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.

Finite differences and recursions

Thi is pure applied math.  I am selecting a recursion of the form:
y^2 -a^y -1 = 0, which then generates a power series in 1+y+y^2+y^3.... and from that I compute tanh and the first two differences. Tanh is the blue and the red and yellow the first two differences.  I want to see how smoothly the selected ratio converges.

For example, I am starting with something simple, the fourth order Taylor expansion of a circle volume when the center is estimated. The I try out ratios that might accurately model that polynomial.  Physics has a lot of dimensionless ratios, 127,1836,119*pi are particular examples.  These ratios have more to do with the spectrum of a sphere then anything else, and ultimately they are all generated by the quarks which are likely just maintaining a power series. I remove pi from the ratios as that is likely the unknown.  Space impedance, for example, tells me that the density of charge will make a sphere is it exceeds a certain ratio, otherwise light leaks out along a straight, one degree of freedom line. Hence the Maxwell equations treated as a spectral problem with local wave action, no relativity, no time, no distance.

Hyperbolic and recursion

[sinh(x) + cosh(x)]*[sinh(y) + cosh(y)] = sinh(x+y) + cosh(x+y)

But we can define some subset of hyperbolic angles such that:

r-1/r = a then we have a recursion, often trivial, of the form:
r^n = a*r^(n-1) + r^(n-2).  The recursion have the r = a/2+root(a^2+4)/2 and 1/r = -a/2 +root(a^2+4)/2

so, r-1/r = sinh(log(r)) =a and log(r) =  k is the base hyperbolic angle.
and sinh(k) = a and cosh(k) = root(a^2+4)  .

1/2(s(k) + c(k)) = r, and (s(k)+c(k))^n = [(1/2)*r]^n= s(nk)+c(nk)
1/2(s(k) - c(k)) = 1/r, and (s(k)+c(k))^n = [(1/2)/r]^n= s(nk)-c(nk)

I likely have messed up the factor of two.

sinh(k)^n =  a^n
cosh(k)^n =  root(a^2+4)^n
tanh(k)^n =a^n /root(a^2+4)^n

The recursions work fine, and easy to make but are discrete. I can  make a recursion in e.  All the derivatives are with respect to integer quants, and all the ratios go to rational fractions with an integer power. I can use tanh series to approximately solve differentials then let my ratio go to e in the limit.

Take 137, for example

It is 136+1 which is 17*8+1. OK, So I think if 137 is the sample limit of light, then knows a short power series r^3 + r^2 + r + 1 with r = 2*17^(1/3).
This gets me 17*4 and 17*108, makes my day.  I can make a recurser to build powers of 17.  Everything gets back to small powers of at most third order.

Why 17?  It was me, I used the Fibonnaci tool to align my number set with the physicists.  In the real vacuum rational numbers, likely all integers, all of them part of a recursive sequence.

So, the vacuum ends up with three aggregates of the vacuum, a precomputed set of three numbers anywhere, and defined by recursive adds.  The number is three because this is sphere packing. The rational ratio, p/q, which makes the recursion, makes the entropy of exchange, defines its spectrum. That spectrum is the exchange variation rate of the original tiny set of three bubbles. It is the variation of motion at the tiniest level. The 17 is me, matching that ratio, but the vacuum has run through the lower value counts of sets, or independent ways to combine aggregates. It only needs the first few separable groups to make the sphere match the tinies  because something in the universe keeps the bubbles polished. It has matched both ends, uncertainty and motion needed to maximally pack a sphere. This is just trying to do physics with finite bandwidth and using combinatorics.
The assumption is minimal redundancy. The aggregates should form a digit system that counts path length through a minimal spanning tree.

Force coupling constant and sphere packing

Alpha here is the called the fine structure constant in physics, e is electric charge. As you can see, its purpose is to estimate the perfect sphere by estimating pi.
This is easy to understand from maximum entropy considerations. The proton needs to match the band limit of light to the quant ratio so that pi is maximally accurate.  It only knows pi as a rational fraction.

