Mathematicians create a tetrahedron that always lands on the same side

A form with four sides which will always rest on the same side, whatever the side to which it begins was built by mathematicians, decades after the first proposal to exist.

Mathematicians have long been fascinated by autonomous “monostable” forms, which have a favorite rest point when placed on a flat surface. A famous example is the Gömböc, a curved object in the shape of a turtle shell which has a precise weight distribution and will swing from one side to the other until it reaches the same stable rest place.

In 1966, the mathematician John Conway worked on the way in which the balance of forms with straight edges and proved that a form on four sides, or tetrahedron, with a uniform distribution of the mass would be impossible. However, he said to his colleagues at the time that a tetrahedron unalcomed unevenly balanced could be possible, but never proven it.

Now Gábor Domokos at Budapest University of Technology and Economy, Hungary, and his colleagues have built a single tetrahedron, which they call the ball, using carbon legs and a ultra-denounced tungsten carbide plate. The name comes from the Hungarian word for TIP, Billen.

They first started working on the problem when Domokos asked his student, Gergő Almádi, to search for the Tetrahedron in Conway by searching for brute force with powerful computers. “You check each tetrahedron, and with luck, you find it, or with time, or with [computing power]Or a mixture of these, ”explains Domokos.

As Conway predicted it, they found no single tetrahedron with a distribution of uniform weight, but they found unequal candidates and continued to prove their existence mathematically.

Domokos and his team wanted to build a real example, but it turned out to be “a more difficult order of magnitude,” he said. Indeed, according to their calculations, the difference between the density of the weighted and not weighted parts of the objects was to be approximately 5000 times, which means that the object should be essentially made from air but always rigid.

To make the form, Domokos and his team have joined an engineering company and spent thousands of euros to specifically design the carbon fiber forces at a tenth of a millimeter and make the basic tungsten plate less than a tenth gram.

When Domokos saw the functioning of the ball in real life, he felt that he “levied 1 meter above the ground,” he said. “It is a great pleasure to know that you have achieved something that would make John Conway happy.”

“There is no model, previous example or nothing in the nature that [have suggested to Conway] that this form exists, “says Domokos.” It was in such an obscure corner of reality that no human [could] Reach it “so far,” when you have powerful computers and you are ready to pay thousands of dollars “.

The shape they built on a specific tilting path between its sides, explains Domokos, based from B to A, C to A, and D to C and C to A. There is another type of single tetrahedron which indicates that they would need a material which is once and dense than the sun.

Domokos hopes that their work will help engineers modify the geometry of lunar moles to make them less likely to fall, as several recent spacecrafts have done so. “If you can do it with four sides, you can do it with any other number of faces.”