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An Exact Value For The Fine Structure Constant.


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#137 Turtle

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Posted 07 June 2012 - 02:22 PM

[quote name='pascal' timestamp='1339058645' post='318534']
Here is the set of generalized Fib number sequences stacked one atop the other that I used previously with quantum number and periodic system constructional parameters:

-1.....1.....0.....1.....1.....2.....3.....5.....8....13
/0.....1.....1.....2.....3.....5.....8....13....21....34
+1.....1.....2.....3.....5.....8....13....21....34....55
+2.....1.....3.....4.....7....11....18....29....47....76
+3.....1.....4.....5.....9....14....23....37....60....97
+4.....1.....5.....6....11....17....28....45....73...118

Note differences between any two contiguous values within any column is fixed and Fib, and the differences from left to right grow as Fib.
The fourth row is what I mean by Lucas sequence.

...
Jess Tauber
[/quote]

thanks for the table jess. :thumbs_up

if we expand the table infinitely, all natural numbers occur at least 3 times in the first 4 columns, which seems to make possible any arbitrary associations.

do you limit the table as you have because 118 is some maximum in the periodic table of elements? if so, i notice what is missing, i.e. {10, 12, 15,...} and have to wonder if these numbers are also missing in some fashion from the periodic table. ??

anyway, take heart that i am paying attention and making an honest effort to get a handle on your analysis. :turtle: i may be slow and hard-shelled, but i do good work.

Edit: PS [quotename='pascal']Note differences between any two contiguous values within any column is fixed and Fib, and the differences from left to right grow as Fib.
The fourth row is what I mean by Lucas sequence.
[/quote]

on the boldened: the correct terminology is to say "the differences from left to right grow as Fib a Lucas sequence.
The fourth row is what I mean by Lucas sequence the set of positive Lucas numbers.

Edited by Turtle, 07 June 2012 - 03:09 PM.


#138 pascal

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Posted 07 June 2012 - 03:48 PM

No, I mean that all differences between contiguous same-column numbers is a fixed Fibonacci number, and these differences grow, from column to column, left to right, in the same order and number as Fib numbers: i.e. differences 1,1,2,3,5... in successive columns left to right. The Lucas thing was entirely separate, identifying a ROW.

As for the periodic table ref, yes this list can be extended indefinitely, for an idealized quantum mechanical tabularization of the Janet Left-Step kind, but only some of the columns seem to matter (the one containing 118 is out of bounds), so far. But because of the FSC, and stability, in practice we don't need all that many rows either. The mapping of relative contributions to the total mass/energy of the universe may use the same stack of sequences, but is something entirely different otherwise. If it actually means anything it would be interesting to know if other, parallel diagonals have some similar physical meaning in cosmology. By the way, it turns out that if you SUM the numbers in the diagonal here, the running sums fall along a sequence of integers x every other Fib number: 0=0=1x0; 0+2=2=2x1; 0+2+7=9=3x3; 0+2+7+23=32=4x8; 0+2+7+23+73=105=5x21 etc. One can use this relation to figure out further members of the diagonal, even if they don't get used, without having to go through the trouble of laying out parts of the sequences. Thus the next sum should be 6x55=330, subtract 105, yielding 225, which is correct.

The cube-plane-rotation idea I've used to map charge of fermions- leptons and quarks, matter and antimatter. But I don't like special reference frames divorced entirely from others cut from the same cloth. There are 4 axes in any cube crossing the center and intersecting two vertices. And one can also rotate planes along any of them. And there are other trigonometric relations to the other parts of the cube. All sorts of things to explore. I keep hoping one day I'll hit on a mapping of the fermion rest masses this way, but will probably have to build a physical model to do it (not up to a computer graphical model that can be manipulated- though I bet one of you have these skills).



Jess Tauber

#139 pascal

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Posted 26 June 2012 - 12:10 AM

Finally worked out the mapping of the powers of Metallic Means to the (2,1)-sided Pascal Triangle. Got some help here by the mathematician Vera de Spinadel. Turns out that the you have to get the factors for the equations of these powers by summing terms along the shallow diagonals of the Pascal system of shape X(M^N), where X is the term from the diagonal, M is the basis of the Metallic Mean (0 for 1.0000, 1 for Phi, 2 for the Silver Mean, etc.), and N is the power/dimension term of the deep diagonal passing through X, parallel to the 2's side of the Triangle.

You calculate all the X(M^N) along the shallow diagonal and sum them- but this is only half the story.

Then for all ODD powers of the Metallic Mean, you add this sum to it's reciprocal, S+1/S, which gives the power of that Metallic Mean, but for all EVEN powers, you subtract the reciprocal, so S-1/S.

This works for all powers of any Metallic Mean.

Jess Tauber

#140 pascal

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Posted 26 June 2012 - 12:19 AM

I've also been thinking about the FSC and the relation to 89. FSC as approx. 1/137 which we've established as a special fraction with a large repeating palindrome. But 137 is also 1/3 of 411, which shows up in the stack of generalized Fib sequences as in earlier posts, in that column where differences are all 89. So the column is 55,144,322,411,500,589, etc.

The triplet here of 411,500,589 is interesting. 589 is 19x31, both terms from another generalized Fib sequence (half of 38 and 62 as approximations of numbers related to 100Phi: 38,62,100,162,262, etc.).

We also know that 1/89 has some special properties, but just as a member of a larger set of fractions.

All this is making me think that perhaps 137 is related, through 411, as a kind of complement to 589, which is more directly related to Phi-based phenomena. A 'shadow' if you will. Can 137 be seen as only subdimensional, relating to one spatial dimension at a time?

There are also all sorts of interrelations I've found between these numbers and 121, the square of 11, and as we know the Pascal systems show powers of 11. And 11 shows up versus 89 in a number of places. Something very strange in all this.

Jess Tauber