In 2023, Domokos – together with his students graduated Gargő Almádi and Krisztina Regős e Robert Dawson Of the Saint Mary’s University in Canada, proper to it is really possible to distribute the weight of a tetrahedron so that it sit on one face. At least in theory.
But Almádi, Dawson and Domokos wanted to build the thing, a task that proved to be much more demanding than they expected. Now, in a preprint published online yesterday, they presented the First functioning physical model of the form. The tetrahedron, which weighs 120 grams and measures 50 centimeters along its longest side, is made of light carbon fiber and dense tungsten carbide. To work, it was to be designed at a level of precision within a tenth by one gram and a tenth of a millimeter. But the final construction always puts on a face, exactly as it should.
The work demonstrates the important role of experimentation and the game in research mathematics. It also has potential practical applications, such as in the design of space vehicles.
“I didn’t expect more work to be released on Tetraedri,” said Papp. Yet, he added, the search for the team allows the mathematicians to “really appreciate how much we didn’t know and how deepened our understanding is now”.
Turning point
In 2022, Almádi, therefore a diplant who aspires to become an architect, enrolled in the Domokos mechanics course. He didn’t say much, but Domokos saw a great worker in him that was constantly in deep thought. At the end of the semester, Domokos asked him to invent a simple algorithm to explore like the balance of tetraedri.
When Conway originally posed his problem, his only option would have been to use the pencil and paper to demonstrate, through abstract mathematical reasoning, that there are single -protective tetrahedra. It would have been almost prohibitively difficult to identify a concrete example. But Almádi, who worked decades later, had computers. He could do a research with brute force through a huge number of possible forms. In the end, Almádi’s program found the coordinates for the four leaders of a tetrahedron who, when assigned some weight distributions, could be rendered unique. Conway was right.
Almádi found a single -mostable tetrahedron, but presumably there were others. What properties did they share?
Although this may seem like a simple question, “a statement like” a tetrahedron is unique “cannot be easily described with a simple formula or a small series of equations,” said Papp.
The team realized that in any monostable tetrahedron, three consecutive edges (where couples of faces are encountered) would have had to form obtuse corners, between measuring over 90 degrees. This would guarantee that one face would hang on another, allowing it to overturn.
The mathematicians have therefore shown that any tetrahedron with this function can be made unique if its mass center is positioned within one of the four “loading areas”, many smaller tetrahedral regions within the original shape. Until the center of the mass falls into a loading area, the tetrahedron will balance on one face.
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