James Gleick (1987) Chaos, published by Vintage 1998, quote from page 173

It was the same number. Incredibly, this trigonometric function was not just displaying a consistent, geometric regularity. It was displaying a regularity that was numerically identical to that of a much simpler function. No mathematical or physical theory existed to explain why two equations so different in form and meaning should lead to the same result.

Feigenbaum called Paul Stein. Stein was not prepared to believe the coincidence on such scanty evidence. The precision was low, after all. Nevertheless, Feigenbaum also called his parents in New Jersey to tell them he had stumbled across something profound. He told his mother it was going to make him famous. Then he started trying other functions, anything he could think of that went through a sequence of bifurcations on the way to disorder. Every one produced the same number.