To measure how rapidly the diameters of the period-doubling components shrink, we look at the ratio of successive diameters. That is,
|cn+1 - cn| / |cn+2 - cn+1|
The values of the first few ratios are
| 4.233738275 ... |
| 4.551506949 ... |
| 4.645807493 ... |
| 4.663938185 ... |
| 4.668103672 ... |
| 4.668966942 ... |
| 4.669147462 ... |
| 4.66916224 ... |
| 4.66919003 ... |
This sequence approaches a limit, the Feigenbaum constant.
The same limit is obtained for the period-doubling cascade of the logistic map, and also for the analogous sequence of many, many other functions.
Return to Scalings in the Mandelbrot set.