Latest news with #tetrahedron
WIRED
4 days ago
- Science
- WIRED
This New Pyramid-Like Shape Always Lands With the Same Side Up
Aug 10, 2025 7:00 AM A tetrahedron is the simplest platonic solid. Mathematicians have now made one that's stable only on one side, confirming a decades-old conjecture. The original version of this story appeared in Quanta Magazine. In 360 BC, Plato envisioned the cosmos as an arrangement of five geometric shapes: flat-sided solids called polyhedra. These immediately became important objects of mathematical study. So it might be surprising that, millennia later, mysteries still surround even the simplest shape in Plato's polyhedral universe: the tetrahedron, which has just four triangular faces. One major open problem, for instance, asks how densely you can pack 'regular' tetrahedra, which have identical faces. Another asks which kinds of tetrahedra can be sliced into pieces that can then be reassembled to form a cube. The great mathematician John Conway was interested not only in how tetrahedra can be arranged or rearranged, but also in how they balance. In 1966, he and the mathematician Richard Guy asked whether it was possible to construct a tetrahedron made of a uniform material—with its weight evenly distributed—that can only sit on one of its faces. If you were to place such a 'monostable' shape on any of its other faces, it would always flip to its stable side. A few years later, the duo answered their own question, showing that this uniform monostable tetrahedron wasn't possible. But what if you were allowed to distribute its weight unevenly? At first, it might seem obvious that this should work. 'After all, this is how roly-poly toys work: Just put a heavy weight in the bottom,' said Dávid Papp of North Carolina State University. But 'this only works with shapes that are smooth or round or both.' When it comes to polyhedra, with their sharp edges and flat faces, it's not clear how to design something that will always flip to the same side. Gábor Domokos discovers and builds new shapes to understand the world around us. Photograph: Ákos Stiller Conway, for his part, thought that such tetrahedra should exist, as some mathematicians recall him saying. But he ended up focusing on the balancing acts of higher-dimensional, uniformly weighted tetrahedra. If he ever wrote up a proof of his off-the-cuff 3D conjecture, he never published it. And so for decades, mathematicians didn't really think about the problem. Then along came Gábor Domokos, a mathematician at the Budapest University of Technology and Economics who had long been preoccupied with balancing problems. In 2006, he and one of his colleagues discovered a shape called the gömböc, which has the unusual property of being 'mono-monostatic'—it balances on just two points (one stable, the other unstable, like the side of a coin), and no others. Try to balance it anywhere else, and it will roll over to stand on its stable point. But like a roly-poly, the gömböc is round in places. Domokos wanted to know if a pointy polyhedron could have a similar property. And so Conway's conjecture intrigued him. 'How was it possible that there was an utterly simple statement about an utterly simple object, and yet the answer was far from immediate?' he said. 'I knew that this was very likely a treasure.' In 2023, Domokos—along with his graduate students Gergő Almádi and Krisztina Regős, and Robert Dawson of Saint Mary's University in Canada—proved that it is indeed possible to distribute a tetrahedron's weight so that it will sit on just one face. At least in theory. But Almádi, Dawson, and Domokos wanted to build the thing, a task that turned out to be far more challenging than they expected. Now, in a preprint posted online yesterday, they have presented the first working physical model of the shape. The tetrahedron, which weighs 120 grams and measures 50 centimeters along its longest side, is made of lightweight carbon fiber and dense tungsten carbide. To work, it had to be engineered to a level of precision within one-tenth of a gram and one-tenth of a millimeter. But the final construction always flip-flops onto one face, exactly as it should. The work demonstrates the important role of experimentation and play in research mathematics. It also has potential practical applications, such as in the design of self-righting spacecraft. 'I didn't expect more work to come out on tetrahedra,' Papp said. And yet, he added, the team's research allows mathematicians to 'really appreciate how much we didn't know and how thorough our understanding is now.' Tipping Point In 2022, Almádi, then an undergraduate aspiring to become an architect, enrolled in Domokos' mechanics course. He didn't say much, but Domokos saw in him a hard worker who was constantly in deep thought. At the end of the semester, Domokos asked him to concoct a simple algorithm to explore how tetrahedra balance. When Conway originally posed his problem, his only option would have been to use pencil and paper to prove, through abstract mathematical reasoning, that monostable tetrahedra exist. It would have been almost prohibitively difficult to pinpoint a concrete example. But Almádi, working decades later, had computers. He could do a brute-force search through a huge number of possible shapes. Eventually, Almádi's program found the coordinates for the four vertices of a tetrahedron that, when assigned certain weight distributions, could be made monostable. Conway was right. Krisztina Regős helped discover new properties of tetrahedra. Photograph: Courtesy of Krisztina Regős; Ms. Tara Inman Robert Dawson helped discover new properties of tetrahedra. Almádi found one monostable