The curve is the real logarithmic spiral r(θ) = a·e^(bθ). Its defining property is that it grows by a constant multiplicative factor for every fixed turn of angle — here parameterized as growth k per quarter turn (90°), so b = ln(k)/(π/2). Setting k = φ = (1+√5)/2 ≈ 1.618034 gives the true "golden spiral". The 3D view rises the curve along a time axis (years per turn) purely as a visualization choice — a "spiral through time" — it does not change the 2D shape of the curve itself.
The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, …) is defined by F(n) = F(n−1) + F(n−2). The ratio of consecutive terms F(n+1)/F(n) is a genuinely computed, live-updating sequence that provably converges to φ as n grows — this is a real, provable numerical fact, not a decorative animation. Click + next term or Autoplay to watch the ratio tighten around φ on the chart below.
Honest note: claims that the golden ratio governs shell shapes, galaxies or human anatomy are a popular myth not supported by measurement — this simulator sticks to the two facts that are actually provable: the geometry of the logarithmic spiral, and the numerical convergence of Fibonacci ratios.
r(θ) = a·e^(bθ) b = ln(k)/(π/2)
F(n) = F(n-1)+F(n-2) F(n+1)/F(n) → φ as n→∞