Imagine trying to spin a pencil in every direction while covering the least possible space. It sounds simple, but this problem—known as the Kakeya conjecture—has puzzled mathematicians for over a century. Now, researchers Hong Wang of NYU’s Courant Institute and Joshua Zahl of the University of British Columbia have finally cracked the three-dimensional version of this problem, marking a monumental moment in mathematical research.
Originally posed in 1917 by Sōichi Kakeya, the problem was first studied in two dimensions, where the answer was unexpectedly intricate: a set of bizarre, infinitely small twists and turns known as a Besicovitch set. But when the problem moved to three dimensions, the challenge became exponentially harder, evading solutions for over five decades.
Enter Wang and Zahl. Following ideas laid out by Nets Katz and Terence Tao, they developed a novel approach, proving that even the most space-efficient way to rotate a thickened version of the pencil still requires a certain minimum volume. Their work established that the Minkowski dimension of any Kakeya set in three dimensions must be exactly three—a seemingly mild constraint that turned out to be an extraordinary hurdle to prove.
This result doesn’t just settle an old problem—it reshapes modern harmonic analysis, a field that connects geometry to the Fourier transform, one of the most powerful mathematical tools in physics and engineering. Many key mathematical questions rest on the foundation of Kakeya, and now that it has been confirmed in three dimensions, researchers can climb the mathematical tower that rests upon it.
For those wondering, the four-dimensional Kakeya conjecture remains open. But mathematicians are optimistic—if the leap from two to three dimensions was the hardest, Wang and Zahl’s breakthrough might just be the key to unlocking even higher-dimensional mysteries.
For a deeper dive, check out the research article here: https://arxiv.org/abs/2502.17655
Quanta Article: Solving the Three-Dimensional Kakeya Conjecture.
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