In 2023, Domokos predicted that it is indeed possible to distribute the weight of the tetrahedral so that it can only sit on one face. At least in theory.
But Almádi, Dawson and Domokos want to build this thing, and the task proved to be more challenging than they expected. Now, in a preprint released online yesterday, they introduce the first working physics model of the shape. The tetrahedron weighs 120 grams and measures 50 cm along the longest side, and is made of lightweight carbon fiber and dense water-carbide carbonate. To work, it must be designed to within one gram and one tenth of a millimeter. However, the final construction is always dragged onto a face the way it should be.
This work proves that experiments and plays an important role in studying mathematics. It also has potential practical applications, such as in the design of self-implying spacecraft.
“I didn’t expect more work on the tetrahedron,” Papu said. However, he added that the team’s research allowed mathematicians to “very well understand how much we don’t know and how thorough we understand.”
Critical point
In 2022, Almádi was an undergraduate aspiring to become an architect and took a mechanics course in Domokos. He didn’t say much, but Domokos saw a hardworking worker in him, who he often thought deeply. At the end of the semester, Domokos asked him to fabricate a simple algorithm to explore the equilibrium of the tetrahedron.
When Conway initially asked his question, his only option was to prove the existence of unit tetrahedrons through abstract mathematical reasoning using pencil and paper. It is almost very difficult to identify a specific example. But Almádi, who worked decades later, owned a computer. He can do brute force searches through a large number of possible shapes. Finally, Almádi’s plan found the coordinates of the four vertices of the tetrahedron, which could be made single if some weight distribution was assigned. Conway is right.
Almádi found a single tetrahedron, but presumably there were other tetrahedrons. What property did they share?
While this seems to be a simple question, “A statement like ‘tetrahedrons are single statements’ cannot be easily described with simple formulas or a small portion of equations.”
The team realized that in any single tetrahedron, three consecutive edges (face-to-face pairs) need to form obtuse angles, which is one of over 90 degrees. This will ensure that one face hangs on the other side so that it can tip over.
Then, mathematicians showed that any tetrahedron with this feature could be united if its mass center is located in one of the four “loading areas” (four smaller tetrahedral regions within the original shape). As long as the mass center falls within the loading area, the tetrahedron will be balanced on only one face.
