Kakeya's Needle, from Zero

The needle in the picture is one unit long. As it turns, it eventually points in every possible direction, yet it never leaves the ragged green region. That region covers less than a quarter of a square unit.

There is no positive lower limit on the area. The region can be made smaller than any positive size you choose. A related construction even has area exactly zero, while still containing a needle in every direction.

No mathematical background is assumed. Each section adds one idea, with the pictures carrying as much of the explanation as possible.

An open repository accompanies the article. It contains the figure code and tests that recompute the numerical examples with exact arithmetic. Claims that depend on published mathematics rather than those tests are identified as such.

1 · The needle and its directions

Begin with a straight piece of wire one unit long; this is the needle. Its position tells us where it lies, while its direction tells us how it is turned. Direction is the important part of this problem.

A needle pointing east occupies the same line as one pointing west because it has no marked head or tail. Its distinct directions therefore fill half a turn, from 0 degrees up to 180 degrees, before repeating.

One half-turn is enough for the needle to pass through every distinct direction in the plane.

We will keep two related questions separate:

1. Does the shape contain a needle in every direction? If it does, it is a Kakeya set.
2. Can one needle move continuously through those directions without leaving the shape? If it can, the shape is a Kakeya needle set.

The second condition asks for more. Separate parking spaces can hold a car at many angles, but a connected road is needed to drive between them. The same distinction matters here.

2 · Turning around

The simplest room is a disc: place the needle across its centre and spin it. A unit needle needs a disc of diameter one, covering about 0.7854 square units.

Smaller rooms are possible. The next picture shows the complete motion inside three different choices.

The triangle. An equilateral triangle of height one works and covers about 0.5774 square units. The needle makes a three-point turn: it pivots 60 degrees around a corner, slides along a side, and pivots around the next corner. Three such pivots complete a half-turn. Julius Pal proved in 1921 that no smaller room without inward dents can work.

The deltoid. A room with inward dents can be smaller. Roll a circle with radius one quarter inside a circle with radius three quarters and follow one point on the smaller circle. The resulting three-cusped curve has a unit-length chord along every tangent direction. The needle follows those chords as it moves around the room, which covers about 0.3927 square units.

Soichi Kakeya asked in 1917 for the smallest possible room and suspected that the deltoid was optimal. The actual answer follows from two simple geometric observations.

3 · How much space

When two shapes overlap, their combined area is not found by simply adding their separate areas. The shared patch belongs to the union only once.

The small-area constructions rely on making many pieces share the same ground.

Split a triangle down the middle, then slide its right half over the left. Neither piece is resized or removed, and translation does not change the directions contained in either half. Even so, the two pieces now cover less of the page because part of their area is shared.

At the best offset, the two halves together cover exactly two thirds of the original triangle. The exact calculation in the repository confirms that this minimum occurs when the slide is two thirds of the chosen unit.

The article computes area in the same visual way. A horizontal line moves up the figure; at each height, the code measures the covered width and counts overlaps once. Those widths are then added using exact fractions rather than sampled estimates.

Overlapping the pieces reduces their combined area without removing any of their directions. It lets several directions use the same patch of space.

4 · The free slide

A sideways-moving needle paints out a rectangle. If it moves along its own line instead, it covers only that line, which has no area.

In area terms, the needle may travel any distance along its own direction at no cost. A pivot around one endpoint does cost area: it sweeps a pie-shaped region whose area is half the size of the turning angle when that angle is measured in radians.

These two motions can be combined to transfer the needle from one line to another parallel line.

First slide far along the original line. Make a very small pivot, then slide along the new, slightly slanted line until it reaches the target line. A second small pivot restores the original direction. Only the two pivots add area, and their combined cost can be made as small as desired. This manoeuvre is a Pal join.

Smaller pivots require a longer journey. For a fixed gap, halving the pivot angle roughly doubles the distance travelled. The path may therefore become extremely long and thin while its area continues to decrease.

5 · The Perron tree

The same overlap can be repeated after cutting the triangle into many pieces.

