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 attainable to distribute a tetrahedron’s weight so that it’s going to sit on only one face. Not less than in concept.
However Almádi, Dawson, and Domokos wished to construct the factor, a job 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 facet, is fabricated 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 last development at all times flip-flops onto one face, precisely because it ought to.
The work demonstrates the essential position of experimentation and play in analysis arithmetic. It additionally has potential sensible functions, comparable 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 group’s analysis permits mathematicians to “actually respect 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 develop into an architect, enrolled in Domokos’ mechanics course. He didn’t say a lot, however Domokos noticed in him a tough employee who was continually in deep thought. On the finish of the semester, Domokos requested him to concoct a easy algorithm to discover how tetrahedra steadiness.
When Conway initially posed his downside, 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 attainable shapes. Ultimately, 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, “a press release like ‘A tetrahedron is monostable’ can’t be simply described with a easy method or a small set of equations,” Papp mentioned.
The group realized that in any monostable tetrahedron, three consecutive edges (the place pairs of faces meet) would wish to kind obtuse angles—ones that measure over 90 levels. That will be certain that one face would cling over one other, permitting it to tip over.
The mathematicians then confirmed that any tetrahedron with this characteristic will be made monostable if its middle of mass is positioned inside one in every 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 steadiness on just one face.
