What's the smallest area you need to spin a needle all the way around?
Take a needle of length 1 and try to rotate it a full 180 degrees, back to its starting line but pointing the opposite way, while sweeping through the smallest possible area of the plane. A circle of diameter 1 obviously works, but is it the smallest region that will do the job? Mathematician Soichi Kakeya posed the question in 1917, confident that some minimum, fixed area must be required.
Reveal the answer
Surprisingly, no minimum positive area is required at all. In 1919, Abram Besicovitch showed that using cleverly overlapped star-shaped regions, now called Besicovitch sets, the needle can be rotated through a region of area as close to zero as you like. The result stunned mathematicians and became a foundational example in geometric measure theory, still generating open research questions today.
— Soichi Kakeya, Abram Besicovitch, Kakeya set — Problem posed 1917, resolved 1919