Use a triangle with a unit base and an apex one unit above it. Divide the base into a power-of-two number of equal intervals and connect each interval to the apex. The resulting thin triangles are called slivers. Together they retain the original triangle's full fan of directions. They can now be overlapped in stages: first neighbours, then neighbouring pairs, then groups of four, and so on.

Because every sliver is only translated, not turned or resized, the final pile still contains every direction in the original fan. This arrangement is a Perron tree. Oskar Perron gave this pictorial form of Besicovitch's construction in 1928.

Repeatedly using the best two-piece overlap is not enough. If every stage keeps two thirds of each merged pair, the process eventually stops improving.

This steady squeeze produces a bowtie. Every sliver passes through one point, but the solid triangle beneath that point remains. The total area is one sixth plus a small extra amount that shrinks with each cut, so the area approaches one sixth and goes no lower. Even an enormous number of slivers leaves one third of the original triangle's area.

The useful construction varies the overlap. It squeezes more where the slivers are thin and less where they are wide. At a stage numbered j, it retains the fraction obtained by dividing j plus one by j plus four.

The decrease is very slow. For a tree made from n slivers, the best possible area falls roughly like one divided by the logarithm of n, and known results show that this rate cannot be improved for such trees. Reaching an area below one percent would require roughly the number of slivers obtained by doubling one a hundred times. Small drawings therefore reveal the trend, but not how far the construction can ultimately go.

6 · Area zero

Every picture contains the same fan of directions while covering less ground. Besicovitch proved in 1919 that the area can be pushed below any positive amount. A limiting version has area exactly zero and still contains a full unit needle in every direction. This is called a Besicovitch set.

A single tree covers only the triangle's fan of about 53 degrees, not the full half-turn. Four rotated copies can cover four different parts of the direction circle. Their total area is still arbitrarily small, and together they supply every direction. The same four-copy repair will be used again in section 12.

Besicovitch found the construction while working in Perm, Russia, on Riemann integration rather than on Kakeya's original room problem. The unusual set was a tool for that separate question.

Containing all directions does not yet provide a continuous motion between them. Pal joins supply the missing connections.

Within each sliver, the needle pivots around the apex and passes through that sliver's small range of directions. A Pal join then moves it to the next sliver with an arbitrarily small addition of area. Combining the tree with these joins gives Besicovitch's 1928 result:

Choose any positive area, however small. There is a set below that area in which a unit needle can turn continuously through 180 degrees.

Consequently, there is no smallest room. Given any room that works, another one with less area can be constructed. The deltoid is therefore not an optimum once inward dents and complicated shapes are allowed.

7 · Zero, but big

Area can no longer distinguish these sets: a Besicovitch set, a dot, and a line segment can all have area zero. Dimension gives a finer comparison by recording how quickly a set fills space when viewed at smaller scales.

Cover a set with equal small boxes and count how many are needed. A dot continues to need one box. If the box width is made ten times smaller, a segment needs about ten times as many boxes, while a square needs about one hundred times as many. Those growth rates correspond to dimensions zero, one, and two.

The Cantor set lies between these familiar examples. Start with a unit interval, remove its middle third, then remove the middle third of every remaining piece, and continue forever. Each round keeps two thirds of the previous length, so the completed set has length zero.

At the natural scale for each round, making the boxes three times narrower requires exactly twice as many of them. A full line would require three times as many, while a dot would still need only one. Converting that growth rate into a dimension gives about 0.6309. The Cantor set therefore has zero length but a dimension between that of a dot and a line.

Roy Davies proved in 1971 that every Besicovitch set in the plane has dimension two, the same dimension as a solid square, despite having zero area.

8 · The conjecture

The ingredients are now in place. A Kakeya set contains a unit needle in every direction and may still have zero area or volume. Dimension detects size that area and volume miss. In the plane, Davies's theorem says that such a set must nevertheless have the largest possible dimension.

The corresponding question can be asked in space with three, four, or any larger number of dimensions. Volume may still fall to zero, but can the dimension fall below the dimension of the surrounding space?

in a space with any number of dimensions,
a set may have zero volume while
holding a unit needle in every direction,
but its dimension must still fill that space

This statement is the Kakeya set conjecture.

The conjecture allows the volume to be squeezed away, but not even a fraction of the dimension. It claims that only the maximum dimension is possible.

