The year 2025 has brought a historic breakthrough to the world of mathematics. Quanta Magazine has called 2025 "a historic year for mathematics," highlighting the "proof of the Kakeya conjecture in three dimensions" as one of three major advances.

This problem was first proposed in 1917 by Japanese mathematician Sōichi Kakeya. Despite its deceptively simple appearance, it has confounded mathematicians worldwide for over a century. In February 2025, this century-old puzzle was finally solved in three-dimensional space.

What is the Kakeya Problem? A Simple Question About Rotating a Pencil

The Kakeya problem begins with a surprisingly simple question:

Place a pencil (a line segment of length 1) on your desk. How can you rotate this pencil to point in every direction while sweeping over as little area as possible?

The most intuitive answer is to spin the pencil around its center, tracing out a circle. However, with clever maneuvering, you can actually cover a much smaller area.

Kakeya himself initially suggested the Reuleaux triangle as a solution, but his colleagues soon discovered that rotation was possible within an even smaller equilateral triangle.

A Surprising Discovery: Rotation in Zero Area

In the 1920s, Russian mathematician Abram Besicovitch made a remarkable discovery that would reshape the problem entirely. He proved that "the area required to rotate a line segment of length 1 can be made arbitrarily small." Moreover, he demonstrated that "a set containing line segments in every direction can have zero area."

This discovery shifted mathematicians' focus from "minimum area" to "dimension." Even if the area is zero, the concept of "dimension," measuring how complexly a shape occupies space, became crucial.

The dimensions referred to here are mathematically rigorous concepts known as "Hausdorff dimension" and "Minkowski dimension." Unlike integer dimensions, these can take fractional values like 2.5. Complex shapes like fractals often possess such intermediate dimensions.

The Kakeya Conjecture: Does the Dimension of the Set Equal the Dimension of the Space?

The modern "Kakeya conjecture" states:

In n-dimensional space, any set containing a unit line segment in every direction (a Kakeya set) must have dimension n.

In other words, a Kakeya set in 2D space must have dimension 2, and a Kakeya set in 3D space must have dimension 3.

The conjecture is trivially true for one dimension. Roy Davies proved the two-dimensional case in 1971. However, for three dimensions and beyond, the problem remained unsolved for over 50 years, challenging the brightest mathematical minds.

The Historic 2025 Proof

In February 2025, Hong Wang, an associate professor at New York University's Courant Institute of Mathematical Sciences, and Joshua Zahl, an associate professor at the University of British Columbia, finally proved the 3D Kakeya conjecture completely.

Fields Medalist Terence Tao praised the achievement, writing, "There has been some spectacular progress in geometric measure theory!" Eyal Lubetzky, chair of the Mathematics department at the Courant Institute, called it "one of the top mathematical achievements of the 21st century."

Quanta Magazine described this proof as a "once-in-a-century" result, emphasizing its historic significance.

The Key Idea: Multi-Scale Analysis

The method employed by Wang and Zahl is called "multi-scale analysis."

They conceptualized line segments (needles) as thin tubes of radius δ, then observed them progressively from microscopic to macroscopic scales, much like adjusting magnification on a microscope. They rigorously proved a property called "non-clustering," showing that multiple tubes cannot concentrate excessively in any particular location in space.

Central to the proof is a multi-scale, self-similar structure of Kakeya sets known as "stickiness," whose importance dates back to the groundbreaking 1999 work of Nets Katz, Izabella Łaba, and Terence Tao. Wang and Zahl themselves first proved, in 2022, that this special class of "sticky Kakeya sets" must have dimension 3, and in 2025 they reduced the general case to the sticky case, completing the proof.

Impact on Our Lives: Future Applications

While this is a result in pure mathematics, it has the potential to influence our daily lives in the future.

Contributions to Harmonic Analysis and Fourier Transforms

The Kakeya conjecture is intimately connected to a cluster of problems forming the foundation of harmonic analysis. The Fourier transform, a technique for expressing any signal as a sum of sine waves, underpins modern mobile phones and digital communications. This resolution is expected to drive advances in these theories.

Signal Processing and Communication Technology

The proof contributes to better computational models that accurately describe how signals like radio waves propagate and overlap in space. This could help optimize 5G and next-generation wireless networks.

Cryptography and Computer Science

The geometric understanding related to the Kakeya conjecture may be applied to improving cryptography and data compression algorithms.

Medical Imaging

Image reconstruction algorithms for MRI and CT scans are based on harmonic analysis techniques. The new mathematical methods emerging from this achievement may lead to more precise medical imaging technologies.

Future Prospects: The Challenge of Higher Dimensions

With the 3D proof complete, mathematicians now set their sights on four dimensions and beyond. The proof also opens pathways to the Kakeya maximal conjecture, the restriction conjecture, and the Bochner-Riesz conjecture, a hierarchy of challenging problems in harmonic analysis.

These conjectures form a layered structure, with the Kakeya conjecture as the foundation. This proof makes climbing this "mathematical tower" a realistic possibility.

About Sōichi Kakeya

Sōichi Kakeya (1886–1947) was a mathematician from Fukuyama City, Hiroshima Prefecture, Japan. After graduating from Tokyo Imperial University, he served as an associate professor at Tohoku University, a professor at Tokyo Higher Normal School, and a professor at Tokyo University of Literature and Science. In 1928, he delivered an invited lecture at the International Congress of Mathematicians in Bologna and received the Japan Academy's Imperial Prize the same year.

An interesting anecdote about the problem's origin survives: When mathematician Kentaro Yano asked Kakeya how he conceived the problem, Kakeya reportedly answered that "samurai carried their spears even when entering the toilet. If they had to fight in such a confined space, they would need to wield the spear in the smallest possible area."

The fact that a question posed by a Japanese mathematician 108 years ago has been solved through the efforts of mathematicians worldwide symbolizes the international nature of mathematical inquiry.

What Famous Math Problems Come from Your Country?

Just as the Kakeya conjecture, born in Japan, led to a worldwide breakthrough, mathematical problems transcend national boundaries to connect researchers globally. What famous mathematical problems originated in your country? Are there any challenging problems still being studied today? Share your thoughts in the comments!

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