The Kakeya needle

How far down can we go? — a century of suspense

In 1917, the Japanese mathematician Sōichi Kakeya asked the question of the most economical half-turn. In 1928, Abram Besicovitch stunned everyone : there is no floor. With enough tricks (finer and finer zigzag back-and-forths), you can turn around in an area smaller than any threshold — 0.01, 0.000 001, as close to zero as you like.

The real riddle remained, in three dimensions : can a set that contains a needle in every direction be a « dust », a ghostly object with almost no thickness ? The dimension measures how much an object fills space : a line lives in dimension 1, a sheet in dimension 2, and some fractal objects slip in between. In 2025, Hong Wang and Joshua Zahl closed the question, open for a century : no — such a set may well have tiny volume, yet it is necessarily of dimension 3. A proof over a hundred pages long, hailed as one of the great results of the century. On 23 July 2026, at the opening of the International Congress of Mathematicians (ICM) in Philadelphia, Hong Wang received the Fields Medal — mathematics' highest honour, awarded every four years to mathematicians under 40.

The central question — can you turn around without being seen?

Almost. What melts toward 0 is the trace — the surface repainted by the needle's passage. With enough tricks, the half-turn repaints less than a stamp, less than a speck of dust : it becomes almost invisible… to area.

But never entirely. The area approaches 0 without ever reaching it — and the finer the trace, the more wildly the motion zigzags. Discretion is paid for in contortions.

And that's where she comes in. Even « invisible to area », the object does not disappear : its dimension stays full — that is Hong Wang's theorem. You can hide from area. Not from dimension. What you don't see from one angle is revealed when you change your point of view.

Her blackboard, decoded — project, then sort into boxes

In the interview, you can glimpse a sheet of her work : a cloud of points, a line, and three lines of notation. Here is what they say — it's the favourite move of her proofs :

• L = π̃(T) — π̃ is a projection : you flatten the figure in one direction, like a shadow on the ground. The line L is the shadow of a tube T (a thickened needle).
• F = π̃(π̃⁻¹(L) ∩ E) — you take everything that projects onto L, keep the points of E that live there, and look at their shadows : that's F.
• pigeonhole to choose F′ ⊆ F — the pigeonhole principle : if many shadows spread across few boxes, one box must hold a lot. You pick that well-filled drawer : F′, a piece of F (F′ ⊆ F, « F′ included in F »), smaller but richer — and the proof carries on with it.

Try it yourself : set the projection direction and watch the shadows sort into the boxes. When many shadows fall into the same box, it's no accident — the projection has just unmasked a hidden structure (aligned points). That is exactly why mathematicians project.

What she said — three messages for students

• 🎓 Thank you to the teachers. Trained at Peking University, then in France — at the École polytechnique and Orsay —, she made a point, on receiving her medal, of paying tribute to her French teachers. Behind every medal there are teachers — and behind every student too.
• ⚖️ Women, men: no difference. She is the third woman to win the Fields Medal in 90 years (after Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022), and the first Chinese female mathematician. In the interview she is clear : for her, there is no difference between women and men in mathematics. The variables that really move a proof forward are elsewhere : the work, the time, and the teachers you had — the difference of gender is an epsilon.
• ⏳ The long game. She speaks of « hard problems » that kept her busy for years. Nobody « sees » the solution at first glance — not even a Fields medallist. Searching for a long time is not a sign of weakness : it is the very craft of mathematics.

Hong Wang (王虹), 35, born in Guilin, entered Peking University at 16, is a permanent professor at the IHÉS (Bures-sur-Yvette, near Paris) and a professor at New York University (Courant Institute). ▶ the interview (CGTN Français, 1 min, in French)