But while it’s ostensibly a simple question about needles, “the geometry of these Kakeya sets underpins a whole wealth of questions in partial differential equations, harmonic analysis and other areas,” said Jonathan Hickman of the University of Edinburgh. Mathematicians have yet to prove this statement, known as the Kakeya conjecture. While they know that such sets can be small in terms of area (or volume, if you’re arranging your needles in three or more dimensions), they believe the sets must always be large if their size is measured by a metric called the Hausdorff dimension.

Mathematicians call these arrangements Kakeya sets. If you don’t require the needle to move in one continuous motion, and instead simply lay down a needle in every direction, you can construct an arrangement of needles that covers no area at all. But rotate it more cleverly, and you can cover an arbitrarily small fraction of space. Earlier this summer, Quanta reported on one such problem: What is the smallest area that you can sweep out while rotating an infinitely thin needle in all possible directions? Spin it around its center like a dial, and you get a circle. In mathematics, a simple problem is often not what it seems.

Circle geometry markets