Kakeya's problem, or how to move a needle in the minimum of space

Research Article published on 10 September 2025 , Updated on 10 September 2025

What is the smallest area in which a needle can rotate? For over a century, mathematicians have been interested in this question and have attempted to characterise these areas precisely. This is known as Kakeya's conjecture. Hong Wang, permanent professor of mathematics at IHES and professor of mathematics at New York University, and her collaborator Joshua Zahl, who recently joined Nankai University (China), have just taken an important step forward on this question.

Some mathematical problems are surprisingly simple to state, yet even the most brilliant minds have failed to solve them. Kakeya's problem is undoubtedly one of them. To understand it, all you need is a needle placed on a table: by pinching and rotating the needle between your fingers, it turns completely around itself, tracing the contours of a circle. However, is it possible for the needle to complete a 360° turn while tracing a smaller shape? If so, what would be the smallest surface area? Hong Wang, permanent professor of mathematics at IHES and professor of mathematics at New York University, has just made a breakthrough on this question with her colleague Joshua Zahl. By looking at what happens when the needle is no longer on a table but floating in three-dimensional space, they have solved a question that researchers have been asking themselves for more than fifty years.

From the deltoid to a spiky shape

In 1917, Japanese mathematician Soichi Kakeya was the first to question the surfaces on which a needle can rotate. In doing so, he identified shapes with an area smaller than that of a circle, such as a deltoid – a kind of equilateral triangle with sides rounded inwards. As a result of his work, these objects were named “Kakeya sets”.

Two years later, Abram Besicovitch, a Russian mathematician, made a breakthrough: "He constructed a Kakeya set with an arbitrarily small volume," explains Hong Wang. He demonstrated that it was possible to construct surfaces as small as desired that contained a needle in every direction – provided that the needle was thin enough. If we imagine a needle that is nothing more than a line with no thickness, the surface area would even be zero.

The implications of the mathematician's work go even further. Whether scientists look at the problem of a needle lying on a table – in two dimensions – or floating in the air – in three dimensions – or in larger dimensions that are difficult for humans to visualise, the answer is the same: the size of the smallest area occupied by the needle is zero if the needle is tiny.

“The shape found is in a way all spiky,” explains Hong Wang. In two dimensions, this shape looks like a pile of scattered Mikado sticks (the sticks overlap and point in all directions), but more cleverly arranged than by chance. For instance, scientists stack as many sticks as possible pointing in similar directions so that they take up less space.

Kakeya set whose area approaches zero

Abram Besicovitch did not end research on Kakeya sets though. There is something unsatisfactory about arriving at a range of sets with zero measure because it is not precise enough. “Zero-measure sets have different sizes,” says Guy David, professor at Université Paris-Saclay and mathematician at the Orsay Mathematics Laboratory (LMO – Univ. Paris-Saclay/CNRS). For example, consider a single point on the table and Abram Besicovitch’s Kakeya sets are all of zero measure. However, it is clear that the other sets – with all their peaks – are larger.

Mathematicians have other tools to measure the size of objects more precisely, such as Minkowski and Hausdorff dimensions. “These dimensions give us insight into the size of sets, even if they have zero volume,” explains Hong Wang, who describes what these measurements are: “If you cover the set with balls, the Minkowski and Hausdorff dimensions tell us, approximately, how many balls are required when their radii approach zero.”

It is difficult to calculate these dimensions for Kakeya sets. In 1971, Roy Davis took a first step: he demonstrated that the Minkowski and Hausdorff dimensions were 2 in the case of Kakeya sets existing on the table, i.e. in two-dimensional space. Researchers then assumed that the other dimensions followed this logic: that the answer was 3 for three-dimensional space, 4 for the fourth dimension, and so on. This is Kakeya's conjecture.

“What is interesting about this conjecture is that its statement is very simple. But it takes a lot of effort to prove it,” says Guy David. Several scientists have tackled the conjecture for the case of dimension 3 – the needle in space – but without completely resolving it. That was until early 2025, when Hong Wang and Joshua Zahl presented their work: in an article that has not yet been peer-reviewed – but is considered highly credible – they demonstrate that Kakeya's conjecture is true for the third dimension.

Immersion in the third dimension

In the course of her career as a mathematician, it was quite natural for Hong Wang to turn her attention to the Kakeya conjecture. In 2013, while studying at Université Paris-Saclay, she heard about a related problem called the restriction conjecture. And then, as she explains: “My thesis supervisor, Larry Guth, was interested in Kakeya's conjecture. I watched him think about the subject during my PhD.” At the end of her postdoctoral studies in 2021, she finally immersed herself in this problem.

A year later, she and her co-author Joshua Zahl presented their first breakthrough: they had solved the question for a special case of sets called “sticky” sets. Hong Wang describes them: "Sticky sets make the additional assumption that if two needles form a restricted angle, then they must be in close proximity.” These sets appeared in several previous works and seemed to constitute an important special case. “Studying them turned out to be much more difficult than we thought,” recalls Hong Wang.

Then, the two mathematicians tackled the general case. ‘To do this, they used several techniques established by others before them, and also had to develop new ones,’ describes Guy David. In 2014, Larry Guth described Kakeya sets in particular if the conjecture proves to be false. In this case, the sets must be granular, a sort of ‘union of small tiles,’ explains Hong Wang. Based on this observation, the researcher and Joshua Zahl demonstrated that it was impossible for such a Kakeya set to exist if its Minkowski and Hausdorff dimensions were less than 3.

Not only does this research answer a question that has been on the table for some fifty years, it also has links to other fields in mathematics. One example is harmonic analysis, which investigates functions by breaking them down into sine waves. ‘Kakeya's conjecture provides information about the geometry of waves,’ explains Hong Wang. The link between these two questions is the restriction conjecture, which the mathematician became interested in during her university studies.  
“In this field, we often have to cover sets with rectangles, see how many are needed and find ways to optimise that number,” explains Guy David. Thanks to these techniques, it is possible to accurately approximate the correct properties of certain functions. “These thin rectangles overlap and intersect,” points out the lecturer and researcher. This resembles Kakeya sets.

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