Around 1975, Feigenbaum began to study period-doubling in the logistic map. First he developed a complicated generating function theory to predict the value of where a -cycle first appears. To check his theory numerically he programmed his hand-held computer to calculate the first several He noticed a simple rule: the converged geometrically, with the distance between successive transitions shrinking a constant factor of about 4.669.

Perhaps a month later he decided to compute the in the trancendental case numerically. Again, it became apparent that converged geometrically, the convergence rate was the same 4.669.

In fact, the same convergence rate appears no matter what unimodal map is iterated. In this sense, the number

is universal. It is a new mathematical constant, as basic to periodic-doubling as is to circles.