# The Needle and the Nothing > The Kakeya needle problem asks how small an area a needle can sweep while turning to point in every direction. The surprising answer is an area as close to zero as you like, found by Besicovitch in 1919. That puzzle grew into the Kakeya conjecture: such a set has zero area yet must have the full dimension of its space. This is the story of the needle, the deltoid, Besicovitch sets, why dimension is the right measure, and the once-in-a-century 2025 proof in three dimensions by Hong Wang and Joshua Zahl, who built on Wolff, Guth and the method of grains. It ends with the tower of conjectures Kakeya holds up, and Hong Wang's 2026 Fields Medal. Interactive film: https://www.planetpast.com/stories/kakeya-needle-problem/ Publisher: PlanetPast (https://www.planetpast.com/) · Published 2026-10-13 · Math ## Prologue ### The needle and the nothing *A hundred-year-old puzzle* How little room do you need to turn a needle all the way around? Here is a simple-sounding question. You have a needle lying on a table. You want to turn it so that, at some moment, it has pointed in every single direction. How small an area can the needle sweep while it does that? Spin it around its middle and it sweeps a whole disc. But you can be far cleverer. In 1919 Abram Besicovitch found a shocking answer: you can do it inside an area as close to zero as you like. Almost nothing. That puzzle turned out to hide one of the deepest questions in modern mathematics — and in 2025 it was finally answered in three dimensions. ![A unit needle, free to slide and turn on a table.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c00.jpg) - **1917**: Kakeya asks the question - **→ 0**: smallest possible area - **2025**: solved in three dimensions > A once-in-a-century kind of result. (Nets Katz, on the 2025 proof) Prologue · Why this matters #### The needle and the nothing The **Kakeya needle problem** asks: how small an area can a unit needle sweep while turning to point in every direction? The answer — **as close to zero as you like** — opened a century of mathematics that ended with a landmark 2025 proof. - **Asked by**: Sōichi Kakeya 1917 - **Smallest area**: → 0 Besicovitch, 1919 - **Solved in 3D**: 2025 Wang & Zahl - **Fields Medal**: 2026 Hong Wang How to use: pick a chapter below or press ← → to change step · ▶ plays the step · drag to rotate, scroll or pinch to zoom · “Play the film” runs every step full-screen. Sources: [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/); [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set) ## Chapter I · The needle ### The obvious way *Chapter I · The question* Spin it, and you sweep a whole disc. Let the needle have length one. The simplest way to point it in every direction is to spin it around its centre. Doing that, it sweeps out a disc of radius one-half. The area is pi over four, about 0.79. That is the lazy answer. The real game is to do it in less area. And it turns out you can do much, much less. ![Spun about its centre, the needle sweeps a disc of radius ½.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c01.jpg) $$A=\frac{\pi}{4}\approx 0.785$$ The area swept by spinning the needle. Chapter I · Step 1 #### The obvious way A unit needle spun about its centre sweeps a **disc of radius ½**. Its area is π/4 ≈ 0.785 The needle is one unit long. “Pointing in every direction” means that, over the whole motion, the needle lies parallel to every line through the plane at least once. Sources: [Wikipedia: the Kakeya needle problem](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_needle_problem) ### The three-point turn *Chapter I · Do better* Slide as well as rotate. Here is the trick: you are allowed to slide the needle, not just pivot it. Think of a three-point turn in a car. Using clever slides, the needle can make all its turns inside a triangle. An equilateral triangle of height one does the job. Its area is one over the square root of three, about 0.58 — already smaller than the disc. Among all convex shapes, that triangle is the best you can do. But who said the shape has to be convex? Drop that, and the floor falls out. $$A=\frac{1}{\sqrt{3}}\approx 0.577$$ Area of an equilateral triangle of height 1 — smaller than the disc. - Allowed to slide, not only pivot - 1920 · Pál: the triangle is the smallest convex answer Chapter I · Step 2 #### The three-point turn Because the needle may **slide** as well as rotate, it can turn inside an **equilateral triangle of height 1**: area 1/√3 ≈ 0.577 - 1920 Pál: the triangle is the smallest convex