original version of this story Appeared in Quanta Magazine.
In 1917, Japanese mathematician Soichi Kakeya proposed what at first seemed like a fun exercise in geometry. An infinitely thin, 1-inch long needle is placed on a flat surface and rotated in order to point in all directions. What is the smallest area that the needle can clean?
Rotating around the center will give you a circle. But it’s possible to get creative and move the needle to carve out smaller spaces. Since then, mathematicians have presented a related version of this question, called the Kakeya conjecture. In an attempt to solve it, they revealed the following: Surprising relationship with harmonic analysisnumber theory, and even physics.
“Somehow this geometry of lines pointing in different directions is ubiquitous in much of mathematics,” he said. Jonathan Hickman PhD from the University of Edinburgh.
But it’s also something mathematicians still don’t fully understand.In the past few years they have proven variations of the Kakeya conjecture With easier settings, but this problem remains unsolved in normal three-dimensional space. For a while, all progress seemed to have stalled on this version of the conjecture, even though it yielded many mathematical results.
Now, two mathematicians have moved the needle, so to speak.their new evidence break through big obstacles This situation has been going on for decades, and there is renewed hope that a solution may finally be in sight.
What is a small deal?
Mr. Kaketani was interested in the set in a plane that contains line segments of length 1 in every direction. There are many examples of such sets, the simplest of which is a disk of diameter 1. Kakeya wanted to know what the smallest such set would be.
He proposed a triangle with slightly concave sides, called the deltoid, with half the area of the disc. But it turns out we can do much, much better.
The right deltoid is half the size of a circle, but both needles rotate in all directions.Video: Meryl Sherman/Quanta Magazine
In 1919, just a few years after Kaketani posed the problem, the Russian mathematician Abram Veshkovich proposed that if the needles were arranged in a very special way, they could be made into barbed aggregates with arbitrarily small areas. We have shown that it is possible to construct (Because of World War I and the Russian Revolution, his results did not reach the rest of the mathematical community for many years.)
To see how this works, take a triangle and divide it into thin triangular pieces along its base. Next, slide these pieces so that they overlap as much as possible, but stick out in slightly different directions. By repeating this process many times, you can make the set as small as you want by subdividing the triangles into thinner fragments and carefully rearranging them in space. In the limit of infinity, we can obtain a set that mathematically has no area, but which, paradoxically, can accommodate a needle pointing in any direction.
“It’s kind of surprising and counterintuitive,” he said. Zhang Ruixiang from the University of California, Berkeley. “It’s a very sick set.”
