In 2023, Domokos—alongside along with his graduate college students Gergő Almádi and Krisztina Regős, and Robert Dawson of Saint Mary’s College in Canada—proved that it’s certainly potential to distribute a tetrahedron’s weight so that it’s going to sit on only one face. No less than in concept.
However Almádi, Dawson, and Domokos needed to construct the factor, a activity that turned out to be far more difficult than they anticipated. Now, in a preprint posted on-line yesterday, they’ve offered the first working bodily mannequin of the form. The tetrahedron, which weighs 120 grams and measures 50 centimeters alongside its longest aspect, is manufactured from light-weight carbon fiber and dense tungsten carbide. To work, it needed to be engineered to a stage of precision inside one-tenth of a gram and one-tenth of a millimeter. However the ultimate development all the time flip-flops onto one face, precisely because it ought to.
The work demonstrates the essential function of experimentation and play in analysis arithmetic. It additionally has potential sensible purposes, akin to within the design of self-righting spacecraft.
“I didn’t anticipate extra work to return out on tetrahedra,” Papp mentioned. And but, he added, the workforce’s analysis permits mathematicians to “actually recognize how a lot we didn’t know and the way thorough our understanding is now.”
Tipping Level
In 2022, Almádi, then an undergraduate aspiring to change into an architect, enrolled in Domokos’ mechanics course. He didn’t say a lot, however Domokos noticed in him a tough employee who was continuously in deep thought. On the finish of the semester, Domokos requested him to concoct a easy algorithm to discover how tetrahedra stability.
When Conway initially posed his drawback, his solely possibility would have been to make use of pencil and paper to show, via summary mathematical reasoning, that monostable tetrahedra exist. It will have been virtually prohibitively tough to pinpoint a concrete instance. However Almádi, working a long time later, had computer systems. He may do a brute-force search via an enormous variety of potential shapes. Finally, Almádi’s program discovered the coordinates for the 4 vertices of a tetrahedron that, when assigned sure weight distributions, could possibly be made monostable. Conway was proper.
Almádi discovered one monostable tetrahedron, however presumably there have been others. What properties did they share?
Whereas that may appear to be a easy query, “a press release like ‘A tetrahedron is monostable’ can’t be simply described with a easy components or a small set of equations,” Papp mentioned.
The workforce realized that in any monostable tetrahedron, three consecutive edges (the place pairs of faces meet) would want to kind obtuse angles—ones that measure over 90 levels. That might be certain that one face would dangle over one other, permitting it to tip over.
The mathematicians then confirmed that any tetrahedron with this characteristic may be made monostable if its middle of mass is positioned inside certainly one of 4 “loading zones”—a lot smaller tetrahedral areas inside the authentic form. As long as the middle of mass falls inside a loading zone, the tetrahedron will stability on just one face.