Every direction, in every dimension

The collection described by "every direction" depends on the dimension of the surrounding space.

The rotating camera makes the depth in these scenes visible. In two dimensions, the directions form a flat circle; its flatness is clear when the camera sees it edge-on. In three dimensions, the directions spread over a sphere. Four-dimensional directions cannot be viewed directly, so the final panel rotates them through a fourth axis and projects them into three dimensions, much as a cube can be projected onto paper. The changing projected lengths show movement through that hidden fourth axis rather than an ordinary three-dimensional rotation.

9 · Why it matters

An ideal needle has no thickness. Giving it a small chosen thickness turns it into a tube.

Tubes also appear when analysts describe waves. Over a short period, a wave packet travelling in one direction is concentrated in a tube-like region. Combining wave packets from many directions raises the same geometric question: how much can those tubes overlap?

This connection brings Kakeya estimates into the restriction problem in Fourier analysis, Bochner-Riesz summation, local smoothing for wave equations, dispersive partial differential equations, exponential sums in number theory, and randomness extractors in computer science. Progress on Kakeya can therefore affect several other problems about how directional pieces overlap.

10 · On a grid

In 1999 Thomas Wolff proposed a finite version of the problem. Choose a prime number, call it q, and do arithmetic on a grid whose coordinates wrap around after q steps.

In the two-dimensional version, each point has two coordinates chosen from the whole numbers starting at zero and ending one below q. A line starts at one point and repeatedly adds a fixed direction, producing q points before wrapping around. There is one more direction than the value of q. A Kakeya set contains a complete line in each direction.

For the smallest grids, a computer can search every candidate set:

The minimum occupies more than half of each of these small grids, rather than a vanishingly small part.

Dvir's argument works in any number of dimensions. The rotating example below shows what the finite grid looks like in three.

This grid has twenty-seven points and thirteen directions. A greedy computer search found the highlighted fifteen-point Kakeya set, while Dvir's general lower bound guarantees at least ten points. Rotation makes the grid's depth and the relative positions of its points visible.

The next argument is the most technical part of the article, but it uses only two ideas already available: a polynomial is a sum of terms made from numbers and coordinate variables, and both polynomial terms and grid points can be counted. The repository carries out each step on small grids, providing concrete numerical examples alongside the description.

In 2008 Zeev Dvir gave a short proof for every dimension. Start by calling the Kakeya set K and assume, for the sake of the argument, that it has too few points.

1. Fix a maximum polynomial degree one below q. In a space with n coordinate directions, there is a definite number of possible terms and therefore coefficients. Requiring the polynomial to be zero at one point places one linear condition on those coefficients.
2. If K contains fewer points than the number of coefficients, linear algebra produces a nonzero polynomial, call it g, that is zero at every point of K.
3. The set K contains a full line in each direction. Along any such line, g becomes a one-variable polynomial with q roots but degree below q. It must therefore be zero along the entire line.
4. The highest-degree coefficient on that line is obtained by evaluating the highest-degree part of g in the line's direction. Because every line restriction is zero, this top part is zero in every direction.
5. A highest-degree part that is zero in every direction is zero on the whole grid. A nonzero polynomial of degree below q cannot do that.
6. This contradicts the choice of g. Thus K must have at least as many points as the coefficient count; in dimension n, this is at least the total number of grid points divided by the factorial of n.

The repository checks these steps on small grids. It finds the vanishing polynomial by finite-field linear algebra, restricts it to each line, verifies that all resulting coefficients are zero, compares the leading coefficient with the highest-degree part, and computes the final rank.

11 · Why it was hard

The chart records progress in three dimensions by showing the best lower bound known in each period: the dimension that every Kakeya set in space had been proved to reach.

The 2000 result moved the bound beyond two and a half by only about one ten-billionth. The size of that improvement indicates how resistant the problem was to the available methods.

Wolff improved the count by changing the configuration under study. Many tubes through one point form a bush. He instead selected one tube crossed by many others; this is a hairbrush. Crossings distributed along a whole tube force more occupied space than crossings concentrated at one point. The resulting bound of two and a half remained the main benchmark for decades.

