Plugging the fine structure constant into my recursioon system

I interpret the fine structure constant to be: The amount of motion a balloon full of bubbles need if it want to get 4*pi exactly right.  The constant should be a result of the pure mathematics of packing a sphere with three types of tiny bubbles.

So, I ask myself, what is the optimum recursion ratio if I made a fourth order power series from the constant. So I take the constant, take the square root and the square to get powers  1 and 4.  I construct the tanh with (1-a)/(1+a) for both the first and fourth order.  Then I compute the hyperbolic angles, four of them, but I divided by three.  I divided by three because the sphere does not use the fourth order, but the fourth order is the limit of motion. Go past the fourth order and Higgs will bitch. I get a recursion in base  1.23, and its inverse; then plot the tanh and two orderss of finite differences to see how the ratio handles hyperbolics.  Here are the two plots, one for 1.23 and one for its inverse:

Is this significant? No, I borrowed all the work physicists had done and just created a digit system that counts out the entropy of light. We can see both curves almost converge by the fourth order.  So any  sphere packer who uses local recursion might consider looking at the work done by physicists, they are pretty good at sphere packing.

I have a hunch, however, that these little bubbles might know about the moment of inertia and compute up to the fifth order.  Just a hunch.