The Needle and the Nothing

The Kakeya needle problem asks how small an area a needle can sweep while turning to point in every direction. The surprising answer is an area as close to zero as you like, found by Besicovitch in 1919. That puzzle grew into the Kakeya conjecture: such a set has zero area yet must have the full dimension of its space. This is the story of the needle, the deltoid, Besicovitch sets, why dimension is the right measure, and the once-in-a-century 2025 proof in three dimensions by Hong Wang and Joshua Zahl, who built on Wolff, Guth and the method of grains. It ends with the tower of conjectures Kakeya holds up, and Hong Wang's 2026 Fields Medal.

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