A New Proof Moves the Needle on a Sticky Geometry Problem

The unique 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 route in flip. What’s the smallest space the needle can sweep out?

If you merely spin it round its middle, you’ll get a circle. But it’s attainable 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, known as the Kakeya conjecture. In their makes an attempt to unravel it, they’ve uncovered stunning connections to harmonic evaluation, quantity principle, and even physics.

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

But it’s additionally one thing that mathematicians nonetheless don’t absolutely perceive. In the previous few years, they’ve proved variations of the Kakeya conjecture in simpler 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, despite the fact that it has quite a few mathematical penalties.

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

What’s the Small Deal?

Kakeya was excited by units in the airplane that include a line phase of size 1 in each route. There are many examples of such units, the easiest being a disk with a diameter of 1. Kakeya needed to know what the smallest such set would appear to be.

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

In 1919, simply a couple of years after Kakeya posed his downside, the Russian mathematician Abram Besicovitch confirmed that in case you prepare your needles in a very specific method, you possibly can 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 a variety of years.)

To see how this may work, take a triangle and break up it alongside its base into thinner triangular items. Then slide these items round in order that they overlap as a lot as attainable however protrude in barely totally different instructions. By repeating the course of time and again—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 possibly can get hold of a set that mathematically has no space however can nonetheless, paradoxically, accommodate a needle pointing in any route.

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