tetrahedron, but presumably there were others. What properties did they share? While that might seem like a simple question, 'a statement like 'A tetrahedron is monostable' cannot be easily described with a simple formula or a small set of equations,' Papp said. The team realized that in any monostable tetrahedron, three consecutive edges (where pairs of faces meet) would need to form obtuse angles—ones that measure over 90 degrees. That would ensure that one face would hang over another, allowing it to tip over. The mathematicians then showed that any tetrahedron with this feature can be made monostable if its center of mass is positioned within one of four 'loading zones'—much smaller tetrahedral regions within the original shape. So long as the center of mass falls inside a loading zone, the tetrahedron will balance on only one face. The gömböc, discovered in 2006, can stand on only two points, one stable, the other unstable. Mathematicians have continued to search for other shapes with intriguing balancing properties. Photograph: Gábor Domokos Achieving the right balance between the weight of the loading zone and the weight of the rest of the tetrahedron is easy in the abstract realm of mathematics—you can define the weight distribution without a care for whether it's physically possible. You might, for instance, let parts of the shape weigh nothing at all, while concentrating a large amount of mass in other parts. But that wasn't entirely satisfying to the mathematicians. Almádi, Dawson, and Domokos wanted to hold the shape in their hands. Was it possible to make a monostable tetrahedron in the real world, with real materials? Getting Real The team returned to their computer search. They considered the various ways in which monostable tetrahedra might tip onto their stable face. For instance, one kind of tetrahedron might follow a very simple path: Face A tips to Face B, which tips to Face C, which tips to Face D. But in a different tetrahedron, Face A might tip to Face B, and both Face B and Face D will tip to Face C. The loading zones for these different tetrahedra look very different. The team calculated that to get one of these 'falling patterns' to work, they would need to construct part of the shape out of a material about 1.5 times as dense as the sun's core. While studying to be an architect, Gergő Almádi was drawn to a decades-old geometry problem. Photograph: Réka Dolina They focused on a more feasible falling pattern. Even so, part of their tetrahedron would have to be about 5,000 times as dense as the rest of it. And the materials had to be stiff—light, flimsy materials that could bend would ruin the project, since it's easy to make a round or smooth shape (like the roly-poly) monostable. In the end, they designed a tetrahedron that was mostly hollow. It consisted of a lightweight carbon fiber frame and one small portion constructed out of tungsten carbide, which is denser than lead. For the lighter portions to have as little weight as possible, even the carbon fiber frames had to be hollow. With this blueprint in hand, Domokos got in touch with a precision engineering company in Hungary to help build the tetrahedron. They had to be incredibly accurate in their measurements, even when it came to the weight of the tiny amounts of glue used to connect each of the shape's faces. Several frustrating months and several thousand euros later, the team had a lovely model that didn't work at all. Then Domokos and the chief engineer of the model spotted a glob of stray glue clinging to one of its vertices. They asked a technician to remove it. About 20 minutes later, the glue was gone and Almádi received a text from Domokos. 'It works,' the message read. Almádi, who was on a walk, started jumping up and down in the street. 'Seeing the lines on the computer is very far from reality,' he said. 'That we designed it, and it works, it's kind of fantastic.' 'I wanted to be an architect,' he added. 'So this is still very strange for me—how did I end up here?' In the end, the work on monostable tetrahedra didn't involve any particularly sophisticated math, according to Richard Schwartz of Brown University. But, he said, it's important to ask this kind of question in the first place. It's the kind of problem that's often easiest to overlook. 'It's a surprising thing, a leap, to conjecture that these things would exist,' Schwartz said. At the moment, it's not clear what new theoretical insights the model of the monostable tetrahedron will provide—but experimenting with it might help mathematicians uncover other intriguing questions to ask about polyhedra. In the meantime, Domokos and Almádi are working to apply what they learned from their construction to help engineers design lunar landers that can turn themselves right side up after falling over. In any case, sometimes you just need to see something to believe it, Schwartz said. 'Even for theoretical math, geometry especially, people are kind of right to be skeptical because it's quite hard to reason spatially. And you can make mistakes, people do.' 'Conway didn't say anything about it, he just suggested it—never proved it, never proved it wrong, nothing. And now here we are, I don't know, 60 years later,' Almádi said. 'If he were still alive, we could put this on his desk and show him: You were right.' Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
Gizmodo
05-07-2025
- Science
- Gizmodo
This Weird Pyramid Always Lands on the Same Face, Confirming 40-Year-Old Theory
