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’ll sit on only one face. At the least in idea.
However Almádi, Dawson, and Domokos needed 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 introduced 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 degree of precision inside one-tenth of a gram and one-tenth of a millimeter. However the remaining development all the time flip-flops onto one face, precisely because it ought to.
The work demonstrates the necessary position of experimentation and play in analysis arithmetic. It additionally has potential sensible functions, equivalent 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 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 out to be an architect, enrolled in Domokos’ mechanics course. He didn’t say a lot, however Domokos noticed in him a tough employee who was consistently 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 way of summary mathematical reasoning, that monostable tetrahedra exist. It will have been virtually prohibitively troublesome to pinpoint a concrete instance. However Almádi, working many years later, had computer systems. He might do a brute-force search by way of 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, 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 which may appear to be a easy query, “an announcement 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 want to type obtuse angles—ones that measure over 90 levels. That may make sure that one face would cling over one other, permitting it to tip over.
The mathematicians then confirmed that any tetrahedron with this function will be made monostable if its middle of mass is positioned inside one in all 4 “loading zones”—a lot smaller tetrahedral areas throughout the authentic form. As long as the middle of mass falls inside a loading zone, the tetrahedron will stability on just one face.
