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?

In the event you merely spin it round its heart, you’ll get a circle. However it’s attainable to maneuver the needle in creative methods, so that you simply carve out a a lot smaller quantity of area. Mathematicians have since posed a associated model of this query, known as the Kakeya conjecture. Of their makes an attempt to unravel it, they’ve uncovered surprising connections to harmonic analysis, quantity idea, and even physics.

“Someway, this geometry of traces pointing in many various instructions is ubiquitous in a big portion of arithmetic,” mentioned Jonathan Hickman of the College of Edinburgh.

However it’s additionally one thing that mathematicians nonetheless don’t totally perceive. Previously 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, although 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} resolution would possibly lastly be in sight.

What’s the Small Deal?

Kakeya was fascinated with units within the aircraft that comprise a line section of size 1 in each route. There are a lot of examples of such units, the best being a disk with a diameter of 1. Kakeya needed to know what the smallest such set would appear like.

He proposed a triangle with barely caved-in sides, known as a deltoid, which has half the world of the disk. It turned out, nonetheless, that it’s attainable to do a lot, a lot 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 specific manner, you may assemble a thorny-looking set that has an arbitrarily small space. (As a result of World Battle I and the Russian Revolution, his outcome wouldn’t attain the remainder of the mathematical world for quite a lot of years.)

To see how this would possibly 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 attainable however protrude in barely totally different instructions. By repeating the method again and again—subdividing your triangle into thinner and thinner fragments and thoroughly rearranging them in area—you can also make your set as small as you need. Within the infinite restrict, you may get hold of a set that mathematically has no space however can nonetheless, paradoxically, accommodate a needle pointing in any route.

“That’s sort of shocking and counterintuitive,” mentioned Ruixiang Zhang of the College of California, Berkeley. “It’s a set that’s very pathological.”