Meet The Bille, A Self-Righting Tetrahedron That Nobody Was Sure Could Exist

Theorizing the Bille

Monostable shapes are, kind of like the entire field of number theory, one of those kooky mathematical trivialities that secretly has major applications for the real and futuristic world we live in. They were first posited by math’s trickster god John Conway back in the 60s, with various people coming up with concrete examples every few years since then – but there was one glaring exception to the collection. A four-sided monostable shape – the simplest possible in terms of sides – would be impossible to create.

At least, that’s what Conway said. To be fair, he was talking specifically about a perfectly balanced tetrahedron – a weighted version, he said, should be possible. Unfortunately, like Fermat before him, he never seems to have formalized any proof of that or come up with an example to demonstrate it – so the mathematical world was simply left to wonder.

But for Gábor Domokos, a Professor of Geometric Modeling at the Budapest University of Technology and Economics, that wasn’t good enough. He already had one famous hypothesized monostable shape under his belt – the turtlish Gömböc, which he and colleague Péter Várkonyi presented back in 2006 – and now, he was going to find another.

“I had little doubt,” that such a shape should exist in theory, Domokos tells IFLScience. “I am fully aware of Conway’s super-human mental powers, and I knew that he was convinced about this[.] That eliminated any doubts I may have had.”

But actually finding the shape? That was another question entirely. Other than a strong hunch that it was out there, Domokos, along with student Gergő Almádi, had basically nothing to go on: “We did not theorize any shapes, because we could not,” he says. “And what beats me still completely, [is] how Conway could have any sort of intuition about this object which is at the border of physical existence.”

“He certainly did not compute anything.” 

The search is on

So, armed with modern computers, a lot of gumption, and not a lot else, then, the pair proceeded to search. 

“Almádi did a rather educated, but still brute force search,” Domokos tells IFLScience. “This is nothing special with today’s computers, still, it is a formidable task for an undergrad in architecture!”

Whether through fate or fortitude, eventually, a couple of potential tetrahedra were found. “We burned the CPU for a couple of days,” Domokos says, “but […] you probably need only a couple of hours.” 

It must have been an elating result – right up until they read the specifics. Actually building these shapes, it seemed, would be a task somewhere in between “extremely difficult” and “impossible”. One solution would only work if it were built out of some material “an order of magnitude denser than any material found on Earth,” the team report in a yet-to-be peer-reviewed preprint paper describing the Bille, so that was probably out; even the more feasible option needed about a 5,000-fold difference in density between the weighted and unweighted parts of the shape.

Basically, they needed to build a shape that was all but made from air itself, but still rigid. It was a task daunting enough to make Domokos hesitate: “I thought that if it had been possible to engineer such an object, somebody would have already done it,” he tells IFLScience. 

“I knew how utterly beautiful this question was,” he says, “[and] I knew that there are plenty of brilliant mathematicians and engineers out there, much better than I am. So the fact that it apparently did not exist made me insecure.”

Bille-ding the Bille

Eventually – and after a little trial and error – a solution was found. It came in the form of ultra-light carbon fiber tubes, which were used to form the shape’s skeleton, coupled with tungsten carbide for the weighted parts – a compound whose very high density has traditionally made it a go-to material for tools and ammunition. The result: a fully-functional, non-homogeneous, monostable tetrahedron – a.k.a., Bille.

 

Well, almost. “The brilliant chief engineer, Ákos Török, who created the full technology for Bille, called me on the phone and told me that they built it, but there is a slight problem: it has two stable equilibria,” Domokos recalls. “I was devastated.” 

“[I] asked whether there could have been any mistake in the super carefully planned production process,” he tells IFLScience. “He assured me that they went ‘by the book’ and, beyond a millimeter-size additional blob of glue, the Bille is perfect.”

“We agreed that they remove the blob. And then it worked. I could hardly believe it.”

It had taken six months, thousands of Euros of Domokos’s own money, and no small amount of perseverance, but at last, the Bille was here.

A shape for the future

We know what you’re thinking: what, outside of a neat mathematical curiosity, is the point of the Bille?

Well, consider this: only a few months ago, space exploration company Intuitive Machines reported that a lunar lander had tipped over after reaching its destination. It marked the second time in two years that a mission had ended in failure after a spacecraft was left with its solar panels out of the sunlight and unable to self-right. 

Shapes like the Bille offer a potential solution to such problems. “While it may not be possible to design objects which can passively self-right on any terrain, designing for self-righting on a horizontal support may be feasible,” they write, “and we hope that for those designs our study could offer insights.”

There are also mathematical implications. Already, the Bille has answered some questions around the potential for so-called mono-monostatic tetrahedra – four-sided shapes with one stable equilibrium point and also one unstable equilibrium point, meaning a point on which it will balance, but only if you’re very careful not to disturb it. “We were hunting for [them]” during the search for the Bille, Domokos tells IFLScience, but “we did not find them, and later proved that they did not exist.”

Outside of the physical applications, however – well, is it so bad to enjoy a puzzle well completed?

“I knew that if it was possible at all, it [would] be very difficult,” Domokos says, “and we [would] need all our geometric and engineering intuition.”

“All in all, we struggled with the model for over 6 months,” he says. “Gergő’s huge enthusiasm was a key driving factor.” 

“I knew that if we [could] pull this off, that could start his career, and he deserves it.”

The paper, which is yet to be peer-reviewed, is posted to arXiv.