A New Proof Strikes the Needle on a Sticky Geometry Drawback

The unique model of this story appeared in Quanta Journal.

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 middle, you’ll get a circle. Nevertheless it’s doable 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 resolve it, they’ve uncovered stunning connections to harmonic evaluation, quantity principle, and even physics.

“By some means, this geometry of strains pointing in many alternative instructions is ubiquitous in a big portion of arithmetic,” mentioned Jonathan Hickman of the College of Edinburgh.

Nevertheless it’s additionally one thing that mathematicians nonetheless don’t absolutely perceive. Previously 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, 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 serious impediment that has stood for many years—rekindling hope {that a} resolution would possibly lastly be in sight.

What’s the Small Deal?

Kakeya was interested by units within the airplane that include a line section of size 1 in each route. There are lots 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 doable to do a lot, significantly better.

In 1919, simply a few years after Kakeya posed his downside, the Russian mathematician Abram Besicovitch confirmed that for those who organize your needles in a really explicit manner, you possibly can assemble a thorny-looking set that has an arbitrarily small space. (As a result of World Warfare I and the Russian Revolution, his end result wouldn’t attain the remainder of the mathematical world for a variety of years.)

To see how this would possibly 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 doable however protrude in barely totally different instructions. By repeating the method over and over—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 possibly can acquire a set that mathematically has no space however can nonetheless, paradoxically, accommodate a needle pointing in any route.

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