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: τ^{i·nτ/ln(τ)} ≡ e^{iτn}. Two ghost traces — cyan for τ (always closes at n=1) and amber for arbitrary base α. Only when α = τ does one step equal one turn.
8-Axis Visibility Atlas
Eight independent axes (A–H) controlling 6,144 distinct mathematical structures. Each axis is a continuous opacity multiplier across sign pairings, trig transformations, and exponent configurations.
Portable Scroll-Animation System
Link any slider to the animation engine with end-value targets and easing curves. Multiple parameters sweep simultaneously with loop and bounce modes.
Cinematic Rendering
UnrealBloomPass, OrbitControls, additive-blended star particles, and Line2 fat-line strands. Performance mode for smooth interaction on lower-end devices.
Technology
Built with vanilla JavaScript and Three.js via CDN importmap. No build step. Mathematical engine operates on [re, im] 2-tuples with τ-native complex arithmetic. WebGL shader-based computation.
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