answer - Next allow non-convex shapes… Sources: [Wikipedia: the Kakeya needle problem](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_needle_problem); [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set) ### The deltoid *Chapter I · Smaller still* A curved three-cornered shape. Let the needle slide along a curved track and it traces a graceful three-cornered curve called a deltoid. The needle stays tangent to it, sweeping only the region inside. That area is pi over eight, about 0.39 — smaller again. Kakeya himself guessed, in 1917, that this deltoid might be the smallest possible. He was wrong — and gloriously so. ![The needle slides tangent to a three-cusped curve.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c02.jpg) - Disc (spin): π/4 - Triangle: 1/√3 - Deltoid: π/8 Chapter I · Step 3 #### The deltoid Sliding along a curve, the needle sweeps a **deltoid** (a three-cusped curve): area π/8 ≈ 0.393 Kakeya conjectured in 1917 that the deltoid was the smallest region of all. Two years later that guess was spectacularly overturned. Sources: [Wikipedia: the Kakeya needle problem](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_needle_problem) ## Chapter II · Almost nothing ### Area almost zero *Chapter II · 1919* Besicovitch’s shock. Abram Besicovitch was working on a different problem when he found the construction that would make him famous. Start with a triangle. Cut it into thin slivers. Now slide the slivers so they overlap as much as possible, letting them poke out in a spray of fine spikes. Each sliver still carries needles pointing in its own small range of directions, so together they still cover every direction. But the overlapping means the total area shrinks. Repeat the cutting and sliding, and the area can be driven below any number you choose. You can cover every direction inside an area as close to zero as you like. ![Thin slivers slid to overlap, spraying into fine spikes.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c03.jpg) - **→ 0**: total area - **all**: directions still covered Chapter II · Step 4 #### Area almost zero Cut a triangle into thin slivers and slide them to overlap. Repeat. The set still contains a needle in **every direction**, yet its area can be made **smaller than any number**. - **Besicovitch set area**: → 0 can be made as small as you like - **Directions covered**: all of them These spiky figures are called Besicovitch sets. In the limit a Besicovitch set of area exactly zero exists. Sources: [Wikipedia: Besicovitch set](https://en.wikipedia.org/wiki/Besicovitch_set); [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/) ### Turning in nothing *Chapter II · 1928* The needle really can turn around. That was about covering directions. But can a needle actually turn continuously through all of them inside such a tiny area? In 1928 Besicovitch showed: yes. You can connect the spikes with thin corridors, and shuffle the needle from one to the next. A needle can make a full turn inside an area as small as you please. There is no smallest answer at all — only smaller and smaller. - Connect the spikes with thin corridors - Shuffle the needle from one to the next - A full turn in an area as small as you like - **→ 0**: no smallest area — only smaller and smaller Chapter II · Step 5 #### Turning in nothing In **1928** Besicovitch showed a needle can turn **continuously** through every direction inside an area as small as you like. There is no minimum — only an endless race toward zero. So the original Kakeya question, taken literally, has a strange answer: “as little area as you want.” The interesting question had to be asked differently. Sources: [Wikipedia: Besicovitch set](https://en.wikipedia.org/wiki/Besicovitch_set); [Wikipedia: the Kakeya needle problem](https://en.wikipedia.org/wiki/Kakeya_set#Kakeya_needle_problem) ## Chapter III · The real question ### Every direction at once *Chapter III · A better question* A Kakeya set. Mathematicians stopped thinking about one moving needle, and pictured every position at once: a set that contains a line segment pointing in every direction. Such a set is called a Kakeya set, or a Besicovitch set. We have seen it can have zero area. So if area cannot tell these sets apart from nothing, we need a sharper ruler. How “big” is a set that has zero area but still contains a needle in every direction? ![Every needle position at once: a segment in every direction.