'Bille' is the first-ever monostable tetrahedron, or a pyramid-like shape with four triangular faces that has one stable resting position. What this means is that Bille, no matter how you throw it and how it lands, will flip back on exactly the same side every single time. In a recent preprint submitted to arXiv, mathematicians revealed the first physical model of Bille, closing a decades-old theory proposed by the renowned British mathematician John Conway. Made of lightweight carbon fiber and dense tungsten carbide, Bille represents an array of ridiculously sophisticated engineering decisions—making this as much a technological achievement as a mathematical one. It's no surprise, therefore, that its self-righting property additionally hints at some exciting applications for the spaceflight industry—which notably experienced two recent landing mishaps with toppled-over lunar landers. In his initial conjecture, Conway surmised that a tetrahedron with unevenly distributed weight across its sides would always flip to the same side, although a few years later Conway himself rejected the idea. Some mathematicians still thought there could be something to it, however, namely study co-author Robert Dawson, who almost succeeded in proving Conway right in the 1980s using lead foil and sticks of bamboo. 'But my recollection was that this only almost worked because of angular momentum,' Dawson, now a mathematician at Saint Mary's University in Canada, told Gizmodo. 'In the way that if a car comes across a bump in the road and it's already moving, it'll get over it thanks to angular momentum. But it might have a hard time starting up against that bump.' Ideally, the monostable tetrahedron shouldn't need another push to flop back on the 'base' side. For a while, it seemed like Conway's theory would end up in a box of really-cool-but-unlikely math ideas—until about three years ago, when mathematician Gábor Domokos and his student, Gergő Almádi at the Budapest University of Technology and Economics, reached out to Dawson. Domokos, a long-time expert on tricky balancing problems in geometry, had already discovered the gömböc, a roundish object that balances only on two points like a roly-poly toy. While an impressive discovery, the gömböc, with its mostly round, multi-sided design, features relatively easy conditions for self-balancing, Domoko told Gizmodo. The fewer sides a figure has and the smaller the angles are on each side, the harder it is to make that figure monostable, he said. Picture the common six-sided die. 'If it is a fair die, it will land on each face with equal probability,' Domoko explained. Even if someone cheats and modifies the die by putting some extra weight on a couple of surfaces, the probability will shift slightly, but it should still be possible for the die to stand on all its faces. In that sense, the tetrahedron, with its pointy corners and tiny acute angles across its four sides, makes it the 'most difficult problem, the highest category' of shapes in terms of monostability—barring some kind of engineering miracle. Which really happened. After deriving a theoretical model to calculate Bille's dimensions, Almádi, an architecture student, spearheaded the quest to build a structure that, somehow, had one side made from a 'really heavy material, the lighter parts almost air, and an almost empty skeleton,' Domokos said. The team settled on carbon tubes for the skeleton and, for the base, dense tungsten carbide—a metal alloy twice as heavy as steel. Even after all that, an issue remained: For some reason, Bille kept landing on two different sides, not the one intended side. 'Then we looked at it, and there was a very small glob of glue which was sticking to one end!' Domoko exclaimed. Despite the chief engineer's assurances that it made no difference, Domoko insisted on removing the tiny blob of glue—the density and shape of which were also calculated with ridiculous precision. And—voilà. Bille made mathematical history. That said, the engineers played a huge role in making this possible, Domokos clarified. 'They were all part of the creation process—the geometry, engineering, and technological design. They all needed to click. If you take out any of these, it doesn't work.' To make sure Bille wasn't just a one-time dud, Domokos' team succeeded in making a second model—though this probably isn't something one could easily make at home. 'We wish good luck to anyone doing it,' Domokos joked. 'But somebody doing it now has a huge advantage compared to us, because we didn't know whether it would work.' Domokos is particularly excited to see what might become of Bille further down the line. One reason Domokos didn't want to stop at merely modeling Bille was because of gömböc, he explained. Like many aesthetically pleasing mathematical breakthroughs, gömböc got a lot of love from artistic communities and natural scientists drawing parallels between turtle shells and gömböc—which Domokos more or less expected. What he didn't expect was that Novo Nordisk, in collaboration with MIT and Harvard, would take interest in gömböc's design principles for an insulin capsule that self-rights itself once inside a stomach, eliminating the need for needle injections. 'And it sounded so outlandish—like science fiction,' Domokos said. 'Gömböc taught me that physical objects are crucial—there are many bright people out there who are not mathematically minded, but they can look at something and it will reflect in their minds many other things.' Still, it'll probably be a while—if ever—before Bille ends up in the blueprint for the latest lunar lander, which Domokos knows will be extremely challenging. 'When you develop something, you have to wait and technological innovation will catch up. Sometimes it takes 100 years, sometimes it takes 10 years. Mathematics is always a little bit ahead.'
