Hold Your Balance

I remember when I was a lot younger, I used to fall over constantly. I’d come home everyday to a mother shocked at discovering yet another graze on my leg. I’m not sure if I was just a wild brawler back in the day, or simply didn’t quite have an aligned center of mass. I’m gonna guess it was probably the latter. You know, maybe I wouldn’t have toppled over as much if I’d been shaped more like a gömböc…

If you’ve ever played around with an egg before, you’ll have realised that, if you let it roll around for a bit it will eventually come to rest lying on its side. You can nudge it a bit, but it settles down pretty soon. The more adventurous of you may have tried balancing said egg on one of its ends. Even if this is accomplished, the slightest of disturbances will cause it to fall back down onto its side. Yeah, I’m sorry, you wasted your time.

These two tips of the egg are known as unstable equilibrium points, which means that it is possible (contrary to experimental data) to balance it on these points, but any disruption, however small, will cause the egg to topple. The circumference around the ‘belly’ of the egg consists of stable equilibrium points which are, as you’d expect, stable. This means that if the egg lands on one of these points, it’ll tend to stay there.

It turns out that mathematicians really like shapes, and more so eggs. And so, of course, they set out to find the ultimate three-dimensional shape that has just one stable and one unstable point of equilibrium. It was also to be convex, meaning that the shape doesn’t bulge inwards, as well as homogenous, meaning that it is made of the same material throughout ,with equal density. These last two conditions make this shape extra hard to discover. The shape, if it exists, would self-right itself to the stable equilibrium point, regardless of how you placed it on the table. No matter how you knocked it down, it would always come back up, kind of like a Weeble or Rocky Balboa.

Two Hungarian scientists by the names of Gábor Domokos and Péter Várkonyi proved the existence of this mystical shape in 2006. It turns out that this shape is not unique, and there are many variations on the shape they found, all of which kind of resemble a sphere if you have a wild imagination. It was thus named the gömböc, which is Hungarian for ‘tiny sphere’. However, if you go into a Hungarian restaurant and order gömböc, you may be unpleasantly surprised when you receive a dish of seasoned pork filled in pig-stomach. Just putting that out there.

 

gomboc2

Now, you may be wondering, ‘That’s great and all, but why does this matter?’

Well, first of all, you should be in awe at the sheer mathematical beauty of this shape and its complex simplicity. But it turns out that the mathematicians were beaten to finding this shape by the one and only, Nature. Some species of turtle and beetle have adapted their shells to help automatically right themselves when they have their belly exposed. The mathematics used in finding the gömböc has also been useful in modelling how asteroids evolve their shape, somehow.

indian_star_tortoise_tennoji
Image by Wikimedia Commons user Kuribo.

If you’re looking to buy a new paperweight, I’d lower your hopes now. The gömböc’s measurements must be strictly precise in order for it to function properly, within tens of microns, which is a tenth the thickness of human hair. The technical processes required to construct this holy grail of shapes gives it quite a hefty price tag. I know what I’m asking for Christmas though this year…

Harvey

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