Dvir's grid proof does not transfer directly to ordinary space. Two finite-grid lines share at most one point, and containment of a line is exact. In ordinary space, nearly parallel tubes can overlap along much of their length, and a set may lie very close to many lines without containing them exactly. The polynomial argument has no direct way to measure this kind of approximate structure.

12 · The proof

On 24 February 2025, Hong Wang and Joshua Zahl posted their 127-page paper, Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions. The main theorem states:

Every Kakeya set in three-dimensional space has Hausdorff dimension three and Minkowski dimension three.

These are two ways of measuring dimension. Minkowski dimension is the box-counting ruler introduced in section 7. Hausdorff dimension is more sensitive and can assign a smaller value to irregular sets. Wang and Zahl proved that even this finer measurement reaches three.

A central ingredient is a volume estimate for thin tubes. If no convex region is allowed to contain too many of them, their union must occupy nearly the largest volume permitted by the estimate. The Kakeya conclusion is then derived from this tube statement.

The rotating view shows that the tube directions spread across an entire sphere. A fixed viewpoint can make them look as though they lie in a flat circle.

The flat tree from section 5 can be extended into space by giving it height, producing a thin slab.

Take any spatial direction described by three components: sideways, forward, and upward. The unit needle's shadow on the base plane is no longer than one unit. If that shadow points into the tree's fan, it fits within a segment already held by the tree, while the slab supplies the needed height. Repository tests sample the blue needle point by point and verify that it remains within both a sliver and the slab.

One slab is not a complete spatial Kakeya set. It inherits the original tree's limited fan and works only for directions whose shadows fall inside that fan. A repository test explicitly rejects directions outside it. As in section 6, four rotated copies cover the missing shadow directions. Adding height can be done without preventing the volume from tending to zero; controlling dimension is the difficult part.

The proof is specific to three dimensions. It uses the fact that two directions determine a plane, leaving only one further spatial direction to control. Four and higher dimensions allow more configurations, and current lower bounds remain below the full dimension there.

The conjecture is therefore proved in dimensions one, two, and three, but remains open from dimension four onward.

What you can check yourself

The repository does not claim to verify Wang and Zahl's 127-page proof. It checks the smaller constructions used in this guide: exact Perron-tree areas and the one-sixth limit for a fixed squeeze, the deltoid's unit chord, the triangle's three-point turn, the Cantor set's box dimension, minimum Kakeya sets on grids through seven by seven, and the finite computations illustrating each step of Dvir's proof.

make venv && make test && make figures

The original problem asked for a minimum area. Besicovitch showed that no positive minimum exists, which shifted attention from area to dimension. That replacement question remained unresolved in three dimensions for roughly another century and became closely connected with major problems in Fourier analysis.

Wang and Zahl's proof builds on several decades of methods for describing geometric structure across many scales. Their work made those methods strong enough to reach the full three-dimensional bound.

For the physical needle problem posed in 1917, the conclusion remains striking: continuous turning is possible inside regions of arbitrarily small positive area.

The plain words, and the real ones

The guide introduced several ideas in everyday language. The table pairs those phrases with the standard terms used in mathematical writing; it adds vocabulary, not new concepts.

Where to go next

• The companion repository, github.com/muchmirul/conjectures, has the code for every figure, the exact-arithmetic geometry, Dvir's proof run step by step, the full test suite, and research notes listing which claims are checked here and which are quoted.
• H. Wang and J. Zahl, Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions (arXiv:2502.17655), the 2025 proof itself.
• T. Tao's blog post, The three-dimensional Kakeya conjecture, after Wang and Zahl, a readable account of the strategy, written the day after the paper appeared.
• J. Zahl, A survey of the Kakeya conjecture, 2000 to 2025 (arXiv:2512.09397), for where the field stands and what is still open.
• Z. Dvir, On the size of Kakeya sets in finite fields (2008), the half page proof of section 10.
• K. Falconer, The Geometry of Fractal Sets, for Besicovitch sets and dimension built up from the ground.
• Quanta Magazine, 'Once in a Century' Proof Settles Math's Kakeya Conjecture (March 2025), the story with no equations in it.
• The Wikipedia article Kakeya set, for current status and further references.