⚙️ Instrumentum

A precision tool that reveals golden π (4/√φ = 3.144606...) through the identity 4y²x / ((y²+x²)√(y²-x²)) — exact only when y/x = √φ. Explores the relativistic spacetime interval √(y²−x²) and the Lorentz factor γ = φ at the golden ratio.

Open full-screen instrument ⚙️

About the Instrument

Adjust the r = y/x slider to see how the computed value of π changes:

• Slide toward √φ ≈ 1.27202 — the computation converges on π = 4/√φ = 3.144606
• At any other ratio, the result diverges from both golden π and conventional π
• The green target marks √φ — the exact golden ratio root
• The red dashed line marks the light cone (r=1) — below it, the value becomes complex
• Relativity readouts show β = v/c, γ (Lorentz factor), τ/t (proper time ratio), and ψ (rapidity)
• At r = √φ, γ = φ exactly — proper time dilates to τ = t/φ

Includes a geometric visualization canvas with light cone boundaries, real-time readouts comparing golden π vs conventional π, relativistic quantity displays, and a complete algebraic proof.