Welcome to Math Mutation, the podcast where we discuss fun, interesting, or weird corners of mathematics that you would not have heard in school. Recording from our headquarters in the suburbs of Wichita, Kansas, this is Erik Seligman, your host. And now, on to the math.

If you heard the title of this podcast, you’re probably asking yourself, “What did he say?” Today we are discussing a 3-dimensional shape that was first discovered only in the 21st century, by a pair of Hungarians named Gábor Domokos and Péter Várkonyi,. It’s called the Gomboc, spelled g-o-m-b-o-c, with two dots over each of the o’s due to its Hungarian origin. It looks kind of like a spherical stress ball whose top has been partially squeezed in; we can’t really do it justice in a verbal description, but you can find links to articles with pictures in the show notes at mathmutation.com. But most importantly, this new shape has the amazing property that it has only one stable equilibrium position: no matter what position you put it down in, it will roll around and right itself.

Now at first, you might not think this is very interesting. When you were a child, you very likely played with a toy like that, a small egg-shaped doll, which would always pull itself upright no matter how you placed it. These are called roly-poly toys in some places, but in the US during my childhood, they were marketed as “Weebles”, with the advertising slogan that “Weebles wobble but they don’t fall down”. They worked because of a weighted bottom: the bottom half of the toy weighed significantly more than the top. The Gomboc, on the other hand, is made of a single material with uniform density, and miraculously still has this self-righting property.

Another interesting fact about the Gomboc is that, until a few decades ago, it was not obvious that such a shape should exist at all. If you look at the 2-dimensional analog of the problem, trying to find a two-dimensional shape of uniform density that will always right itself when placed on a line that has a gravitational pull, there is no solution. Any convex shape you create will have a least two stable equilibrium positions. Domokos had originally set out to try to prove the 3-dimensional analog of this theorem, which would have demonstrated that nothing like the Gomboc could actually exist. But after a conversation with a Russian mathematician named Vladimir Arnold, he realized that things were a bit different in three dimensions, and the theorem might not hold.

Domokos and Varkonyi then started working with computer modeling and trying to figure out what such a self-righting shape would look like. At one point, Domokos came up with the idea that maybe nature had solved the problem already, and started experimenting with pebbles found at a beach, to see if any had naturally assumed a self-righting shape. After checking over 2000 pebbles, he was disappointed. Their breakthrough came when they started defining some key parameters of the shapes they were creating, called “flatness” and “thinness”, and realized they would have to minimize both together to come up with their desired shape.

Continuing their computer modeling, but with these parameters in mind, they were finally able to start describing 3-D self-righting shapes. The first one they came up with was very close to a sphere— disappointingly, so close that they could not manufacture one in practice. Because it only differed from a true sphere by a factor of 10^-5, and even microscopic variations from their intended design would kill its self-righting property, attempts to manufacture it would just lead to the creation of ordinary spheres. But by realizing that they could sacrifice smoothness, and allow some sharp edges between sphere-like segments, they were then able to come up with a more practical shape. It’s still incredibly challenging to manufacture: to get one you can hold in your hand, it has to be made with a precision down to the width of a human hair. Apparently interest is now wide enough that they are being mass-produced, and can be ordered at the “gomboc shop” website.

Another interesting thing about this shape— Domokos realized afterwards that his insight about nature developing in this direction wasn’t totally off base. He couldn’t find it in pebbles, because the tight margin for error meant that any Gomboc pebble would quickly wear away at one of its edges into a non-self-righting shape. But evolution painstakingly comes up with precise designs over millions of years— and after carefully searching species of tortoises, he did find two with shell designs very close to his Gomboc. A tortoise stuck on its back is very vulnerable, so it really does make sense that they would evolve an improved ability to right themselves as needed.

And this has been your math mutation for today.

References:

- https://gomboc.eu/en/mathematics/
- https://www.wired.com/2008/02/the-gomboc-the/
- https://www.youtube.com/watch?v=rvVF5QWSYF4
- https://www.maplesoft.com/company/casestudies/Professional/Gomboc.aspx
- https://gomboc-shop.com/
- https://plus.maths.org/content/story-goumlmboumlc
- https://findwords.info/term/weeble