In 1975, physicist Mitchell Feigenbaum was studying the logistic map xₙ₊₁ = r·xₙ(1−xₙ) on a calculator at Los Alamos. As r increases, the system's behaviour doubles in period: from a fixed point, to a 2-cycle, to a 4-cycle, to an 8-cycle: before collapsing into chaos.
Feigenbaum noticed the ratio between successive doubling steps converged to a constant: δ ≈ 4.669. He then tried a completely different equation: the sine map: and found the same number. It was universal. Any smooth 1D map with a single hump produces the same Feigenbaum constant.
The universality was later explained by renormalisation group theory. δ has since been measured in fluid turbulence, electronic circuits, heart rhythms, and dripping faucets. It is one of the few mathematical constants discovered by numerical computation rather than algebra.