Kakeya Conjecture

Imagine a needle — that is, a straight line of length 1. You want to rotate it within a plane by 360 degrees so that it ends up pointing in its original direction again. How small can the area be that you need for this? That is the basic question behind the Kakeya problem, named after the Japanese mathematician Sōichi Kakeya, who posed it in 1917. The surprising answer: the area can be made arbitrarily small — theoretically even approaching zero. But what happens in three, four, or even more dimensions? That is precisely what remains not fully resolved to this day, and that is exactly what the Kakeya conjecture is about.

Why mathematicians have been working on it for over 100 years

The conjecture sounds like a nice puzzle from school geometry. In reality, it is a junction where many areas of mathematics converge. Whoever proves it automatically gains insights into seemingly completely different problems — for example, how waves propagate or how numbers are distributed evenly.

Specifically, the conjecture is connected to so-called harmonic analysis, a branch of mathematics that describes oscillations and signals. It also appears in number theory — that is, the study of whole numbers and their patterns. A proof would therefore not only answer a single question but would also bring about a whole series of other theorems that currently only hold under the assumption that the conjecture is true.

The problem in higher dimensions

In two dimensions — that is, in the plane — the needle problem has been solved: the area can indeed become arbitrarily small. This was proven in 1928 by the Russian mathematician Abram Besicovitch. To do so, he constructed a kind of jagged, fractal structure in which the needle is maneuvered through many small loops without touching much area.

The actual conjecture concerns higher dimensions. It states: a set in n-dimensional space that contains a line segment of length 1 in every direction must have the full dimension n. This means that although it may be tiny, it must not be so strangely shaped that it appears, in a sense, “flatter” than its dimension would suggest. In two dimensions this is trivial. From three dimensions onward, it remains unproven to this day.

In February 2025, a breakthrough caused a stir: the mathematicians Hong Wang and Joshua Zahl published a proof for the three-dimensional case. This proof was regarded by the expert community as one of the most significant mathematical advances in years. The general case for arbitrarily many dimensions, however, remains open.

Where the Kakeya conjecture appears in the news

The Kakeya conjecture is not a term from the technology industry — it does not appear in products or apps. It surfaces in the media whenever a mathematical breakthrough is reported. In early 2025, science magazines and daily newspapers worldwide reported on the proof by Wang and Zahl, because it concluded decades of work by many mathematicians.

Indirectly, however, the conjecture does have points of contact with applied mathematics. The harmonic analysis to which it is linked underlies image processing, signal technology, and data compression. So anyone who opens a compressed image or streams music indirectly benefits from mathematics originating from the same field of research.

A common misconception: the Kakeya conjecture is sometimes confused with the question of how to rotate a needle in a space-saving way — as if it were an optimization problem for engineers. It is not. It concerns the mathematical structure of sets and dimensions, not practical design. Its value lies in the connections it reveals — between disciplines that, at first glance, seem to have nothing to do with one another.