The original version on this story shows the Quanta Magazine.
In 1917, the Japanese mathematician Sōichi Kakeya demonstrated what at first seemed like a fun exercise in geometry. Place an infinitely thin, inch-long needle on a flat surface, then rotate it so that it points in every direction. What is the smallest area the needle can sweep?
If you just rotate it in the middle, you get a circle. But it’s possible to move the needle in inventive ways, so you can carve out a smaller space. Mathematicians have since presented a related version of this question, called the Kakeya conjecture. In their attempts to solve it, they discovered surprising connections with harmonic analysis, number theory, and even physics.
“In a way, this geometry of lines pointing in many different directions is in many areas of mathematics,” says Jonathan Hickman of the University of Edinburgh.
But this is also something that mathematicians still do not understand. In the last few years, they have proven variations of the Kakeya hypothesis in simpler settings, but the question remains unresolved in normal, three-dimensional space. For some time, it seemed that all progress had stalled on that version of the hypothesis, even though it had great mathematical results.
Today, two mathematicians are moving the needle, so to speak. Their new proof overcomes a major obstacle that has stood for decades—reviving hope that a solution may finally be found.
What’s a Small Deal?
Kakeya is interested in sets of planes that have a line segment of length 1 in each direction. There are many examples of such sets, the simplest being a disk with a diameter of 1. Kakeya wants to know what the smallest set looks like.
He proposed a triangle with slightly concave sides, called the deltoid, with half the area of the disk. It turns out, however, that it is possible to do much, much better.
The deltoid on the right is half the size of the circle, although both needles rotate in all directions.Video: Merrill Sherman/Quanta Magazine
In 1919, just a few years after Kakeya stated his problem, the Russian mathematician Abram Besicovitch showed that if you arranged your needles in a particular way, you could make a thorny tan- this set has a small area. (Because of World War I and the Russian Revolution, his results did not reach the entire mathematical world for many years.)
To see how this works, take a triangle and divide it at its base into thinner triangular pieces. Then slide the pieces around so they overlap as much as possible but come out in slightly different directions. By repeating the process—dividing your triangle into thinner and thinner pieces and carefully arranging them in space—you can make your set as small as you like. In the infinite limit, you can get a set that mathematically has no place but can still, paradoxically, accommodate a needle pointing in any direction.
“That’s strange and counterintuitive,” said Ruixiang Zhang of the University of California, Berkeley. “It’s a very pathological set.”
