This number appears in every chaotic system—logistic maps, fluid turbulence, electrical circuits, even dripping faucets. Discovered by Mitchell Feigenbaum in 1975 using only a pocket calculator. Why is this universal? Nobody fully knows.
As a parameter increases, stable fixed points become unstable. The system "doubles" its period: 1→2→4→8→16→... until chaos. The bifurcation points r₁, r₂, r₃... converge geometrically.
The ratio (rₙ - rₙ₋₁)/(rₙ₊₁ - rₙ) approaches δ ≈ 4.669 as n→∞. This means each period-doubling interval is ~4.67× smaller than the last. At r∞ ≈ 3.5699..., chaos begins.
ANY smooth function with a single quadratic maximum shows the same δ! Logistic, sine, Gaussian, polynomial—they all give 4.669... This is like π appearing in every circle, regardless of size.
Feigenbaum's constant has been measured in: fluid convection, laser systems, chemical reactions, heart rhythms, population dynamics, and electronic oscillators. Nature knows this number!