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 doable to distribute a tetrahedron’s weight so that it’ll sit on only one face. At the least in concept.
However Almádi, Dawson, and Domokos wished to construct the factor, a job that turned out to be far tougher than they anticipated. Now, in a preprint posted on-line yesterday, they’ve offered the first working physical model of the form. The tetrahedron, which weighs 120 grams and measures 50 centimeters alongside its longest facet, is fabricated from light-weight carbon fiber and dense tungsten carbide. To work, it needed to be engineered to a degree of precision inside one-tenth of a gram and one-tenth of a millimeter. However the remaining building at all times flip-flops onto one face, precisely because it ought to.
The work demonstrates the vital function of experimentation and play in analysis arithmetic. It additionally has potential sensible functions, similar to within the design of self-righting spacecraft.
“I didn’t anticipate extra work to come back out on tetrahedra,” Papp mentioned. And but, he added, the workforce’s analysis permits mathematicians to “actually admire 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 turn into an architect, enrolled in Domokos’ mechanics course. He didn’t say a lot, however Domokos noticed in him a tough employee who was always 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 choice would have been to make use of pencil and paper to show, by summary mathematical reasoning, that monostable tetrahedra exist. It could have been nearly prohibitively troublesome to pinpoint a concrete instance. However Almádi, working many years later, had computer systems. He might do a brute-force search by an enormous variety of doable shapes. Finally, Almádi’s program discovered the coordinates for the 4 vertices of a tetrahedron that, when assigned sure weight distributions, might 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 seem to be a easy query, “an announcement like ‘A tetrahedron is monostable’ can’t be simply described with a easy formulation 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 wish to type obtuse angles—ones that measure over 90 levels. That might be sure that one face would cling over one other, permitting it to tip over.
The mathematicians then confirmed that any tetrahedron with this function could be made monostable if its heart 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.
