In the Mandelbrot set, all points of the main carodioid are asymptotic (that is, the iterations of c^2 + c approach a constant). In contrast, it seems that all bulbs have a periodicity greater than 1, that is, the iterations settle into a cycle with a certain period.
There are several questions to be asked here:
- Is it the case that the largest bulb off any bulb has a periodicity that is double the periodicity of its parent bulb. For instance, see:
All bulbs that I've tried this with, this seems to be the case. Any proof or reason would be welcome.
- It appears that the second largest bulb has a periodicity that is triple that of its parent bulb - see below.
Similarly, I had conjectured that for any n, the nth largest bulb has a periodicity that is n+1 times larger than its parent bulb. (this is not counting the negative unreal half of the Mandelbrot set, as it is a symmetrical to the positive unreal half) However, I soon realised that there were two different period 5 bulbs off the main cardioid, of different sizes:
Similarly, there were two bulbs of period 10 off the '2' bulb, and two bulbs of period 15 off the '3' bulb. Any thoughts? What makes a number, like 5, get more bulbs of different sizes?