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Kakeya Sets: From Vanishing Area to a 3D Breakthrough
Manage episode 507203369 series 3690682
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Imagine you must rotate a line segment through every direction in the smallest possible space. The Kakeya problem began in 1917, provoking Besicovitch’s startling zero-area sets and a shift from area to dimension via Minkowski dimension. We trace the arc from intuitive puzzles to counterintuitive constructions—Perron trees, Paul joins, and the polynomial method—including the finite-field version and its famous resolution. In March 2025, Hong Wang and Joshua Azal announced a complete solution in three dimensions, a milestone with deep implications for higher dimensions and analysis. We unpack the ideas, the breakthroughs, and what lies ahead for n ≥ 4—and connections to physics and computation.
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