Separate-volume total
Adding individual tube volumes overcounts overlap
If several tubes share a region, adding their full volumes counts that same region several times.
2026 Fields Medal · an interactive guide for non-specialists
The 3D Kakeya problem: volume can vanish; dimension cannot.
At a finite scale, an ideal line segment looks like a thin tube. Packing explains how tubes pointing throughout space can occupy very little volume. The breakthrough answers the harder question left after that: can the limiting set lose dimension? In ℝ³, it cannot.
01 · Pack the thick needles
Keep every tilt. Change only where each tube sits.
The paper studies δ-tubes: the δ-scale neighborhoods of unit line segments. Moving a tube does not change its direction. Good placement lets different tubes reuse some of the same space.
Separated positions
drag to rotate
The tubes have many tilts, but little shared space.
What is allowed to change?
Each tube’s tiltfixed
Each tube’s lengthfixed at 1
Each tube’s locationfree to move
tubes drawn100a finite sample
tube scaleδvisible thickness
rotations made0directions survive
Union volume
Measure shared space only once
The union counts a region once, whether one tube or ten tubes pass through it.
02 · Count the cubes
Freeze that same packing. Put a three-dimensional mosaic over it.
A cube lights up exactly when it meets one of the displayed thick needles. Counting those cubes is the spatial version of covering a picture with tiny squares.
1Same packed tubesNo new geometric object
→
2Add a δ-cube gridOne ruler for all three axes
→
3Count every touched cubeOverlaps are counted once
Same packing · δ-scale microscope
drag to rotate
Tubes and the cubes that touch them are shown together.
Choose the ruler
one unit, divided into 10
10 × 10 = 100distinguishable direction samples
What should remain visible?
The counting rule
Touch once → count once. A cube still counts only once when several tubes pass through it.
Twenty-two exact rulers at once
Count, occupied fraction, and finite exponent—from the same cubes.
All counts were computed offline. The first ten match the interactive ruler choices above; twelve data-only checkpoints fill the gap to 1/100, then extend the same geometry to 1/1000.
The third curve uses d2→m = log(Nm/25) ÷ log(m/2). Every plotted count is exact for this finite model and was precomputed offline. JavaScript builds a 3D cube surface only for the interactive ruler currently selected. None is the final dimension. Wang–Zahl prove that every genuine Kakeya set in ℝ³ has limiting dimension 3 over arbitrarily fine rulers.
Occupied fraction · one ruler size
How much of the enclosing frame is colored?
This 100-tile gauge copies the percentage from the three-dimensional counter above. Each tile represents one percent of the available cube positions—not a flat slice of the tubes.
904 touched÷3,375 available=26.79%
Two measurements, two different questions
Volume is the occupied amount in each sharper snapshot
N(δ) × δ³ ≈ 0.9
At each scale, count the cubes and multiply by one cube’s volume. Ordinary volume is what this occupied amount approaches as δ tends to zero.
Finite exponent from the actual counts
log(904/25) ÷ log(10/2) = 2.229
From 25 cubes at 1/2 to 904 cubes at 1/10. This is a measured finite-scale slope; dimension is its limiting analogue over arbitrarily finer rulers.
Wang–Zahl · three dimensionsResolved
The academic statement
Every Kakeya set in ℝ3 has dimension 3.
Volume can shrink to zero. Dimension cannot.
The theorem does not say that a Kakeya set must have positive ordinary volume. It may have volume zero. It says that no arrangement can make its small-scale covering count grow with an exponent below 3.
In plain language: overlap can shrink the amount of space occupied, but it cannot make the set behave like a surface, a curve, or anything genuinely lower-dimensional.
Minkowski dimension3count equal-sized cubes
Hausdorff dimension3allow flexible tiny coverings
ordinary volumemay be 0full dimension does not forbid this