∫ The World’s Coolest Paperweight

This is a Gömböc:

What is a Gömböc, you ask? It is the answer to a surprisingly tricky math problem, it is the secret to how many tortoises survive, and it would be one cool toy to have on your desk.

But perhaps put most plainly, the Gömböc (that last “c” is pronounced like a “tch,” by the way) is the only convex three-dimensional shape* which will balance in only one position by virtue of it’s shape alone. I’m struggling here to describe it’s properties both accurately and non-technically, but if you’ll bear with me, I’ll show you what I mean.

Suppose you wanted to construct a 3-dimensional shape that would balance in only one spot. That’s easy enough to do: just take a plastic easter egg, open it up, and pour some glue into the bottom, like so:

As I’m sure you can imagine, the extra weight at the bottom will mean that you cannot get the egg to balance on it’s side anymore, because of course gravity will always pull the end with the glue back towards the ground, since it’s the heaviest bit. Too easy.

Well mathematicians don’t like easy puzzles, so after solving that simple problem they decided to make it a little harder. Suppose we can’t add or remove any weight from the shape, but that it has to be the exact same density throughout (the mathematical word is “homogeneous”). If so, could we still make a shape that will balance in only one position?

It turns out this one is only  little bit harder. You can compensate for the inability to add weight to the shape by simply deforming it so that it naturally has less bulk on one side. For me the easiest way to imagine doing this is to take a clay sphere, and then pinch a little stem up from the surface on one side, like a top:

That’ll do the trick, and of course there are plenty of others that will do. But somehow, this stills to simple. So what can we do to up the difficulty once more? Well the mathematicians pondering this stuff realized that all of the obvious solutions at this point require some sort of “pinching” of some kind, so that a portion of the shape is left hanging awkwardly away from the majority of it’s mass. So they figured the next question to ask was “can we do this without adding weights and without pinching the shape in weird ways? In mathematician speak, this means insisting that the shape be “convex,” which is simply enough to understand. Any section you look at on the surface of the shape must be curving outward toward you, not inward like my little top does where the neck meets the rest of the shape.

The first person to pose this particular challenge was the brilliant (and also impressively articulate) Russian mathematician Vladmir Arnold, in 1995. He guessed, correctly, that it would still be possible to find such a shape, and of course he was right: you just have to build a Gömböc. But it took mathematicians and engineers 11 years to figure out just what this shape should look like. It was only in 2006 that two Hungarian scientists and engineers (Gábor Domokos and Péter Várkonyi from Budapest University of Technology and Economics) designed the figure you see above, and proved that is satisfied the necessary properties.

It may come as some surprise to you that this last step took so long. After all, we solved the two “simpler” puzzles in a matter of minutes. Could it really have taken eleven years to figure out how to do it with a convex shape? On the surface this doesn’t seem like a terrible restrictive condition. But as a matter of fact, it very much is.

To get a taste of why it was so hard for Domokos and Várkonyi to do this, try doing it yourself in just two dimensions. That is, draw a convex shape on a piece of paper that would balance stably at just one point on its surface (assuming the shape was cut from a uniform material). After just a few minutes of doodline, I’ll bet you’ll start to get rather frustrated. The fact of the matter is, there just aren’t very many different ways to draw a convex shape! All the “interesting” doodles eventually curve back in on themselves at some point, leaving you with not much more to work with than various ellipses and some football-looking things. And manifestly, none of these will do the trick: Every stable balance point is matched by another one somewhere on the other side of the figure.

If you’re still doodling and starting to get frustrated, I encourage you to give up now. It has, in fact, been proven that there is no convex, homogeneous shape in two dimensions that has just one stable balance point, and I hope this sheds some light on why solving the problem in three dimensions was particularly daunting. Of course in some ways you have a lot more options in three dimensions, but at the same time the shapes all look quite a bit alike, in the sense that they’re all just a bunch of outwardly-curving surfaces clued together at the edges. Somewhere in that huge, huge list of  possible shapes, there is one that magically makes things work out. And yet if you’re not careful, you might miss it, because there are also infinitely many shapes that look almost exactly like it but don’t quite work. As a matter of fact, when the Gömböc shape was originally discovered, it took quite some time before a working model could be built in practice, precisely because the smallest changes in it’s dimensions destroy it’s special properties. If you want to build one that is four inches tall, for example, you need the edges to be no more than a millimeter away from their planned locations.

So that, mathematically, is what a “Gömböc” is: a three-dimensional, convex and homogeneous shape with only one balance point. But I promised to say something about turtles and paperweights, didn’t I? As it turns out, in the years since its discovery, a number of tortoise species have been discovered to have shells that strongly resemble the upper half  of a Gömböc (the Indian Star Tortoise, for example). Most such turtles have legs and necks that are much too short to be able to roll themselves back upright should they ever find themselves flipped over on their backs, but since they’re shaped like Gömböcs, it’s nearly impossible for this to happen. Remember, the whole point of a Gömböc is that it will balance in only one position. That means that if you try to flip one over and leave it in any other position, it will topple over and roll about until it eventually winds up back where it started.

This brings me to why it would be such fun to have on on your desk: you could just toss it randomly on top of a stack of papers, confident that no matter what you did, it would eventually roll back into an upright position. It is the “cat-in-mid-air” of inanimate objects: it always finds a way to wind up right-side up. Unfortunately, I could not find a good youtube clip of a Gömböc in action, but if you follow the link at the very top of the post, you’ll find a video embedded in the webpage which shows it rolling about in this way.

By the way, if you’ve been thinking really cleverly, you’ll have noticed I’ve been lying to you somewhat this whole time (an inevitability when you translate something from the precise language of mathematics into the fuzzy form of English prose). I told you, for example, that the weighted egg and the funny top-looking thing would only balance in one place each, but as you may have realized, it would actually be possible to balance these on the narrow end or the top of the neck, if you were extremely careful about it. But balance points like these are what mathematicians and physicists call “unstable” balance points. You can balance these shapes on an unstable point, technically, but if anything at all happens to disturb them, they will fall over. On a stable balance point, by contrast, if you poke them a little bit or jiggle the table they will just roll back and forth about the balance point until they settle back down to a standstill. Since this is what most people mean by balance points (no one would really claim that the corners of a six-sided die are all balance points in a normal conversation, would they?) I’ve used the term “balance point” above to mean “stable balance point.” A shape with just one stable balance point is called “monostatic,” and this is the correct term to use to describe the Gömböc. But in fact, the Gömböc is even more interesting in that it has also only one unstable balance point (right at the top of its “fin”) and thus is called “mono-monostatic” by mathematicians (who sometimes seem to name things based on their childhood speech impediments).

__

*Actually, this is a lie that I’m going to repeat throughout the post. “Gömböc” refers to a specific family of shapes which all have the properties outlined here. Not all the different branches of the Gömböc family tree are well-understood at this time, and the one pictures above is only the prototypical example. But in fact people say “shape” all the time when they really mean “family of shapes,” it just seems more significant in this case because “Gömböc” is not a family of shapes we’re used to. However, if I had said “Johannes Kepler had famously proved that all stable planetary orbits come in just one shape: an ellipse” then you probably wouldn’t have quibbled, even though there are in fact infinitely many different types of ellipse: fat ones, skinny ones, etc.

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About Colin West
Colin West is a graduate student in quantum information theory, working at the Yang Institute for Theoretical Physics at Stony Brook University. Originally from Colorado (where he attended college), his interests outside of physics include politics, paper-folding, puzzles, playing-cards, and apparently, plosives.

One Response to ∫ The World’s Coolest Paperweight

  1. S says:

    That is truly awesome. What have you been up to?

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