried out the gömböc.”
Arnold died three years later. Domokos is relieved that he was able to present the gömböc to the mathematician who conjectured its existence while he was still alive, rather than waiting until his 75th birthday.
“We were important extras in the development of the gömböc, but no more than extras” Domokos says in respect for the brilliance of the Russian thinker.
For further information about the gömböc see www.gomboc.eu.