Feigenbaum Delta

Take almost any smooth map with a single hump — the logistic map x → rx(1−x) is the standard example — and turn up the parameter. A stable fixed point splits into a cycle of period 2, then 4, then 8, and the intervals between successive splittings shrink geometrically. The ratio of consecutive intervals converges to δ = 4.669201609102990, and then the doublings pile up at a finite parameter value and the system goes chaotic.

Mitchell Feigenbaum found this at Los Alamos in 1975, working it out by hand on an HP-65 programmable calculator, and the astonishing part was universality: the same 4.6692 appears for the sine map, for any map with a quadratic maximum, regardless of the specific function. It is a property of the route to chaos, not of any particular equation. The result was so unexpected that his paper was rejected for three years before publication. Experiment vindicated it: Albert Libchaber measured the same constant in convecting liquid helium in 1979–80, and it has since been found in dripping taps, oscillating chemical reactions and driven electronic circuits. The digits given here come from rigorous computer-assisted computation, not measurement, so the constant carries no experimental uncertainty.