The Gömböc

A shape and its creator.

Algebra is important. Without it, science would suck. Every piece of software is written in the language of variables, functions, matrices—a.k.a. in slightly different terms it’s your good pal algebra. Consequently, a tremendous amount of energy is devoted to getting kids to learn more of this stuff.

Geometry, on the other hand…listen, I love it. But admittedly it seems less important than ever in the long history of the subject—which is actually what makes it fascinating. Other areas of math feel essential and timeless. Often enough, geometry is covered with the messy fingerprints of its creators.

Which brings us to Gábor Domokos and his remarkable, self-resettling Gömböc.

The gömböc was discovered in 2006. Its name roughly means “sphere-ish” in Hungarian. All on his own, Gábor Domokos had been searching for this unusual shape for a decade and a half.

Domokos is more engineer than cutting-edge mathematician. (He says his “dark secret” is that he’s an architect.) He teaches Civil Engineering in Hungary, but as he describes in “My Lunch With Arnold,” the path that brought him to the gömböc began on a stay in America.

In America, Domokos had a mathematician friend named Andy Ruina. One day, Andy mentioned that his buddy, Jim Papadopulos, asked an interesting question. Jim at the time was designing machines that refilled laser toner cartridges. Wikipedia now says that he specializes in “bicycle science.” Jim seems like a cool dude.

Jim asked, suppose you drew a convex curve on a plank. Then suppose you cut it out with a saw, then stood the wooden cutout on a table. In how many orientations would the cutout stay balanced?

For an ellipse, the answer is four. (See above.) Two of those orientations are unstable—if the wind’s blowing, the ellipse will topple over. Two of those orientations are stable—the shape will just wobble around if you nudge it, before settling back in.

It’s easy to find shapes that have more than four balanced orientations. (Just roll a hexagon onto each of its six faces.) But could a shape have fewer than four? Domokos, Andy, and Jim thought there couldn’t, that four was an absolute minimum for a convex curve. In 1994, they proved this.

The proof is a relatively straightforward consequence of the Four-Vertex Theorem, which had been known since 1909. They published their result in an obscure engineering journal called “Journal of Elasticity.” Jim had no academic affiliation, so the paper just lists his address in Green Bay.¹

Life moved on. Domokos returned to Hungary. But he kept thinking about his problem.

Domokos wanted to know if their result—that every convex curve has at least two stable, two unstable orientations—would apply to 3D solids. He was starting to suspect that it did not.

Domokos is a guy who thinks about things. He’s written papers about pebble erosion and the elasticity of beams. He wanted to know if there was some thing that was unlike the ellipse—a solid that would only have one stable orientation, not two.

Well, there is such a thing, he soon discovered. Take a paper towel roll and cut off the sides. It has only one stable orientation—that’s it. Neat!

The only thing is that it’s hollow, which is somewhat of a cheat. There are lots of toys that cheat in a similar way. You know those inflatable clown toys that you can knock down, and they come swinging back up? That clown only has one stable orientation (hence the swing back) but it only works because there’s a big weight at the bottom.

I mean, can you imagine if it weren’t hollow? If it was solidly made of wood or metal, with no holes or shenanigains, but then it somehow jumped right back up when you punched it over? How weird would that be?

VERY weird, right?

Let’s cut to the chase: this is precisely the gömböc. It only has one stable orientation. (It also has an unstable orientation, but that’s it.) You knock it down, it gets up again. You aint never going to keep it down. Even though it’s solid all the way through. You could even try this with the 4.5 ton metal gömböc the Hungarians built, though I wouldn’t necessarily recommend it.

Good news: we’ve reached the point in the article where I pretend that the gömböc is also incredibly useful! New applications are being found daily! There is apparently someone who thinks you could make a gömböc pill—like, to swallow. Pretty soon we’ll all be eating gömböcs!

With that out of the way, it’s time to tell you a story about who Domokos really is, what makes him tick, and what energized him to search for his answer. It’ll give you a better appreciation for the shape than talk of applications.

The Story

In 1995, a major applied mathematics conference met in Hamburg. Domokos attended this conference, but soon found himself on the edge of despair. His own talk had gone poorly. He couldn’t understand any of the sessions he attended. He felt depressed and lost.

Along with the rest of the conference goers, he shuffled into the main auditorium to hear a plenary talk from the famous mathematician V.I. Arnold. He knew he wouldn’t understand it, but what else could he do? He was from a minor university and attending had been a financial stretch. He’d worked hard to even attend. He couldn’t walk away.

Domokos struggled to understand anything in Arnold’s talk. It mostly involved corners of the mathematical world that Domokos found incomprehensible. But there was one thing he got. Because throughout the talk, Arnold bizarely kept returning to the number four.

“He said these topics were examples of a theorem created by the great nineteenth-century mathematician Jacobi,” Domokos wrote later, recalling the experience. “He said Jacobi’s theorem had many applications, and that always something had to be bigger or equal to four. He covered one topic or another that would be familiar to each person in the audience, always coming back to the number four.”

Suddenly, while leaving the talk, Domokos had a realization—he also had a theorem about the number four! It must be related…and he had to tell Arnold.

There was no way to reach him after the talk. But Domokos saw a poster advertising an opportunity to eat lunch with Arnold, organized by the conference. So nice. To attend, all you had to do was pay an exorbitant fee.

“Although my budget was tight and my mathematics is not at the level of Arnold,” wrote Domokos. “I could calculate that if I reduced my eating from two hotdogs a day to one I could afford a lunch ticket with the great Professor.” Oh, Domokos.

Domokos paid for lunch with Arnold, breaking his budget. So had several others. The other lunchers wouldn’t shut up. They bored Arnold to tears with talk of their under-appreciated papers, their groundbreaking results. Domokos was embarrassed on Arnold’s behalf. The whole thing was a disaster.

Domokos, depressed again, said nothing during lunch. Arnold tried to get him to talk, but the Hungarian “architect” insisted he had paid his great fee just for the chance to listen. He walked away and ate a hotdog.

Later, at the train station, Domokos saw Arnold. Arnold recognized him, and insisted that Domokos sit and talk. There must be something he wanted to say. Nobody pays money for nothing.

Reluctantly, Domokos began to explain. He told Arnold about the plywood result, how there had been four orientations that left the curve balanced. Four! Couldn’t that be connected to the fours of Arnold’s talk?

Arnold grew silent.

Domokos, feeling proud, offered to explain how they proved their result. Arnold waves his hand. No, no. The proof is trivial. He quickly rattled off an argument identical to the one Domokos, Andy, and Jim had published.

“That is not what I am thinking about,” said Arnold. “The question is whether your result follows from the Jacobi theorem or not.”

It turns out that this question itself depends on the 3D case.

“Send me a letter when you find a body with less than four equilibria in the three-dimensional case,” Arnold said. And then he left.

Domokos went home.

Domokos says that, to this day, he doesn’t understand the Jacobi theorem. He built his life on the edges of mathematical abstraction, playing with real physical things instead of abstract mathematical structures. He is not a cutting-edge mathematician. Not even close.

But ten years after this meeting with the famed mathematician—years spent playing with the little things that caught his interest—Domokos finally did it. He invented and named the gömböc, a shape that, no matter how you knock it down, picks itself back up.

Can algebra do that?

1. 2597 Sherry Ln, Green Bay, WI 54302. According to its Zillow listing, it has 2 beds, 2 baths, and was built in 1985, so Jim would have been its first resident.