The way I work the problem is that there exists an angle x such that tanh(n*x),  are linear for n=1,2,3.

Anyway, these angles are multiples of the log of the exponent in the quant ratio.  The maximum Compton matchi is at wave 91, and 3/2 * 91 = 137.5.   And
4/3 * 137 gets you to 182 where I find a near perfect Compton match with ratio 3/2 at 216.  Coincidence?  Well, I am using the first Lagrange,Phi, and 3/2 is a known ratio for sphere packing. So I am going to find these matches.  The question is, does the vacuum count the same way my spread sheet does? This makes 182 the band limit of light, as an exponent of phi, much higher than I would thing.

As an angle on the hyperbolic then the universe counts from pi/2 (not including pi/2)  by sixths, a quadratic root for each of three cubic roots.

Why I use exponents:

 I cheat, I approximate values using Phi, which nature does not know.  But I cheat and get the property to the left. So when the physicists report a prime number, then they have cancelled common terms, and I know the number they report is a common multi0ple, so I know how far up the Fibonacci chain the universe has gotten.  I do not find Fibonacci numbers, but I look for Fibonacci exponents.  This is primitive, yes, but good enough for me.  The real step is finding the ultimate rational ratio that the unioverse has settled on to approximate pi and e. That is not a Phi thing, but a sphere thing, and the numbers keep coming back to about 2/3 or 3/4, I dunno. The length of the power ies the universe uses seem to be about 6, but as combi8nations of two and three, that comes to a 12-16 bit computer.

How is the universe so stable then? It has matches the relative bubble sizes in the salad bowl to a 12 bit computer. The real stability comes from having the bandwidth of light match the sphere, allowing motion between unit spheres. The real trick of the universe is to recycle the vacuum bubble, polish them up, resize them, and recycle them.

Wythoff arrays can pack sphere

Here I am working with a rational ratio and arranging packs of quants thare are  exponents of that ratio= r. I am limited to two a few exponents at any level of the sphere, and I have to be hyperbolic.  The circles sum the (1/r) and outside the circle I have motion as r So I have the worm gear evvect, the circles are tracking sum (1/ratio) = 1 and exchange r with the exterior. But volume to area reaches a critical point going from the perimeter to the center, so I jump to the next row of the Whythoff array:

 Anyway, this is where I am at.  I got here my messing with recursions and discovered the Wythoff did that for me.  The balance is matching motion and unit circles going toward the center, and switching to the next row at the critical moment.  The Wythoff work because they are minimally redundant, one can make the most quants with the least redundancy. Hyperbolics and finite differences and finite sums of local aggregates are the way to go.

So, all you sphere packers, carry on! I am trying to stop this, really, but it has become a sort of fun compulsion.

Local knowledge

Consider a local element in an aggregation with no global knowledge which moves toward minimum density.   It has no uniform knowledge of any dx, and can only make globally stable movements if all elements were of standard size and queued up fairly.  Normally elements avoid queueing and that makes spheres happen as queuing decreases toward the perimeter.

It is not sufficient for all elements to be the same size, they also need to have the same time constant in their movements. Otherwise unnecessary movements starts, and that is not maximum entropy. So element size and time constant have to be equalized.

Now, what happens if there is no optimum move? The element still needs to maintain time constant and will make a random move, I would think. The element itself must always have a self induced imbalance so that its current position in not optimum.

Do these exchanging elements  pack spheres better than the inerts? I would think so, I would think that aggregating empty positions allows more packing by movement, though I cannot prove this. If I had to prove this I would look at a perturbed system, one not in equilibrium. Then, stable configurations must come and go  as it moves toward equilibrium.

If movements are quantized, then is their a background grid system?  Not if the elements are making moves in the second differential and are perturbed.  So empty space is sub optimal, entropy is not always increasing in a static discrete space.

The last point is why any system needs inert nulls plus one active element.  How many other elements does the system need? As many as needed to work the differential order.  If the perturbations are not linear, then a single active element results in repeated movements, entropy is not increasing.

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.

