Can a Shape with No Volume Still Be Fully Three-Dimensional? The Math Behind a Fields Medal Breakthrough

Andrei Mihai

A set can occupy no volume at all and still be fully three-dimensional. It sounds contradictory and visually it can be very hard to wrap your head around it. Unsurprisingly, this concept sits at the heart of one of mathematics’ most stubborn problems. It is also one of the contributions that helped earn Hong Wang a 2026 Fields Medal.

In 2025, Wang and mathematician Joshua Zahl finally proved the three-dimensional Kakeya conjecture, addressing a problem that had resisted generations of mathematicians. Mathematicians had already known that a set containing a unit-length line segment pointing in every possible direction can have zero ordinary volume. But Wang and Zahl proved that such a set nevertheless cannot have dimension less than three.

Geometry and Needles

Start with an infinite collection of unit-length needles. Give one every possible orientation in three-dimensional space. You may slide each needle wherever you want. How tightly can you pack them?

That is essentially the modern Kakeya problem. More precisely, mathematicians call a compact set a Kakeya set if it contains a line segment of unit length pointing in every possible direction. The segments do not have to meet at a common point. Nothing requires one physical needle to travel continuously between them. Another way to define the problem is:

A compact subset of Rⁿ is called a Kakeya set is if it contains a line segment of length 1 in every possible direction.

Nowadays, mathematicians are not really thinking about needles, but the problem was inspired by a literal needle puzzle.

In 1917, Japanese mathematician Sōichi Kakeya asked how little area was needed to turn a unit needle through every direction in the plane. Turning it around its midpoint sweeps out a disk, but cleverer motions can use much less space. In 1919, Abram Besicovitch constructed extraordinarily thin sets of zero area containing a line segment in every direction. His construction was later adapted to show that a needle could actually be turned continuously through 180 degrees inside regions of an arbitrarily small area.

So, mathematicians already knew that Kakeya sets could be extraordinarily thin. The mystery was how thin. Could a three-dimensional Kakeya set have zero volume because, underneath all those directions, it was actually behaving like a two-dimensional sheet? Could it have some strange fractional dimension?

Wang and Zahl proved the answer.

How Can Something with Zero Volume Be Three-Dimensional?

In everyday speech, “three-dimensional” and “volume” are generally used interchangeably. But in mathematics, that is not the case. In a Euclidean sense, volume is a three-dimensional measure of the internal space of an object. But in this context, volume measures how much three-dimensional space a set occupies. Dimension, in the sense relevant here, describes how the complexity of a set scales as you examine it at finer and finer resolutions.

We can look at it another way: Volume asks how much space something occupies, while dimension asks how its complexity changes as you examine it at finer scales.

A simpler way to think about this is if we are covering a three-dimensional object with a grid of tiny cubes, how many cubes are needed?

Suppose you cut the width of those cubes in half. For a line, you will need twice as many cubes to cover it (2¹). For a flat sheet, you need four times as many (2²). For a solid block, you need about eight times as many (2³). The exponent tells you how the amount of detail grows as the scale gets finer: one-dimensional for the line, two-dimensional for the sheet, three-dimensional for the solid.

More generally, if the cubes have width δ, the number needed grows roughly like 1/δ for a line, 1/δ² for a surface, and 1/δ³ for a three-dimensional object. This way of measuring dimension is called the Minkowski dimension.

A Kakeya set might contain enormous amounts of empty space. Its ordinary three-dimensional volume can even be zero. But Wang and Zahl showed that when you probe it at finer and finer scales, its complexity still grows at the three-dimensional rate. Their theorem establishes full dimension under both the Minkowski dimension and a measure called the Hausdorff dimension, also used to quantify the dimension of complicated or fractal sets. In essence, even though a Kakeya set may have zero volume, its fine-scale complexity cannot have the scaling behavior of a surface, a curve, or a lower-dimensional fractal.

The Third Dimension Is Tough

Mathematicians solved the two-dimensional version decades ago. In 1971, Roy Davies proved that every planar Kakeya set has dimension two; Antonio Córdoba later obtained a similar result through a different approach.

Three dimensions introduced far more room for geometrical mischief.

Wang made these kinds of questions central to her career. Much of her research sits where geometry meets harmonic analysis, a branch of mathematics concerned, roughly speaking, with breaking complicated signals into waves. In July 2026, she became one of four recipients of that year’s Fields Medals and only the third woman in the award’s history to receive one. The International Mathematical Union cited a much broader body of work, including her contributions to Fourier restriction, Falconer distance sets, Furstenberg sets, the local smoothing problem for the planar wave equation, and the Kakeya problem in three dimensions.

Yet the Kakeya proof stands out even in that body of work.

Of course, Wang and Zahl’s paper runs 127 pages and it is impossible to even attempt to do it justice in a few paragraphs. Yet while the proof is enormously technical, its basic logic is surprisingly neat.

To attack the problem, mathematicians make the infinitely thin lines slightly thicker. They replace each line segment with a very narrow tube. This turns an object with zero volume into something whose volume can actually be measured. This makes the question more about how much these tubes pointing in all these different directions can overlap. If they could overlap too efficiently at every scale, the underlying Kakeya set might have dimension below three.

Years earlier, Wang and Zahl had solved a special case involving “sticky” Kakeya sets. These are arrangements where nearby thin tubes repeatedly bunch together inside broader tubes as you zoom out, creating a recognizable hierarchy across scales.

Wang and Zahl analyzed the problem across many different scales, replacing the infinitely thin line segments with narrow tubes and studying how those tubes can overlap. They split complicated tube arrangements into smaller rectangular structures known as “grains” and established new bounds on how efficiently the tubes can cluster together inside convex regions.

The approach is an iterative one. Earlier work showed that a three-dimensional Kakeya set could not have dimension too far below three. Wang and Zahl developed a mechanism that takes such a lower bound and improves it. Once improved, the argument can be run again, producing a still stronger bound, and then again. Repeating this process pushes the minimum possible dimension all the way up to three.

More broadly, the proof exploits a recurring principle in the problem: The harder one tries to force line segments pointing in every direction into too little space, the more geometric structure the arrangement is forced to develop. Eventually, that structure becomes too restrictive for a lower-dimensional Kakeya set to exist.

This is also why Kakeya grew into something much larger than a puzzle about needles. In harmonic analysis, complicated waves can often be decomposed into “wave packets” concentrated along long, thin tubes. Understanding how such tubes overlap is therefore connected to major questions about Fourier transforms and partial differential equations, which is important for a larger network of problems.

And the Kakeya story itself is not finished: Wang and Zahl settled the conjecture in three dimensions, while the analogous problem remains open in four and higher dimensions.

Still, there is something especially beautiful about this result. Kakeya’s problem began with an almost childlike question about how tightly a needle could be turned. More than a century later, that question led mathematicians through fractal geometry, harmonic analysis, and some of the most sophisticated multiscale arguments in modern mathematics. Even without grasping the full complexity behind it, one can appreciate the elegance.

The post Can a Shape with No Volume Still Be Fully Three-Dimensional? The Math Behind a Fields Medal Breakthrough originally appeared on the HLFF SciLogs blog.