An interactive Three.js proof that τ (tau = 2π) is the unique constant where one unit of counting equals one full rotation — and a live atlas of every mathematical structure that emerges from that single axiom.
Overview
Leonhard Eulers Day Off is a mathematical visualisation tool built around a single observation: when you rewrite Euler's formula in τ-native form, the generalised rotation τ^{i·nτ/ln(τ)} closes at exactly n=1. For any other base, one step is not one turn. The constant τ is forced by the geometry of counting and closure.
The project takes this proof and renders it as a live, interactive 3D atlas using Three.js — 6,144 distinct mathematical structures across eight independent visibility axes, all driven by a single set of parametric equations.
The Origin
It started with a Desmos notebook. Three expressions side by side: Euler's identity (e^{iτn}), the generalised rotation (α^{i·nα/ln(α)}), and the τ-native form (τ^{i·nτ/ln(τ)}). The third expression is algebraically identical to the first. That equivalence — visible in real time — is the proof.
The Atlas
The atlas takes the core expression f = k₁ · τ^{i·τ^k/ln(τ)} and the primary visual sin(n·f)·k₂ strand points, then overlays every mathematically distinct variant across eight independent visibility axes. Each axis is a continuous opacity multiplier:
- A — Strand vs atlas toggle
- B — Sign pairing (positive/negative)
- C — i-sign pairing (±i)
- D — Trig pipeline (id, sin, cos, tan)
- E — Exponent configuration
- F — Secondary modulation
- G — Quadrant coloring (4 sign/i-sign pairs)
- H — Strand count across 8 configurations
The full combinatoric space: A(2) × G(4) × E(3) × D(4) × B(2) × C(2) × F(2) × H(8) = 6,144 distinct items.
Technology
- Rendering: Three.js via CDN importmap — no build step
- Mathematics: τ-native complex arithmetic on [re, im] 2-tuples
- Post-Processing: UnrealBloomPass cinematic mode
- Animation: Portable scroll-animation with link targets, easing, and loop modes
- Architecture: Vanilla JavaScript, zero dependencies beyond Three.js
My Role
Creator and sole developer. The mathematical framework — proving τ is forced by closure, not convention — originated from personal research into generalised Euler identities. The visualisation was built to make that proof visible, interactive, and navigable.