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.
Gábor Domokos discovers and builds new shapes to understand the world around us. 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.