Feigenbaum Constant
Robert P. Munafo, 2002 Dec 30.
A universal constant in mathematics (like pi=3.1415926... and e=2.7182818...) that applies to nearly any parametrized iteration function, such as that used for the Mandelbrot Set. It gives the limit of the ratio between the parameter values at successive period doubling bifurcations in a parameter space.
It is most easily seen as the ratio between the diameters of successive mu-atoms in the sequence R2.1/2a, R2.1/2.1/2a, R2.1/2.1/2.1/2a, ..., R2{.1/2}xNa, ... . The same ratio occurs in all sequences of .1/2.1/2.1/2 mu-atoms that you find in the Mandelbrot Set.
Keith Briggs has computed the value of the constant to very high precision; here are the first 100 decimal places:
4.66920 16091 02990 67185 32038 20466 20161 72581 85577 47576 86327 45651 34300 41343 30211 31473 71386 89744 02394 80138 ...
It can be computed to rather high precision by a method that does not depend on actually iterating the Mandelbrot function and determining the size of mu-atoms.
The second Feigenbaum constant is:
2.50290 78750 95892 82228 39028 73218 21578 63812 71376 72714 99773 36192 05677 92354 63179 59020 67032 99649 74643 38341 ...
It is possible that the first Feigenbaum constant can be used to directly compute the position of the Mandelbrot Set's center of gravity.
See also chaos theory, iteration, Feigenbaum point.
Acknowledgments
Value of constant: Keith Briggs, Department of Applied Mathematics, University of Adelaide, South Australia 5005. See here for more digits.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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