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c04.jpg) - Contains a unit segment in every direction - Can have zero area - So area is the wrong ruler Chapter III · Step 6 #### Every direction at once A **Kakeya set** contains a unit segment pointing in every direction. It can have zero area — so area is the wrong way to measure it. We need a sharper notion of size. Each glowing line is the needle frozen in one of its directions. Together they point every way at once. Sources: [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set); [Wikipedia: Besicovitch set](https://en.wikipedia.org/wiki/Besicovitch_set) ### Area zero, but how full? *Chapter III · The right ruler* Dimension, not area. A single line has zero area, but it is clearly “one-dimensional”. A filled square is two-dimensional. Some sets sit in between, with a fractional dimension. The right ruler is this dimension. Fatten the set into tubes of thickness delta, measure the volume, and watch how fast it shrinks as delta shrinks. That rate is the dimension. A set that is nearly two-dimensional barely loses volume as the tubes get thinner. A thin, wispy set loses volume fast. So the sharp question is: a Kakeya set has zero area, but can it be wispy — of dimension less than the space it lives in? $$\text{volume}\sim \delta^{\,n-d}$$ - $\delta$: tube thickness - $n$: dimension of the space - $d$: dimension of the set Fatten into tubes of thickness δ; how fast the volume shrinks gives the dimension d. - A line — dimension 1 - A filled square — dimension 2 - In between — fractional dimension Chapter III · Step 7 #### Area zero, but how full? Even with zero area, a set has a **dimension**. Fatten it into tubes of thickness δ and see how its volume shrinks as δ → 0: slow shrink means a full, near-solid set; fast shrink means a thin, wispy one. - A line dimension 1 - A filled square dimension 2 - In between fractional dimension Sources: [Wikipedia: Hausdorff dimension](https://en.wikipedia.org/wiki/Hausdorff_dimension); [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/) ### It must be full *Chapter III · The conjecture* Zero area — but full dimension. This is the Kakeya conjecture: although a Kakeya set can have zero area, it can never be wispy. It must have the full dimension of the space it lives in. In the plane, that means dimension exactly two. In three-dimensional space, dimension exactly three. In 1971 Roy Davies proved it in the plane. The flat case was hard enough. Three dimensions resisted for another fifty years. It sounds almost obvious — of course something pointing in all directions should be “full”. But the overlaps that make the area vanish are exactly what make it so hard to pin the dimension down. $$\dim(\text{Kakeya set})=n$$ The conjecture: full dimension, even with zero area. - **1971**: proved in the plane (n = 2) - **2025**: proved in space (n = 3) Chapter III · Step 8 #### It must be full The **Kakeya conjecture**: a Kakeya set has zero area, yet must have the **full dimension** of its space — 2 in the plane, 3 in space. - 1971 Davies proves it in the plane (dimension 2) - 3D open for another half-century The conjecture lies beneath a whole tower of problems in the study of waves and the Fourier transform — which is why so many wanted it true. Sources: [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set); [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/) ## Chapter IV · The proof ### Inch by inch *Chapter IV · The long climb* Fifty years of partial results. Nobody cracked three dimensions in one blow. They raised the floor, bit by bit. In 1995 Tom Wolff showed a 3D Kakeya set must have dimension at least two-and-a-half. For years that was the record. Nets Katz and Terence Tao laid out a plan of attack. In 2014 Larry Guth proved that any counterexample would have to be “grainy” — built from many tiny regions where lots of needles pile up. In 2022 Hong Wang and Joshua Zahl handled the worst-behaved, “sticky” Kakeya sets. The final assault was now in reach. - Wolff · 1995: ≥ 2.5 - Katz–Łaba–Tao · 2000: > 2.5 - Wang–Zahl · 2025: = 3 - 2014 · Guth: any counterexample is “grainy” - 2022 · Wang–Zahl handle “sticky” sets Chapter IV · Step 9 #### Inch by inch Three dimensions fell only after fifty years of partial results, each raising the smallest possible dimension a little higher toward 3. Wolff · 1995 **≥ 2.5** Guth · 2014 **grains** Wang–Zahl · 2022 **sticky sets** Wang–Zahl · 2025 **= 3** Sources: [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/); [Wikipedia: Kakeya set](https://en.wikipedia.org/wiki/Kakeya_set) ### Counting grains *Chapter IV · The idea* A perpetual-motion machine. The breakthrough was to stop tracking individual needles and track the grains instead — the tiny blocks where many needles bundle together. Grains are easier to count, and there is a firm limit on how many can crowd through any one point. That limit is the lever. Wang and Zahl set up an induction: if no Kakeya set can be wispy up to some dimension, then none can be wispy a little beyond it either. Each turn of the argument pushes the bound higher. Terence Tao described it as perfecting a perpetual-motion machine — the argument gives back a little more than you put in, again and again, until the bound reaches three. ![The small blocks where many needles bundle together.