Games I am playing with recursive sequences and hyperbolics

Consider r1=3/2+sqrt(5)/2 and r2=3/2-sqrt(5)/2.   The sum is 3 and their product is one.  I get these values from Phi^2 + Phi^-2.

Take the log(r1) = a and then sinh(a) = sqrt(5)/2. Cosh(a) = 3/2. Tanh(a) = sqrt(5)/3. Hence Phi = 1/2+3*tanh(a)/2 =  1/2 + sinh(a).
Then (Phi)^n = Fn*(1/2+3*tanh(a)/2) + Fn-1 for example. and any (3/2)^n = Cosh(a)^n.

Also the derivative of the tanh will  be 1/cosh(a)^2. Neat stuff.  Is this ratio the reason I got the match at Phi^91 and (3/2)^108? Yes, but I caused the ration because I sorted the powers of Phi and 3/2. Hence I forced this angle, I did a Higgs mechanism on my spread sheet. Now I can work all the orbitals using shifted powers,  just like Higgs, and I can change the ratios r1 and r2.

Thuis idea that light is a bubble exchange with two bubble type swapping places with Higgs nulls is correct.  The shift  I see to get the ratio is charge and spin, and all the properties the Standard model has.

So think of it this way.  Light knows a match radius must be three, it is simply working with Higgs to find a grid size agreement that make the radius three, which it must be at the match point. Higgs and light doing the Lattice QCD for real. These shift in the phase difference between the light bubbles will stabilize then their rate of change can be contained by Higgs bubbles. That determine a unit circle. They can all be calculated using hyperbolic differential wave equations, I think. The phase shifts increase toward the center of  the packed sphere and that add angular motion to the degrees of freedom.

This angle is log(sqrt(5)/2 + 3/2.  That is about ,96 radians. Cannot find it on any map, I am still a little befuddled about where these angles are measured from. And I still think my choice for the Higgs bubble size was mostly luck and guessing. I never actually measured one of them.

There is one better recursive relationship

Phi makes the small and large light bubbles to distant. So I tried this:

r -1/r = 1/2  or r = 1/4+SQRT(17)/4 . This actually worked and makes the proton.electron mass much more accurate, 17*108 = 1836.  It also makes sense since a half shift is r^2 + r.  Log(1836,(r^2+1/r)) = 17/2. Sort of a bingo and a half shift is spin. And the Higgs bubbles do not come out so large, and the light bubble are a reasonable  So, I am looking at these bubble sizes for sphere packing. The hyperbolic angle is .2475, and sinh(a) = 1/4 and cosh(a) = sqrt(17)/4.

So the current plan. We have a measuring stick on the left, measuring the two active bubbles. We design it to count in a way that matches the inactive bubble on the right.  I can make any form of 2/3, 3/2, 3/4; to match some criteria represented by a ratio of grid sizes, using sinh(a). Then have fun with tanh, and read lots of papers by mathematicians doing tanh solutions.

Constraints are boundaries on the null grid, written in cosh as fraction and define null quants at the hyperbolic angle specified by the measuring stick. So my constraint is that null bubble be 3/2 times the measuring stick. We can still formulate (3/2)^n, easy enough, so we still get the dandy match at 108, and now I got a 17, whoopie doo.  The measuring stick is not more accurate at angle .24 then Phi at .96.  But we know Phi needed a shift four to match the density.  Its dense, volume to area favors highly curled action by the measuring stick.  The new ratio will need a 16 shift, is my lucky guess. Phi had the 17s if I remember.

Schizophrenic Persian does not blame the Jews

NRO: Iranian president Hassan Rouhani, whom the Obama administration has considered enlisting in the fight against the Islamic State, blamed the growing threat of the jihadist group on the West’s involvement in the Middle East.
During his speech at the United Nations General Assembly on Thursday, Rouhani said, “Today’s anti-Westernism is the offspring of yesterday’s colonialism; today’s anti-Westernism is a reaction to yesterday’s racism,” according to a live translation. “Certain intelligence agencies have put blades in the hands of madmen who now spare no one.”

I should write for the Onion!