What is the Kakeya conjecture?

In higher dimensions, a Kakeya set contains a unit line segment in every direction. The Kakeya conjecture (still open in dimensions ≥ 4) asserts that such sets must have full Hausdorff dimension. In the plane the set can have measure zero; the interesting question is dimension. In 3D, Wang & Zahl proved dimension must be 3.

Who proved the 3D Kakeya conjecture?

Hong Wang and Joshua Zahl proved the three-dimensional Kakeya conjecture in 2025: every Kakeya set in R³ has Minkowski and Hausdorff dimension 3. Wang was awarded a Fields Medal in 2026 for this breakthrough and related harmonic-analysis work.

What is a Kakeya set?

A Kakeya set (or Besicovitch set) in Rⁿ is a set that contains a unit line segment in every direction. Besicovitch showed planar Kakeya sets can have arbitrarily small area. The modern conjecture concerns their fractal dimension in higher dimensions.

Why is the deltoid important in the Kakeya needle problem?

Kakeya suggested a three-cusped deltoid as a candidate for a small turning region. A unit needle can reverse direction while its endpoints slide on the deltoid (parameter a = 1/4, region area π/8) — already far below the naive disk π/4. Besicovitch later showed even the deltoid is not optimal: area can be driven arbitrarily small.

What is the Hausdorff dimension of a Kakeya set?

In the plane, Kakeya sets can have measure zero but still have Hausdorff dimension 2. The Kakeya conjecture predicts full dimension n in Rⁿ. This is known for n = 2 (classical) and n = 3 (Wang–Zahl, 2025); higher dimensions remain open.

How is my score calculated in this explorer?

Your score is the area of the union of all needle positions as you cover every direction. We paint the needle into an off-screen bitmap (not canvas getImageData), count unique pixels, and convert with a fixed scale of 300 px per unit length. Sliding purely along the needle paints almost nothing; spinning about the midpoint paints a disk of area about π/4. Preset replays use the identical pipeline.

Interactive Math Explorer

Kakeya Needle Explorer How Small Can You Go?

Turn a unit needle through every direction while sweeping as little area as possible. A hundred-year-old geometry puzzle — and the breakthrough behind the 2026 Fields Medal.

Playable needle challengeLive swept-area scoreKakeya's deltoid replayBesicovitch trick demo

The needle challenge

Drag a unit needle through every direction. Score is the union area swept— smaller is better. Body slides, center dot spins, ends pivot. Axial slides are free.

Loading explorer…

Watching: the naive spin. Touch the needle to take over.

Live score

0.00e+0

units² swept

Directions0%

Naive spin: 0.7854
Deltoid (π/8): 0.3927

Presets

Built with ConceptViz · pure canvas math, no AI images

Geometry Snapshots

Deterministic diagrams of the classical constructions — zero AI images

View:

Kakeya's Deltoid Sweep

Unit chords of a three-cusped deltoid turn through every direction. The classical deltoid region has area π/8 ≈ 0.393.

deltoidkakeyageometry

Perron Tree Construction

Besicovitch’s idea in stages: cut a triangle into thin pieces, slide them so they overlap heavily, then rejoin — the area can shrink without bound.

perron-treebesicovitchconstruction

What is the Kakeya needle problem?

In 1917, Sōichi Kakeya asked: if a unit-length needle must turn through every direction in the plane, how small can the region it sweeps be? The first guess is a disk of radius 1/2 (area π/4). The true answer shocked mathematics — the swept area can be made arbitrarily small.

Why can the swept area be arbitrarily small?

Abram Besicovitch proved in 1919 that there is no positive lower bound. Sliding the needle along its own direction costs zero area. By splitting a rotation into many tiny angles and overlapping the thin triangles that each angle sweeps (a Perron tree), the total measure can be driven below any ε > 0.

From deltoid to Besicovitch sets

Kakeya proposed a three-cusped deltoid as a candidate: a unit needle can reverse direction inside a deltoid of area π/8 ≈ 0.393 — already far below the naive disk π/4.
Among convex sets the minimum is larger: Pál’s equilateral triangle of area 1/√3 ≈ 0.577.
Drop convexity and the area has no positive lower bound — that is the classical Kakeya needle theorem in the plane.

The 3D breakthrough: Wang & Zahl

Area can shrink, but dimension cannot. In 2025, Hong Wang and Joshua Zahl proved that every Kakeya set in three dimensions has Minkowski and Hausdorff dimension 3, resolving the 3D Kakeya conjecture. Wang received a 2026 Fields Medal for this and related work.

How is your score measured?

Score = union area of every position the needle occupies while covering directions 0°–180° (undirected).
We software-rasterize the unit needle on a 6×6 world at 300 px per unit, counting newly painted pixels only. Adjacent frames are interpolated so fast drags cannot skip area.
Reference values: naive midpoint spin ≈ π/4 ≈ 0.785; deltoid unit-chord construction ≈ π/8 ≈ 0.393; pure axial translation ≈ 0. Tier thresholds use the same measured pipeline as the live scorer.

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Why can the swept area be arbitrarily small?

From deltoid to Besicovitch sets

Kakeya proposed a three-cusped deltoid as a candidate: a unit needle can reverse direction inside a deltoid of area π/8 ≈ 0.393 — already far below the naive disk π/4.

Among convex sets the minimum is larger: Pál’s equilateral triangle of area 1/√3 ≈ 0.577.

Drop convexity and the area has no positive lower bound — that is the classical Kakeya needle theorem in the plane.

How is your score measured?

Score = union area of every position the needle occupies while covering directions 0°–180° (undirected).

We software-rasterize the unit needle on a 6×6 world at 300 px per unit, counting newly painted pixels only. Adjacent frames are interpolated so fast drags cannot skip area.

Reference values: naive midpoint spin ≈ π/4 ≈ 0.785; deltoid unit-chord construction ≈ π/8 ≈ 0.393; pure axial translation ≈ 0. Tier thresholds use the same measured pipeline as the live scorer.