A Kakeya set in \(\mathbb{R}^n\) is a compact set containing a unit line segment in every direction.
Conjecture: every Kakeya set has Hausdorff and Minkowski dimension exactly \(n\) — however cleverly the segments overlap, the set cannot be genuinely thinner than full-dimensional.
Born from Kakeya's 1917 needle question and Besicovitch's shocking construction (1928) showing such sets can have measure zero. Davies settled dimension two in 1971. Wolff's landmark 1995 bound (dimension ≥ (n+2)/2) stood for a quarter century in higher dimensions.
In February 2025 Hong Wang and Joshua Zahl proved the full conjecture in dimension three — a 127-page reduction of every case to their earlier 'sticky' theorem, with an induction Terence Tao likened to a perpetual-motion machine. Quanta dubbed it a once-in-a-century result; Eyal Lubetzky called it one of the top achievements of 21st-century mathematics. Dimensions four and above remain open, with counterexamples showing the direct tube-volume statement must change form at n ≥ 4.
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