A New Pyramid-Like Shape Always Lands the Same Side Up
Introduction
In 360 BCE, 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.
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 (left) and Robert Dawson helped discover new properties of tetrahedra.
Courtesy of Krisztina Regős; Ms. Tara Inman
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.
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.
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.”
