What Is Kakeya's Needle Problem, And Why Do We Want To Solve It?

It is a truth universally acknowledged among mathematicians that some of the most challenging problems are those that, on paper, sound incredibly simple. Take Fermat’s Last Theorem, for example: the statement fits into almost a single sentence, but a proof took more than 350 years and the development of a handful of brand-new areas of math to complete.

Measured against that, the Kakeya conjecture – a problem stemming from a 1917 thought experiment by Japanese mathematician Sōichi Kakeya – may not seem too impressive. But that hasn’t stopped it enchanting – and frustrating – mathematicians for more than a century now. 

So, what does it say? And will it ever be solved?

A haystack from a needle

Imagine you have a needle. It’s thin – infinitely so, in the mathematical setup, but you can think of it as just a regular needle – and one unit long, and you can move it in any direction you like. Got it in your head?

Great. So, the question is this: what is the smallest possible space you can cover with this needle and still have it face every direction possible at some point?

This was the origin of what is now called the Kakeya conjecture, and it is, at first, not all that difficult. We start in one dimension – a line – and without moving it at all, the needle is already facing every direction possible. Case closed.

In two dimensions, things get a little more interesting. Let’s consider the “face every dimension possible” requirement first: well, there’s an easy solution here – just spin the needle round on its center.

That definitely fulfills half the remit, but what about the area being swept out? Some simple geometry tells us that we’ve covered π/4 square units of area using this method, which doesn’t seem like a lot – but can it be reduced?

In fact, it can. If, instead of keeping the center of the needle in place as we spin it round, we instead move it around in a circle, we can actually halve the space we sweep out by creating a shape known as a deltoid.

That solution is what Kakeya himself proposed back in 1917, and it’s certainly pretty good – but we can do way better. With a little mathematical ingenuity, it’s actually possible to sweep out a space with zero measure, as Russian mathematician Abram Besicovitch showed just two years later in 1919.

Here’s the thing though: the shapes you get from doing whatever movements it takes to sweep out zero area? They’re very weird. 

“They have very counterintuitive properties. You could say pathological properties,” said Joshua Zahl, an aptly-named associate professor in the University of British Columbia’s department of mathematics and one of the most recent authorities on the Kakeya Conjecture.

Named Kakeya sets, the shapes are “compelling objects,” Zahl told IFLScience back in March 2025. “The first time you read about a Kakeya set, if you're naturally curious, you're going to be pretty intrigued.”

The Kakeya conjecture

The shapes may be strange, but it’s hard to argue they don’t answer the original question – and pretty definitively, too. After all, it’s hard to take up less space than no space at all, right? So… why do we still care about these Kakeya sets?

Well, as Fields medalist and long-time Kakeya conjecture chaser Terence Tao put it way back in 2009, “not all sets of measure zero are created equal.” When we say these shapes have zero area, we’re talking about something called the Lebesgue measure – basically, the thing you think about immediately when you think about “measuring” something. But that’s not the only way to describe the size of a thing, particularly in math. “What about the Hausdorff dimension of Kakeya sets?” Tao asks. What about “other types of dimension, such as [the] Minkowski dimension[?]”

It's this question of dimension, then, rather than size, that forms the Kakeya conjecture itself (yeah, sorry – everything up until now has not actually been the Kakeya conjecture at all, just build up). It says, in short, the following: a Kakeya set in Rⁿ has Hausdorff and Minkowski dimension n.

“It seem[s] very intuitive,” Larry Guth, Claude E. Shannon Professor of Mathematics at MIT and another veteran Kakeya conjecturist, told New Scientist in March. “It seem[s] like it must be true, but then it turns out to be very difficult to prove.”

So it was with much fanfare that a potential proof was greeted when it was uploaded to the arXiv preprint server earlier this year. The work of Zahl and his colleague Hong Wang from NYU’s Courant Institute of Mathematical Sciences, the paper was lauded in a statement as “a wonderful piece of mathematics” by Courant Institute Professor Guido De Philippis, as “not only a major breakthrough in geometric measure theory, but […] also open[ing] up a series of exciting developments in harmonic analysis, number theory, and applications in computer science and cryptography.”

Are such comments overstating the problem’s importance? Not in the least.

Unlocking future problems

It may have started with a simple brainteaser, but the Kakeya problem has applications and connections to a vast range of math problems. 

“The geometry of these Kakeya sets underpins a whole wealth of questions in partial differential equations, harmonic analysis and other areas,” Jonathan Hickman, a reader in analysis at the University of Edinburgh, told Quanta Magazine back in 2023.

Solve the Kakeya conjecture – as Zahl and Wang likely did – and it would be the first step towards toppling three central problems in harmonic analysis: the restriction conjecture, then the Bochner-Riesz conjecture, and finally the local smoothing conjecture. Each one relies on the previous to be true: if the Kakeya conjecture had turned out to be false, then all three would have been.

The fact that it seems to be true, then, is good news. 

“Solving the conjecture is definitely important in the area of harmonic analysis,” Zahl told IFLScience, “and maybe the sub field of harmonic analysis that cares about things like the Fourier restriction conjecture, for example.” 

“Beyond that, I mean – for whatever reason, this result seems to have captured public imagination a bit more than Hong and I were expecting,” he added. “But – I mean, I don't think it's particularly meaningful to try to rank the importance of totally unrelated areas of math, but if one were to do so, I don't think this stands above others.”