I was messing around with the Mandelbrot boundaries on Desmos and came across something interesting, and I don't have enough experience with the math behind this to conclude whether it's a coincidence or not.
In the image below, here is what's going on (my apologies if the terms I use aren't correct, please feel free to correct):
- The black lines plot the boundaries of the period 1 and 2 components of the Mandelbrot Set (I'm only concerned with the cardioid at the moment)
- The red line plots a line through the individual two components of the equation of the boundary of the cardioid, defined as$$\frac {e^{it}}{2} - \frac {e^{2it}}{4}$$ (Note that the line, going through the points e^(it)/2 and -e^(2it)/4, is extended to an arbitrary amount that I defined)
- I have a slider for the parameter t, which the orange dot plots the respective point on the cardioid (currently set to t = /4 - at the base of the 1/4 bulb)
- The blue line represents the vector between the two components (e^(it)/2 and -e^(2it)/4)
As seen in the image, the line seems to point directly towards the solution for the 1/3 bulb, and what makes this interesting is that the corresponding point on the cardioid falls at the base of the 1/4 bulb (not to mention that the line intersects the axes at easy rational numbers). However, when I zoom in on the 1/3 bulb, I see that the slope of the line is slightly at a different angle and does not intersect the point exactly, but perhaps this is a matter of precision error - and I have a hard time believing that this is a coincidence. Is there a relationship between the 1/4 and 1/3 bulbs by this line?
Here is the Desmos graph I made, perhaps exploring it will yield further insight on what's going on (change s to change the parameter t, and d to change the length of the projected line).
Edit:
So I managed to figure out the exact answer for the center of the period 3 bulbs thanks to this paper and added it to my Desmos graph, and so given the image below I'm thinking this whole thing might just be a coincidence... although to be fair I had a hard time understanding anything in that paper and so there is still some room for me to miss something important, and so if anyone could provide clarification on this connection I would be very appreciative!
