The answer to how “constant-width shapes” behave in higher dimensions lies in a surprisingly simple construct
In 1986, when the Space Shuttle Challenger exploded after 73 seconds of flight, the eminent physicist Richard Feynman was called in to find out what had happened. He later demonstrated that the “ring-shaped” seals, which were supposed to join the sections of the Space Shuttle’s rocket boosters, had failed due to low temperatures, with catastrophic results. But he also discovered other errors.
Among these was how the NASA I had calculated the shape of the rings. During pre-flight testing, agency engineers repeatedly measured the width of the seals to see if they had deformed. They argued that if the rings flattened slightly – if they became, for example, oval, rather than maintaining their circular shape – then they would no longer be the same diameter all around.
These measurements, Feynman wrote, were useless. Even if engineers took an infinite number of measurements and found that the diameter was exactly the same every time, there are many “constant-width bodies,” as these shapes are called. The circle is just one of them.
The best-known constant-width noncircular body is the Reuleaux triangle, which you can construct by taking the central region of the overlap of a three-circle Venn diagram. For a given width in two dimensions, the Reuleaux triangle is the shape of constant width with the smallest possible area. A circle has the largest possible area.
In three dimensions, the largest body of constant width is a ball. In higher dimensions, it is simply a higher-dimensional ball, the shape that appears when you hold a needle in one point and let it spin freely in all directions.
But mathematicians have long wondered whether it is always possible to find smaller shapes of constant width in higher dimensions. These shapes exist in three dimensions: Although these Reuleaux triangle-shaped bubbles may seem a bit pointy, if you place them between two parallel planes, they will roll smoothly, like a ball. But it is much harder to say whether this is true in general. It may be that, in the higher dimensions, the ball is ideal.
So, in 1988, Oded Schramm, then a graduate student at Princeton University, posed a simple question: Is it possible to build a body of constant width in any dimension that is exponentially smaller than the ball?
Now, in a paper published online in May, five researchers – four of whom grew up in Ukraine and have known each other since high school or college – report that the answer is yes.
The result not only solves a decades-old problem, but gives mathematicians a first look at what these mysterious higher-dimensional shapes might look like. While these shapes are easy to define, they are surprisingly mysterious, said Shiri Artstein, a mathematician at Tel Aviv University who was not involved in the work. “Everything we learn about them, any new construction or calculation, is already very interesting.” Now researchers can finally access a corner of the geometric universe that was previously completely inaccessible.
Since then, the four mathematicians have moved to different cities around the world and pursued different research programs, but meet twice a week via Zoom to work together on difficult geometric proofs.
Constant width shapes were not the order of the day. Last year, the group was trying to answer a question called the Borsuk Conjecture, which has perplexed mathematicians for more than a century. But one idea kept reappearing in meetings: When Schramm posed his question about bodies of constant width in the 1980s, he also suggested that understanding these shapes might provide a way to solve Borsuk’s problem.
Ukrainian mathematicians had taken a different approach, and some of them were reluctant to change focus. But Bondarenko, now at the Norwegian University of Science and Technology, insisted they try, even if it didn’t help much. “He always emphasized that the problem was important in itself,” said Arman, who is now a postdoctoral researcher at the University of Manitoba. Eventually, the rest of the team agreed to give it a try.
Their work provides a surprisingly simple algorithm for constructing an n-dimensional shape of constant width whose volume is at most 0.9n times that of the ball. This limit is somewhat arbitrary, Arman said. It should be possible to find even smaller bodies of constant width. But this is enough to answer Schramm’s question, demonstrating that as the number of dimensions increases, the difference between the volumes of the smallest and largest bodies of constant width grows exponentially. Despite the complex ideas behind the result, Arman said its construction is something any college student can verify.
Move forward
For Hebrew University’s Gil Kalai, there is personal satisfaction in seeing a response for Schramm, his former student, who died in a 2008 mountain hiking accident, after having made significant progress on issues in many different fields. But Kalai is also keen to explore the theoretical consequences of the result. Until then, he said, it was possible that in higher dimensions these shapes simply behaved like balls, at least as far as the volume property was concerned. But “that’s not what happens. So this means that the theory of these bodies in higher dimensions is very complex,” he said.
This theory may also have applications. After all, in the smaller dimensions, bodies of constant width are already surprisingly useful: the Reuleaux triangle, for example, appears in the form of drill bits, guitar picks and tamper-proof nuts for fire hydrants. According to Arman, in higher dimensions, their new shapes could be useful in developing machine learning methods to analyze higher-dimensional datasets. Bondarenko – known in the group for what Arman calls “crazy ideas” – also proposed links to distant branches of mathematics.
The search for the smallest possible body of constant width continues. The team used their construction to investigate a promising candidate in three dimensions, but they were disappointed: it turned out to be a tiny fraction of 1% larger than the smallest known body. For now the mathematicians have decided to abandon the hunt and return to work on the Borsuk problem. In the process, they left behind a world of new forms for others to explore. / TRANSLATION BY RENATO PRELORENTZOU
Original story republished with permission from Quanta Magazine, an editorially independent publication supported by the Simons Foundation. Read the original content on Mathematicians discover new shapes to solve decades-old geometry problems
Source: Terra
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