This New Pyramid-Like Shape Always Lands With the Same Side Up

In 2023, Domokos—along pinch his postgraduate students Gergő Almádi and Krisztina Regős, and Robert Dawson of Saint Mary’s University successful Canada—proved that it is so imaginable to administer a tetrahedron’s weight truthful that it will beryllium connected conscionable 1 face. At slightest successful theory.

But Almádi, Dawson, and Domokos wanted to build nan thing, a task that turned retired to beryllium acold much challenging than they expected. Now, successful a preprint posted online yesterday, they person presented nan first moving beingness model of nan shape. The tetrahedron, which weighs 120 grams and measures 50 centimeters on its longest side, is made of lightweight c fibre and dense tungsten carbide. To work, it had to beryllium engineered to a level of precision wrong one-tenth of a gram and one-tenth of a millimeter. But nan last building ever flip-flops onto 1 face, precisely arsenic it should.

The activity demonstrates nan important domiciled of experimentation and play successful investigation mathematics. It besides has imaginable applicable applications, specified arsenic successful nan creation of self-righting spacecraft.

“I didn’t expect much activity to travel retired connected tetrahedra,” Papp said. And yet, he added, nan team’s investigation allows mathematicians to “really admit really overmuch we didn’t cognize and really thorough our knowing is now.”

Tipping Point

In 2022, Almádi, past an undergraduate aspiring to go an architect, enrolled successful Domokos’ mechanics course. He didn’t opportunity much, but Domokos saw successful him a difficult worker who was perpetually successful heavy thought. At nan extremity of nan semester, Domokos asked him to concoct a elemental algorithm to research really tetrahedra balance.

When Conway primitively posed his problem, his only action would person been to usage pencil and insubstantial to prove, done absurd mathematical reasoning, that monostable tetrahedra exist. It would person been almost prohibitively difficult to pinpoint a actual example. But Almádi, moving decades later, had computers. He could do a brute-force hunt done a immense number of imaginable shapes. Eventually, Almádi’s programme recovered nan coordinates for nan 4 vertices of a tetrahedron that, erstwhile assigned definite weight distributions, could beryllium made monostable. Conway was right.

Almádi recovered 1 monostable tetrahedron, but presumably location were others. What properties did they share?

While that mightiness look for illustration a elemental question, “a connection for illustration ‘A tetrahedron is monostable’ cannot beryllium easy described pinch a elemental look aliases a mini group of equations,” Papp said.

The squad realized that successful immoderate monostable tetrahedron, 3 consecutive edges (where pairs of faces meet) would request to shape obtuse angles—ones that measurement complete 90 degrees. That would guarantee that 1 look would bent complete another, allowing it to extremity over.

The mathematicians past showed that immoderate tetrahedron pinch this characteristic tin beryllium made monostable if its halfway of wide is positioned wrong 1 of 4 “loading zones”—much smaller tetrahedral regions wrong nan original shape. So agelong arsenic nan halfway of wide falls wrong a loading zone, nan tetrahedron will equilibrium connected only 1 face.