The authentic model of this story appeared in Quanta Magazine.

In 1917, the Japanese mathematician Sōichi Kakeya posed what at first appeared like nothing greater than a enjoyable train in geometry. Lay an infinitely skinny, inch-long needle on a flat floor, then rotate it in order that it factors in each path in flip. What’s the smallest space the needle can sweep out?

If you merely spin it round its heart, you’ll get a circle. But it’s potential to maneuver the needle in ingenious methods, so that you just carve out a a lot smaller quantity of area. Mathematicians have since posed a associated model of this query, referred to as the Kakeya conjecture. In their makes an attempt to unravel it, they’ve uncovered surprising connections to harmonic analysis, quantity concept, and even physics.

“Somehow, this geometry of lines pointing in many different directions is ubiquitous in a large portion of mathematics,” stated Jonathan Hickman of the University of Edinburgh.

But it’s additionally one thing that mathematicians nonetheless don’t totally perceive. In the previous few years, they’ve proved variations of the Kakeya conjecture in easier settings, however the query stays unsolved in regular, three-dimensional area. For a while, it appeared as if all progress had stalled on that model of the conjecture, regardless that it has quite a few mathematical penalties.

Now, two mathematicians have moved the needle, so to talk. Their new proof strikes down a major obstacle that has stood for many years—rekindling hope {that a} answer may lastly be in sight.

What’s the Small Deal?

Kakeya was considering units within the airplane that include a line section of size 1 in each path. There are many examples of such units, the best being a disk with a diameter of 1. Kakeya needed to know what the smallest such set would seem like.

He proposed a triangle with barely caved-in sides, referred to as a deltoid, which has half the world of the disk. It turned out, nonetheless, that it’s potential to do a lot, significantly better.

In 1919, simply a few years after Kakeya posed his downside, the Russian mathematician Abram Besicovitch confirmed that when you organize your needles in a really explicit method, you’ll be able to assemble a thorny-looking set that has an arbitrarily small space. (Due to World War I and the Russian Revolution, his consequence wouldn’t attain the remainder of the mathematical world for quite a lot of years.)

To see how this may work, take a triangle and cut up it alongside its base into thinner triangular items. Then slide these items round in order that they overlap as a lot as potential however protrude in barely totally different instructions. By repeating the method over and over—subdividing your triangle into thinner and thinner fragments and punctiliously rearranging them in area—you can also make your set as small as you need. In the infinite restrict, you’ll be able to receive a set that mathematically has no space however can nonetheless, paradoxically, accommodate a needle pointing in any path.

“That’s kind of surprising and counterintuitive,” stated Ruixiang Zhang of the University of California, Berkeley. “It’s a set that’s very pathological.”