The Hungarian Quarterly, VOLUME XLVIII * No. 187 * Autumn 2007

Zoltán Barotányi

Turtles, Eggs and the "Gömböc"

...

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.

...

My Lunch with Arnold

The conference organizers were advertising special lunches. For an exorbitant fee one could buy a ticket to eat with a math celebrity. [...] I calculated that if I reduced my eating from two hotdogs a day to one I could afford a lunch ticket with the great Professor.
The lunch was a disaster, both from my point of view and Arnold's. The organizers had tried to maximize their profit rather than the ticket-buyers' pleasure. At the big round table with Arnold were ten eager young mathematicians. Each was carrying one or two "highly important" scientific papers which were full of "highly relevant" results they wanted to share with Arnold. He could not eat as they held out their papers and made claims about their great original contributions. And unless I was willing to butt into this noisy whining, as each of the people was doing to the others, I could not speak. I sat and tried to look attentive at the pathetic scene.
At the end of the meal Arnold finally asked me, "And what is your paper about?"
I said, "Nothing."
"Surely you have something to ask or say," he said.
But I was depressed by the fray and said no, I had just wanted to listen. I ate one hot dog a day and I went to a hundred fifteen-minute talks that I didn't understand.
On the last day I packed my suitcase and headed for the airport. The main lobby of the conference centre was deserted, maintenance people were taking down posters, the buffet was closed, people were fading out. As I strolled across the big hall I noticed, next to a young Asian man, leaning on a counter near the closed buffet, Professor V. I. Arnold. The young Asian man was talking excitedly in the tone I had noted at the disastrous lunch. As I walked closer, Arnold raised his voice slightly.
"As I told you already several times, there is nothing new in what you are telling me. I published this in 1980. Look it up. I do not want to discuss this further; moreover, I have an appointment with the gentleman carrying the suitcase over there. Good-bye."
The disappointed young mathematician got up to leave and Arnold turned to me. "You wanted to talk to me, right?" Stunned that he even remembered me, but aware of the part I suddenly was supposed to play, I pretended that the discussion was expected. "You sat at the lunch table, right? You must have had a reason. What is it about? Tell me fast. I have to catch my train."
We sat down. I collected my thoughts and explained about the plywood and the wire [constructions] and how they gave the number two, which really meant four. He stared off without saying a word. After five minutes I asked him if he wanted to know how we proved that the plywood had at least four equilibria. He waved me away. "Of course I know how you proved it" and then he breezily outlined the proof in a few phrases. "That's not what I am thinking about. The question is whether your result follows from the Jacobi theorem or not."
He stared off again.
[...] "Send me a letter when you find a body with less than four equilibria in the three-dimensional case," he said, "I have to catch my train. Good-bye, young man, and good luck to you!"

Zoltán Barotányi
is on the staff of the weekly Magyar Narancs.