Turn the Needle
In 1917 Sōichi Kakeya asked: what is the smallest region in which a needle of length 1 can be turned all the way round, so it ends up pointing the other way? Try three answers, then watch the real one.
1 · Three ways to turn a needle
Each shape lets a needle of length 1 turn half a circle and come back reversed. All three are drawn at the same scale. Smaller area is better.
The disk turns the needle about its middle. The equilateral triangle of height 1 is the best convex shape (Pál, 1920): pivot at a corner, slide, repeat. The three-pointed deltoid was Kakeya’s own guess — it is smaller still, but not the end of the story.
2 · As small as you like
Besicovitch’s idea: cut a triangle into thin slivers and slide them so they overlap. Each sliver keeps its directions, so the union still contains a segment in every direction the triangle had — but covers less and less area. Raise the number of slivers.
Shifts found by a simple search for this page, so the shrinking is slow; Besicovitch’s constructions shrink it as close to zero as you like. A few rotated copies of such a tree turn a needle all the way round. There are even sets with a segment in every direction and zero area.
3 · Zero area, full dimension
If the area can be zero, what is left to prove? The Kakeya conjecture says such a set is still, in a precise sense, as thick as the space it lives in: its dimension is 2 in the plane, 3 in space, n in n dimensions. It can have no area and still not be “thin.”
- 1917Kakeya asks the needle questionAsked
- 1920–1928Besicovitch: the area can be as small as you likeProved
- 1971Two dimensions: DaviesProved
- Feb 2025Three dimensions: Wang and Zahl (Fields Medal, 2026)Proved
- Oct 26, 2026Every dimension: an OpenAI model, via local smoothing, with Lean statementsClaimed
“Proved” here means accepted by mathematicians. The 2026 claim is an unreviewed preprint, released as the model wrote it.