Two finite instruments and one universal theorem
The Haystack Cannot Flatten
Compress unit segments pointing in every planar direction, then cross into a live three-dimensional tube census. The finite experiments measure their own limits while two current preprints supply the universal result: every Kakeya set in three-dimensional Euclidean space has full dimension.
Area is not dimension. A planar set can have zero area and still contain a unit segment in every unoriented direction. The current three-dimensional result says every analogous set fills all three dimensions in the Hausdorff and Minkowski senses. It does not say the set has positive volume.
Layer one: compress the witnesses
Fold a triangle into a thicket
The browser cuts each of three source triangles into equal wedges, then translates the wedges with the Perron floor-index rule. The union area comes from the resulting coordinates and every edge intersection. Scrub the angle to ask for one actual unit segment.
Computing the finite Perron construction.
Layer two: count the tubes
Give every direction thickness
Choose a tube radius. The browser greedily selects projectively separated directions, lays down one unit capsule per direction, and scans a voxel grid. A cell wholly inside the union enters the lower count. A cell that might touch the union enters the upper count.
Computing the three-dimensional tube census.
Turn a dimension floor into a volume exponent
For a subset of three-dimensional space with dimension floor d, the corresponding neighborhood-volume exponent is 3 - d. The rail performs only this conversion. Its dimension floors are published results, not outputs of the simulation.
Zero volume and full dimension can coexist
Finite comparison viewComputing the finite product comparison.
computing
Visible apparatus
The check
Everything in the two boxes below is recomputed from the current controls. The theorem sits after the hard divider because the browser does not prove a statement about every Kakeya set.
Finite planar construction
- Perron pieces
- Polygon union area
- Angles sampled
- Witness length
- Containment failures
- Pál-join gaps
Finite three-dimensional census
- Direction count
- One capsule volume
- Summed capsule volume
- Voxel lower bound
- Voxel upper bound
- Boundary-cell interval
- Average multiplicity
- Two-scale slope
Conventions and free choices. Directions are unoriented, so angles are taken modulo pi. Planar length is normalized to one triangle height. The heart ratio is chosen by the reader. The planar area algorithm treats triangle coordinates as exact floating-point inputs, finds all edge-crossing x coordinates, then integrates union slices between them. The 3D tube is the closed radius-delta neighborhood of a unit segment, including spherical caps. Tube centres are either all at the origin or moved by the displayed woven rule. The direction selector is a deterministic greedy sample, not a canonical delta-net.
Approximations and uncertainty. Every finite planar stage has positive area. The angle sweep samples integer degrees rather than a continuum, although the highlighted witness is also checked at the exact slider angle. Voxel cells create the displayed lower-to-upper interval. The two-scale slope changes the finite direction family as delta changes. It is an illustration, not a dimension estimator with a proved error bar. The finite product comparison does not prove a limiting zero-volume statement.
Continuous motion. This page selects orientation witnesses. It does not implement the Pál joins needed to turn the needle continuously between translated positions, so it reports no motion-gap count and makes no continuous-motion claim.
The cited theorem, not numerically reproved here
Wang and Zahl's Theorem 1.1 states that every Kakeya set in R^3 has Minkowski and Hausdorff dimension 3. Guth, Wang, and Zahl give a separate streamlined proof. Both current arXiv records are preprints with no journal reference listed, though the streamlined proof's second version (11 October 2026) says it is to appear in Mathematika. Full dimension does not imply positive three-dimensional Lebesgue measure.
Still open. Wang and Zahl explicitly say their result does not resolve the Kakeya maximal-function conjecture in R^3. Their theorem is specific to three dimensions. A current survey continues to state the set problem as a conjecture in general dimension and develops separate higher-dimensional bounds, so this page does not extend the result to R^n for n >= 4.
- Hong Wang and Joshua Zahl, arXiv:2502.17655, submitted 24 February 2025. Theorem 1.1 and Theorem 1.2. preprint
- Larry Guth, Hong Wang, and Joshua Zahl, arXiv:2601.14411, submitted 20 January 2026. Streamlined proof, 47 pages on its first version; the current record (v2, 11 October 2026) has 49 pages and says it is the final version, to appear in Mathematika. preprint
- Larry Guth, arXiv:2505.07695, submitted 12 May 2025. Expository introduction whose author says he did not check every detail. preprint
- Thomas H. Wolff, DOI:10.4171/RMI/188, Revista Matemática Iberoamericana 11(3), 651-674, published 31 December 1995. peer reviewed
- Nets Hawk Katz, Izabella Łaba, and Terence Tao, DOI:10.2307/2661389, Annals of Mathematics 152(2), 383-446, September 2000. peer reviewed
- Nets Hawk Katz and Joshua Zahl, DOI:10.1090/jams/907, Journal of the American Mathematical Society 32, 195-259, published electronically 29 August 2018. peer reviewed
- Roy O. Davies, DOI:10.1017/S0305004100046867, Mathematical Proceedings of the Cambridge Philosophical Society 69(3), 417-421, May 1971. peer reviewed
- A. S. Besicovitch, DOI:10.1080/00029890.1963.11992093, American Mathematical Monthly 70(7), 697-706, 1963. peer reviewed
- Joshua Zahl, DOI:10.1137/25M1805965 and arXiv:2512.09397, Proceedings of the International Congress of Mathematicians 2026, volume 4, 270-287. The current record lists a proceedings publication; its review procedure was not verified. review unverified
Offline differential verifier:
node research/the-haystack-cannot-flatten/verify-the-haystack-cannot-flatten.mjs
Corrections
Corrected 2026-10-18: The note linking this layer to the coastline layer said the live neighborhood-volume slope is bracketed by voxel boundary cells. Only the volume is bracketed; the slope is read from two midpoint volumes and carries no bracket, as the uncertainty note on this page says. The note now says so; found by the assay line, 2026-10-18.
Updated 2026-10-18: The page said the Guth, Wang, and Zahl record ran to 47 pages on its current version and listed it among two preprints with no journal reference. The page count was true from its first version (20 January 2026) until 11 October 2026, when its second version gave 49 pages and said it is the final version, to appear in Mathematika; the arXiv record still lists no journal reference. Found by the assay line, 2026-10-18.