Ein Architekturstudent hat ein seit 60 Jahren bestehendes mathematisches Rätsel gelöst und ein einseitig stabiles Tetraeder hergestellt, das immer auf der gleichen Seite aufliegt.

Throw it 100 times, and it lands on the same side 99 times.

This "geometric monster" made of carbon fiber and tungsten carbide (aerospace materials) has solved a 60-year-old mathematical mystery.

If this invention had appeared earlier, perhaps the "Athena" lunar lander wouldn't have just toppled over and stayed down (doge).

As early as 1966, mathematician John Conway and his partner Richard Guy proposed the concept of a "uniform monostatic tetrahedron."

They wanted to use a uniform material to make a tetrahedron with an even weight distribution. No matter how this tetrahedron was placed, it would always flip to its stable side facing up.

A few years later, this pair of partners, through continuous attempts, disproved the conjecture of a uniform monostatic tetrahedron: it doesn't exist.

However, what if the weight distribution is uneven?

Later, Conway speculated that a monostatic tetrahedron with uneven weights should exist, but he didn't publish any proof.

Half a century later, this mathematical conjecture was cross - over verified by architectural scholar Gergely Ambrus, and a physical model was also made.

So, how did this architectural scholar excel in this mathematical problem?

From continuous surfaces to polyhedra with sharp vertices

The great mathematician John Conway was very interested in the arrangement and balance of polyhedra.

So, he and his partner wanted to construct a tetrahedron made of a uniform material - its weight was evenly distributed, and no matter how it rolled, it would eventually end up with its stable side facing up.

Unfortunately, after several years of research, they found that such a uniform monostatic tetrahedron doesn't exist.

At this time, Conway changed his original idea: if the model is allowed to have an uneven weight distribution, does such a monostatic tetrahedron exist?

This problem is like attaching a weight under a spinning top toy, which seems feasible.

But some people think that this solution only applies to objects with smooth shapes and may not be suitable for polyhedra.

Conway thought that such a tetrahedron should exist, but he never published any proof.

Moreover, he himself was more focused on studying the balance problems of higher - dimensional, uniformly weighted tetrahedra. So, the exploration of uneven monostatic tetrahedra didn't receive wide attention for decades.

Until 2006, mathematician Gábor Domokos and his colleagues discovered the "gömböc" shape.

This shape has a very unusual characteristic - monostability.

If tossed randomly, it will always roll to its equilibrium point.

It only remains in balance at two points, one stable and the other unstable, just like the flat and side surfaces of a coin.

However, the "gömböc" is round in some places. It is a continuous surface rather than a polyhedron. Domokos wondered if a polyhedron with sharp vertices could also have similar characteristics.

Therefore, Conway's conjecture piqued his interest.

So, Domokos and his team embarked on the journey of exploring the monostatic tetrahedron.

It wasn't until one of his students appeared. Notably, this student wasn't even a math major.

Discovering the monostatic tetrahedron with the help of a computer

This student is Ambrus. He encountered this problem in one of Domokos' elective courses.

In 2022, as a student majoring in architecture, Ambrus took Domokos' mechanics course.

At the end of the semester, Domokos asked him to design a simple algorithm to explore how a tetrahedron balances.

Different from Conway's era when calculations were done with pen and paper, Ambrus, decades later, had a computer as an aid.

So, he used the computer to exhaustively search a large number of possible shapes.

Finally, Ambrus' algorithm program found the coordinates of the four vertices of a tetrahedron. When equipped with specific weights, this tetrahedron had the property of monostability.

This proved that Conway's conjecture was correct.

However, is there only one tetrahedron with this property? Are there other tetrahedra that meet the criteria? What do they have in common?

With these questions, the team used the computer to start a deeper exploration.

They realized that for a tetrahedron to achieve monostability, three connected edges (where two faces intersect) must form an obtuse angle, and the center of mass must fall within one of the four "loading zones".

Simply put, the loading zones are four small tetrahedral regions inside the monostatic tetrahedron.

When the tetrahedron topples, the center of mass moves towards one side of the supporting surface. If the center of mass always stays within the loading zone corresponding to that surface, under the action of gravity, the tetrahedron will always remain stable on that surface. If the center of mass goes beyond the loading zone, the tetrahedron may flip to another surface.

The theory has proven the existence of a monostatic tetrahedron. So, can this structure be made with real materials?

The team simulated the path of the monostatic tetrahedron from toppling to stability.

Through calculations, they found that a part of the shape needed to be constructed with a material with a density about 1.5 times that of the sun's core.

Later, they found a more feasible way. They designed a tetrahedron that was mostly hollow, made of a lightweight carbon - fiber frame, with a small part made of tungsten carbide.

To make the lighter part as light as possible, even the carbon - fiber frame had to be hollow.

After precise manufacturing and high - cost investment, Domokos' team finally completed the model, but they found that it couldn't roll as the theory predicted.

One day, Domokos and the chief engineer noticed a blob of glue sticking to one of the vertices of the model.

They asked the technician to clean it off. Twenty minutes later, Ambrus received a text message from Domokos: "It works."

This made Ambrus, who was out for a walk, start jumping up and down on the street. He excitedly said:

"We designed it, and it works. This is just amazing!"

From manual problem - solving to computer - assisted

John Conway, the proposer of the monostatic tetrahedron conjecture, was focused on studying high - dimensional geometric balance problems.

Fast calculation was one of Conway's signature traits.

In the 19th century, three scientists from the UK and the US created a periodic table of "knots" (an abstract geometric concept, which can be understood as a non - self - intersecting closed curve in three - dimensional space, similar to a knot tied with a rope in real life). In six years, they completed the classification of the first 54 types of "knots."

In 1970, Conway proposed a more efficient method in a paper to do the same thing. It is said that he only took an afternoon.

His method is called "Conway notation," which makes it much easier to draw the tangles and overlaps in "knots."

Someone once said that he was the only mathematician who could solve problems with his own hands, achieving amazing results without the help of tools.

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Decades later, people have computers to assist in solving complex problems.

Architectural scholar Ambrus' cross - over solution to this mathematical problem couldn't have been achieved without the help of a computer.

As netizens said, the shape of this monostatic tetrahedron was calculated and screened out by the computer.

Without the computer's help in the search, perhaps this problem would have persisted for another 60 years.

After completing this mathematical work, Ambrus humorously said:

"I originally wanted to be an architect. Why am I here?"

Perhaps the answer is the computer ~

Currently, Domokos and Ambrus are working on applying this result to aerospace, such as designing a lunar lander that can automatically return to its upright position after toppling.

Paper link: https://arxiv.org/abs/2506.19244

Reference links:

[1]https://www.quantamagazine.org/a-new-pyramid-like-shape-always-lands-the-same-side-up-20250625/

[2]https://news.ycombinator.com/item?id=44381297

[3]https://www.quantamagazine.org/john-conway-solved-mathematical-problems-with-his-bare-hands-20200420/

This article is from the WeChat public account "QbitAI". Author: Wen Le, Shiling. Republished by 36Kr with permission.