Leonhard Eulers Day Off
Explore the τ-Euler Atlas — an interactive 3D visualisation proving τ is the natural constant of rotation, rendered live in Three.js with 6,144 distinct mathematical structures.
Leonhard Eulers Day Off
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.
Core Capabilities
| Prove | Explore | Visualise |
|---|---|---|
| Watch the τ-native unit circle close in exactly one step while every other base fails. Drag the α slider and see the proof in real time. | Navigate 6,144 distinct mathematical items across eight independent visibility axes — every combination of sign, trig, and exponent pairings. | Real-time Three.js rendering with bloom, fat-line strands, and cinematic camera modes across desktop and mobile. |
Features
The Axiom — Live Proof
The central equivalence drives everything: τ^{i·nτ/ln(τ)} ≡ e^{iτn}. This is Euler's formula rewritten in τ-native form. The app renders two ghost traces — a cyan τ-trace that always closes at n=1, and an amber α-trace that only closes when α = τ. Drag the base slider to any value and watch the proof unfold.
8-Axis Visibility Atlas
Eight independent visibility axes (A–H) control which mathematical structures are rendered. Each axis is a continuous opacity multiplier. The full combinatoric space spans 6,144 distinct items — every sign pairing, trig transformation, exponent variant, and strand configuration.
Portable Scroll-Animation System
Link any slider to the animation engine with end-value targets and easing curves. Multiple parameters sweep simultaneously from their base values to their targets, driven by a single global progress cursor with loop and bounce modes.
Cinematic Rendering
UnrealBloomPass post-processing, OrbitControls for 3D and 2D camera modes, additive-blended star particles with per-frame drift physics, and Line2 fat-line strand rendering. Performance mode strips bloom for smooth real-time interaction on lower-end devices.
Technology
Built with vanilla JavaScript and Three.js (via CDN importmap). No build step, no bundler — a single index.html entry point. The mathematical engine operates on [re, im] 2-tuples with τ-native complex arithmetic. The rendering pipeline uses WebGL with shader-based computation.
The Origin
The project started with a single Desmos notebook putting 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 — same curve, same rate, same closure. That equivalence is the proof.
Use Cases
Mathematics Education
Make the relationship between τ, e, and rotation concrete. Students can see — not just calculate — why τ is the natural constant of rotation by manipulating the base slider and watching closure behaviour change.
Research Visualisation
Explore the full parameter space of τ-native mathematical structures. The 8-axis atlas reveals patterns across sign pairings, trig transformations, and exponent configurations that are invisible in symbolic notation.
Mathematical Art
The combination of bloom rendering, strand physics, and animation controls produces visually striking mathematical compositions. Export camera positions and parameter states for presentations and publications.
Related Solutions
- Demonstrate Ideas — Interactive proofs that make abstract mathematics visible
- Present Research — Publish mathematical research with live visual outputs
Related Use Cases
- For Education — Mathematics education through interactive exploration
- For Product — Validating mathematical computation concepts
- For Media — Mathematical visualisation and research publishing