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c07.jpg) > They’re getting more out than they put in. (Terence Tao, on the induction) Chapter IV · Step 10 #### Counting grains Instead of needles, count the **grains** — the small blocks where many needles overlap. A limit on how many grains meet at a point drives an **induction** that lifts the dimension bound, step after step, up to 3. “They’re getting more out than they put in,” said Terence Tao — a mathematical perpetual-motion machine. Sources: [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/); [Wang & Zahl, “Volume estimates for unions of convex sets, and the Kakeya set conjecture” (arXiv:2502.17655, Feb 2025)](https://arxiv.org/abs/2502.17655) ### The proof *Chapter IV · February 2025* And the tower it holds up. In February 2025, Hong Wang of New York University and Joshua Zahl of the University of British Columbia posted their proof: every Kakeya set in three dimensions has full dimension three. Nets Katz called it a once-in-a-century result. It was not just one problem solved — it was the foundation stone of a tower. Above Kakeya sit the restriction conjecture, the Bochner–Riesz conjecture, the local smoothing conjecture — deep questions about how waves and the Fourier transform behave. Had Kakeya been false, the whole tower would have fallen. Now, as one mathematician put it, all of those dreamed-about problems suddenly look approachable. ![Kakeya is the base stone; above it, deep problems about waves.](https://www.planetpast.com/stories/kakeya-needle-problem/img/c09.jpg) - base · the Kakeya conjecture - ↑ the restriction conjecture - ↑ Bochner–Riesz - ↑ local smoothing Chapter IV · Step 11 #### The proof, and the tower In **February 2025**, Hong Wang (NYU) and Joshua Zahl (UBC) proved the **three-dimensional Kakeya conjecture** (arXiv:2502.17655). Kakeya is the base of a tower of problems about waves and the Fourier transform. - base the Kakeya conjecture - ↑ the restriction conjecture - ↑ Bochner–Riesz - ↑ local smoothing “A once-in-a-century kind of result,” said Nets Katz. Sources: [Wang & Zahl, “Volume estimates for unions of convex sets, and the Kakeya set conjecture” (arXiv:2502.17655, Feb 2025)](https://arxiv.org/abs/2502.17655); [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/); [Wikipedia: the Fourier restriction problem](https://en.wikipedia.org/wiki/Restriction_theorem) ## Epilogue ### A needle, a century, a medal *Epilogue* What a hundred-year game was really about. What began as a puzzle about turning a needle became a question about the hidden shape of sets that are invisible to area but full of direction. In 2026, Hong Wang was awarded the Fields Medal, mathematics’ highest honour, in large part for this work. And the game is not over. The Kakeya conjecture in four dimensions and above is still open — the same question, one dimension at a time. From a needle on a table to the frontier of modern analysis: proof that the simplest questions can run the deepest. - **1917 → 2025**: from the question to the 3D proof - **2026**: Hong Wang, Fields Medal - **n ≥ 4**: still open > The simplest questions can run the deepest. (The Kakeya story) Epilogue #### A needle, a century, a medal A puzzle about turning a needle became a deep truth about the hidden dimension of direction. In **2026**, Hong Wang received the **Fields Medal** for this and related work. - 1917 → 2025 from Kakeya’s question to the 3D proof - 2026 Hong Wang, Fields Medal - open dimension 4 and higher The animations here are illustrations of the ideas, not exact mathematical constructions. Sources: [UBC Mathematics: the Kakeya conjecture and the 2026 Fields Medal](https://www.math.ubc.ca/node/4209); [Wikipedia: Hong Wang](https://en.wikipedia.org/wiki/Hong_Wang_(mathematician)); [Quanta Magazine: “Once-in-a-Century” Proof Settles the Kakeya Conjecture (Mar 2025)](https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/)