More games with spheres

I am having fun with recursive sequences and their values of cosh and sinh at multiples of the basic angle.  I made a recursion that generate powers of 3/2. And that led me to t6hinking about sphere packing.

My current observation is that is it done in repeats of two actions. A quadratic root adjusts sphere area and a cubic root adjusts volume. So we have the proton doing cubic, then quadratic, then cubic, then quadratic then cubic.  That is why we get this 3*2*3*2*3 = 108. The system is making every grid as a sphere, then adjusting for surface area, then making spheres out of that, and so on.

Each of the adjustments is an additional set of marks on the measuring stick.

Anyway, I am playing with multiples of angles when the starting angle is the solution to a recursion. The sinh and cosh are simply the two components of a quadratic root so solutions at angles by 2 are just a power series in the quadratic root.