A New Proof Moves the Needle on a Sticky Geometry Problem

The original version of this story appeared in Quanta Magazine.

In 1917, the Japanese mathematician Sōichi Kakeya posed what at first seemed like nothing more than a fun exercise in geometry. Lay an infinitely thin, inch-long needle on a flat surface, then rotate it so that it points in every direction in turn. What’s the smallest area the needle can sweep out?

If you simply spin it around its center, you’ll get a circle. But it’s possible to move the needle in inventive ways, so that you carve out a much smaller amount of space. Mathematicians have since posed a related version of this question, called the Kakeya conjecture. In their attempts to solve it, they have uncovered surprising connections to harmonic analysis, number theory, and even physics.

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

But it’s also something that mathematicians still don’t fully understand. In the past few years, they’ve proved variations of the Kakeya conjecture in easier settings, but the question remains unsolved in normal, three-dimensional space. For some time, it seemed as if all progress had stalled on that version of the conjecture, even though it has numerous mathematical consequences.

Now, two mathematicians have moved the needle, so to speak. Their new proof strikes down a major obstacle that has stood for decades—rekindling hope that a solution might finally be in sight.

What’s the Small Deal?

Kakeya was interested in sets in the plane that contain a line segment of length 1 in every direction. There are many examples of such sets, the simplest being a disk with a diameter of 1. Kakeya wanted to know what the smallest such set would look like.

He proposed a triangle with slightly caved-in sides, called a deltoid, which has half the area of the disk. It turned out, however, that it’s possible to do much, much better.

The deltoid to the right is half the size of the circle, though both needles rotate through every direction.Video: Merrill Sherman/Quanta Magazine

In 1919, just a couple of years after Kakeya posed his problem, the Russian mathematician Abram Besicovitch showed that if you arrange your needles in a very particular way, you can construct a thorny-looking set that has an arbitrarily small area. (Due to World War I and the Russian Revolution, his result wouldn’t reach the rest of the mathematical world for a number of years.)

To see how this might work, take a triangle and split it along its base into thinner triangular pieces. Then slide those pieces around so that they overlap as much as possible but protrude in slightly different directions. By repeating the process over and over again—subdividing your triangle into thinner and thinner fragments and carefully rearranging them in space—you can make your set as small as you want. In the infinite limit, you can obtain a set that mathematically has no area but can still, paradoxically, accommodate a needle pointing in any direction.

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

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