A visual journey from linear change to the hidden circle —
how τ emerges naturally when you integrate the Gaussian.
Four sections, each building from the most ordinary observation to the surprising consequence:
- §1 — Why x²/2 is the natural distance (constant acceleration, the triangle under v = t)
- §2 — How 1/x, log, and exp arise from relative change (scale invariance, the product rule)
- §3 — Why exp(−x²/2) is the unique shape (quadratic energy + independence + the multiplicative product rule)
- §4 — The hidden circle (why ∫ exp(−x²/2) dx = √τ — and where the τ comes from)
Every section ships with interactive HTML5 canvas visualizations. Drag sliders, hit ▶ play, watch the geometry build from zero.
It's a single static page — no build step. Open index.html in any modern browser, or serve the folder:
python -m http.server 8000
# then visit http://localhost:8000/| File | Purpose |
|---|---|
index.html |
English article (single-file, self-contained) |
index_zh.html |
Chinese article (中文) |
logo.svg |
Repo / article hero logo (rich; ≥ 64 px) |
logo-minimal.svg |
Favicon, mobile / inline icon (≥ 16 px) |
og-card.png |
Open-Graph social-share image |
og_card.py |
Script that regenerates og-card.png |
Typos, questions, collaboration: tau@mathtau.com or open a GitHub issue.
Article text and figures: CC BY 4.0 — see LICENSE. Attribution to MathTau appreciated.