This is, roughly, the Kakeya conjecture in three dimensions, a problem that resisted every serious attempt to solve it for half a century. A spatial version of Kakeya’s needle problem kept appearing in that work. Their starting point was a 1995 result by mathematician Tom Wolff, who had shown that no three-dimensional Kakeya set could have a Minkowski or Hausdorff dimension below 2.5. A disproof of the Kakeya conjecture would have brought the entire structure down. “People understood what’s going on in Kakeya-adjacent problems really well in two dimensions, but we lacked the tools to study higher dimensions,” Wang said.