The Needle & The Wave

A single-file browser game that teaches the Kakeya conjecture, its connection to harmonic analysis, and what that connection does (and honestly doesn't) mean for everyday technology. Aimed at a curious high-school student; ~20 minutes.

Play it: quantisan.github.io/kakeya-game, or npm install && npm run build and open dist/kakeya-game.html. Everything — code, fonts, libraries — is inlined; it works offline. Progress saves locally.

The arc (one level per historical beat)

1. 1917 — Spin the needle. Kakeya's question; sweep a needle, read the painted area, bet on convexity (and lose, like Kakeya did).
2. 1919 — The Besicovitch squeeze. Slide triangle slivers together: translation is free, area collapses, every direction survives. Rotating instead goes nowhere — the level lets you try.
3. 1971 — Box-count it. Davies' reformulation: the squeezed set has almost no area but box-counting slope climbing toward 2. Needle (1) and a Sierpinski fractal (~1.58) for contrast.
4. 1971 — Waves are needles. The hinge: drag a wedge in frequency space, watch the wave packet concentrate on a tube. Match the target tube. Pays off the header oscilloscope.
5. 1971 — Break the cutoff. Fefferman's proof as a game: pack near-parallel tubes under your own Level-2 par, apply the sharp cutoff, read the computed blowup ratio.
6. 1990s→2025 — The tower. Kakeya → restriction → Bochner–Riesz → local smoothing. Predict-then-reveal: failure propagates up; a proven foundation proves nothing above it.
7. Today — What's really running. Sort MRI / WiFi / song recognition / JPEG / MP3 into honest buckets. The "waits on the conjectures" bucket is correctly empty.
8. 2008 — Bonus: finite directions. Dvir's finite-field Kakeya as a 5×5 mod-5 puzzle; par computed by the game's own greedy search.

Every number on screen is computed, not scripted: areas are counted pixels, the dimension slope is a least-squares fit of real box counts, the tube shape comes from second moments of the wave field, and the blowup ratio is a pixel-count quotient.

Development

npm install
npm test        # pure-math core tests (Perron areas, box counting, wave moments, finite-field solver)
npm run build   # bundles TypeScript + libraries + fonts into dist/kakeya-game.html
npm run dev     # watch mode
node scripts/drive.mjs  # headless-Chrome playthrough of all levels (needs Chrome)

Architecture: pure math core in src/core/ (no DOM), canvas/DOM shells per level in src/levels/, one shared pixel-counting instrument in src/paint.ts. scripts/tune-*.ts are the numeric experiments that fixed the construction parameters and win thresholds.