Consider a pencil lying on your desk. Try to spin it around so that it points once in every direction, but make sure it sweeps over as little of the desk’s surface as possible. You might twirl the pencil about its middle, tracing out a circle. But if you slide it in clever ways, you can do much better.
In 1917, Sōichi Kakeya posed the problem, but with an infinitely thin pencil. He found a way of sliding the pencil that covered less area than the instinctual circular motion.
Kakeya wondered how small an area the pencil could possibly sweep. Two years later, the Russian mathematician Abram Besicovitch found the answer: a complicated set of narrow turns that, counterintuitively, covers no space at all.
That more or less settled the question until 1971, when Charles Fefferman was studying something apparently unrelated to twirling lines: the Fourier transform, a foundational mathematical tool that lets you reimagine any mathematical function as a combination of waves. In Fefferman’s work, a tweaked version of Kakeya’s problem kept coming up. In this case, the pencil has a thickness and twirls in three dimensions. Here, Kakeya’s question becomes the following: As you change the width of the pencil, how does it affect the volume of space that it traces out?
The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years.
In 1917, Sōichi Kakeya posed the problem, but with an infinitely thin pencil. He found a way of sliding the pencil that covered less area than the instinctual circular motion.
Kakeya wondered how small an area the pencil could possibly sweep. Two years later, the Russian mathematician Abram Besicovitch found the answer: a complicated set of narrow turns that, counterintuitively, covers no space at all.
That more or less settled the question until 1971, when Charles Fefferman was studying something apparently unrelated to twirling lines: the Fourier transform, a foundational mathematical tool that lets you reimagine any mathematical function as a combination of waves. In Fefferman’s work, a tweaked version of Kakeya’s problem kept coming up. In this case, the pencil has a thickness and twirls in three dimensions. Here, Kakeya’s question becomes the following: As you change the width of the pencil, how does it affect the volume of space that it traces out?
The deceptively simple Kakeya conjecture has bedeviled mathematicians for 50 years.
