The 3D Proof of the Kakeya Conjecture: A Japanese Mathematician's 108-Year-Old Problem Finally Solved
Introduction
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.
They also built upon techniques pioneered by Terence Tao and Nets Katz in 2014. These researchers had previously proven that a special class called "sticky Kakeya sets" must have dimension 3, paving the way for this complete 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!
References
- https://gigazine.net/news/20251230-breakthroughs-in-mathematics-2025/
- https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/
- https://www.nyu.edu/about/news-publications/news/2025/march/mathematicians-move-the-needle-on-decades-old-problem.html
- https://terrytao.wordpress.com/2025/02/25/the-three-dimensional-kakeya-conjecture-after-wang-and-zahl/
- https://arxiv.org/abs/2502.17655
- https://levtech.jp/media/article/column/detail_635/
- https://en.wikipedia.org/wiki/Kakeya_set
Reactions in Japan
I'm so moved that a problem proposed by a Japanese mathematician over 100 years ago has finally been solved! Professor Kakeya must be rejoicing in heaven.
Apparently the 3D Kakeya conjecture has been solved. It's amazing that Terence Tao called it 'one of the top mathematical achievements of the 21st century.'
Is it true that the Kakeya conjecture was inspired by 'wielding a spear in a toilet'? Mathematicians have the best sense of humor lol
As a math major, I'm trembling hearing that the Kakeya conjecture proof paper is over 200 pages. How many years will it take to read...
Can someone explain this in a way regular people can understand? When they say 'point a pencil in every direction with minimum volume,' I can't picture it at all.
They say it could be applied to 5G and medical imaging in the future, but honestly I have no idea how it connects lol. Math is mysterious.
Hong Wang graduated from Peking University, got her PhD from MIT, and is now at NYU. Her background is so impressive, she seems like she's from another world.
Apparently solving this pure math problem opens possibilities for solving other unsolved problems in harmonic analysis. It's fascinating how one proof can trigger a chain reaction.
I can't grasp the concept of having dimension 2 or 3 despite having zero area. Math defies intuition way too often.
The global journey of a Japan-originated problem being solved by mathematicians worldwide shows the beauty of academia. A borderless inheritance of knowledge.
Now they're going for 4D and beyond? Huge respect to mathematicians who keep pushing the limits of human intelligence.
Honestly, whether it's useful to society or not, the romance is in solving a century-old mystery. The romance of mathematics.
As someone who gave up on math in high school, seeing news like this makes me wish I had tried harder. Too late now though.
Apparently multi-scale analysis is the key technique. Observing step by step from micro to macro sounds like a physics-inspired approach.
I just learned Sōichi Kakeya was the first director of the Institute of Statistical Mathematics. He was truly an important figure in Japanese mathematical history.
There's a reason Terence Tao called this a 'once-in-a-century result.' This proof is a monument in pure mathematics. Congratulations to Wang and Zahl on this magnificent achievement.
As a math teacher in Spain, I can't wait to explain this proof to my students. The story of a Japanese mathematician posing a problem over 100 years ago that was solved today is a wonderful example of the international nature of mathematics.
As a Chinese person, I'm proud that Hong Wang is a Peking University graduate. Mathematics has no borders, but it's inspiring to see such talent flourishing on the world stage.
As a researcher specializing in harmonic analysis, I'm excited about the ripple effects this proof will bring. Opening the path to the restriction conjecture and Bochner-Riesz conjecture is revolutionary.
In Germany, we value mathematical rigor, and I've heard this proof spans over 200 pages. We should be cautious in our evaluation until peer review is complete.
India had great mathematicians like Ramanujan. Every time a problem like the Kakeya conjecture is solved, I'm reminded once again of the beauty and depth of mathematics.
Honestly, I don't understand specifically how this will help mobile communications or medical imaging. I suppose it takes time for pure math results to lead to applications.
From the tradition of the French Bourbaki school of mathematics, such advances in geometric measure theory are very important. Applications to Fourier transforms are anticipated.
Considering that Besicovitch was a Russian mathematician, this problem had its foundations laid by Japanese and Russian mathematicians. A wonderful example of international collaboration.
The fact that female mathematician Hong Wang achieved this historic proof will provide great inspiration to women working in STEM fields.
Italian mathematics education has emphasized geometry, but problems involving such advanced dimensions are hard for ordinary people to understand. We need more accessible explanations.
In Korea, more young people are excelling in math olympiads. Global breakthroughs like this will inspire the next generation of mathematicians.
Mathematics education in Brazil is still developing, but news of such historic proofs is a great opportunity to convey the appeal of mathematics to younger generations.
I was surprised to hear that the Kakeya conjecture proof could have applications in cryptography. An unexpected connection between pure mathematics and cybersecurity.
As a Danish mathematician, I'm convinced that solving the Kakeya problem will impact the entire field of harmonic analysis. We'll see many related studies published from now on.