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# An Exact Value For The Fine Structure Constant.

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137/128 equals, approximately, 173/162

Why not use an approximation which is both more accurate and has lower denominator, such as 61/57 (both are primes!), 76/71, 91/85, 107/100 (10^2!), and many more ?

Now, what is interesting here is that the denominators are both half and double squares...

Why is that interesting?

Further, if you look at 137 and 173, their difference is 36 (twice 18), with an average value of 155- this latter number just happens also to be 1/4 of 620, relating it to 1000x the rounded value of the lower Golden Ratio.

The "lower Golden Ration" is 0.61803399, 1000x this is 618, not 620.

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To date, by far the most accurate measurement of the fine-structure constant (measured at the scale of the electron mass) was made by Gerald Gabrielse and colleagues from Harvard, Cornell and RIKEN.

Yes, I am the fellow that helped Don calculate w(10^15).   I have extended the calculations in Don's paper, and made some investigations on my own.   The results indicate that the presence of the "fin

It’s good to see you’re still voraciously playing with finding expressions for physical constants (if $\alpha$ actual is a constant, which isn’t certain – see its wikipedia article for more

Hey, I don't make the rules (for rounding, like WHERE in the sequence to do it). Nature may not care how we do things. Besides, it looks as if the motivation comes from the Fib+Lucas approximation,starting with whole numbers. No natural phenomenon I know of uses the EXACT Golden Ratio, they use the Fibonacci and Lucas numbers, generally the early ones.

As for the denominators being half and double squares- well *I* find it just as interesting as the fact that the periods in the periodic table use the same numbers. I remember not too long ago, on another forum, being told bluntly by one of the respondents that my observations about the various Pascal mappings were not only trivial but useless careerwise. Observations of the same rank got Mendeleev and others involved in the development of the periodic relations fame and fortune. But it is true that today nearly anything goes.

I think the problem is that people know too many ways to get to the same result mathematically, that they can't separate what is arbitrary from what is more fundamentally motivated. You should see the mess of attempts to cut up the periodic system in its early days, when nobody could be entirely certain of who was right, since they were using atomic weights instead of atomic numbers as we do now, and there were gaps that had to be filled by further discoveries. It was craziness for another 40 years after Mendeleev. Every one of these gentlemen was a professional, most had prestigious positions, yet most got it wrong. There is an entire laundry list of named 'elements' that got left on the cutting room floor.

Such is the case I believe with 'numerological' attempts to derive the FSC. Which one is right? Too early to tell. I'm not saying my proportion is correct, either in construction or values- I'm just throwing it in the air for consideration. In any case number theory has been kind to me so far, on the atomic side. It doesn't mean there isn't more to do, or to learn, or to find. There are interferences on that side from other phenomena like relativity. It means there isn't any 'one-size-fits-all' equation for all properties of atoms. My tetrahedral treatment works great for the purely quantum aspects of idealized structures that have no spin-orbit couplings and a few other things. But it gets most things right, and what it gets wrong isn't the fault of the model. As I said other things are going on, and the Pascal math is only one motivation.

Finally, first things last, I may have mentioned in my post that there was a webwork of Phi-related numerical connections behind the periodic system. So it may be with the FSC as well. Perhaps something about the combinatorial nature of whatever spacetime is? The numbers you gave are of interest from this perspective: 61 is 100-39, the latter 3x13Fib. 57 is 3x19, while 76Luc is 4x19. 71 is 100-29Luc, 91 is 7Lucx13Fib, and 107 is 200-93, the latter being 1.5x62. 107 is also 144Fib (and square)-37, where the latter is in the next series after Lucas. I've generally found that within this webwork there may many ways to cut the apple, but most relate to the Golden Ratio one way or another, and those that don't are relatively rare. Since I'm not a mathematician I can't say that this just means that numbers related to Phi can generate most others with simple functions. I'm sure many of my wanderings through this vast numerical space have been in vain, but others have been quite fruitful.

Jess Tauber

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Hey, I don't make the rules (for rounding, like WHERE in the sequence to do it)

I know you don't, that is why I remarked that 0.61803399*1000 rounded to an integer is by the rule of rounding 618, not 620 as you claim.

Well, actually, you do not claim it, you say "relating it to" but that is neither an exact nor a scientific statement.

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Welcome to hypography, Jess! :)

137/128 equals, approximately, 173/162. Here of course are the (whole number reciprocal approximation) low and high energy Bohr atom FSC's on the left, and the supposedly more realistic neutral atom limit for the right numerator.

I don’t understand what you mean by the “low and high energy” fine-structure constant. Doesn’t the FSC have a (perhaps changing very slowly, and very different soon after the Big Bang) constant value of about [imath]\frac{1}{137}[/imath]? Under what conditions does the FSC have a value of about [imath]\frac{1}{128}[/imath]?

A quick web search didn’t help me. Please post a link or reference to an explanation what you mean.

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Just so folks are aware- there are lots of newly recognized Phi-based phenomena out there. Besides the usual panoply of phyllotaxis and other biological processes involving packing and growth (for example the commonality of icosahedral viral capsids), 5-fold symmetry of quasiperiodicity (once entirely a mathematicians toy until quasicrystals were first synthesized in alloys involving Fib and Luc numbers of different metals, etc.), planetary orbital resonances, and galactic arm structure, we also now have my observations about the periodic table, others' about the scaling of N/P in the nucleus, and certain higher level electronic interaction, and elsewhere also the maintenance of the relative proportions of nucleotide bases over long stretches of DNA within limits defined by Fib and Luc numbers again, perhaps interactions of brain wave frequencies so they don't interfere, and finally, new evidence that linguistic syntactic structure is governed by Phi-like rules having to do with size and ordering of phrase and clause chunks. I can point to links if any are interested.

The Fibonacci series is a fractal one, by the way. It can be derived from the Pascal Triangle via interactions across diagonals. Other related sequences, like the Lucas, can be gotten just by changing the sides of the Triangle from 1's to other values. And the Golden Ratio, Phi, is also a based on a continued fraction. Then we have, on a bigger scale, the other Metal Means (of which the Golden Ratio is just one). All these also can be equated to continued fractions. Another formula is 1/2(N+sqrt(Nsq+4)), so that the Golden Ratio is that case where N=1, so that the value of the term under the square-root is 5. All these Metal Means have values such that if one subtracts the whole number portion before the fractional one, and takes the reciprocal of the latter, you regenerate the original. So for instance if you take Phi, 1.618033988, subtract out 1, then invert so that you have 1/0.618033988, you end up back at Phi.

I discovered some weeks ago (and am still trying to find out whether this was already known, for publication purposes) that the powers of the Metal Means were intimately related to the (2,1)-sided Pascal Triangle sister, the one that generates both Lucas and Fibonacci numbers on the 2 and 1 sides respectively. See http://en.wikipedia.org/wiki/Silver_mean and also down near the bottom of the discussion page for that entry.

People who look at natural phenomena from the perspective of packing and growth (both of which pertain at every level of structure at least as regards some things) tend to just consider the Golden Ratio only. Yet it turns out that these other Metal Means are capable of doing their part as well. I'm looking into how they might be involved in the areas I've reported on so far.

The reason I emphasize all this is because this kind of math, which is based on the simplest operations, provides a motivated basis for physical phenomena. I don't think Nature is easily capable of executing complex equations, at least unless these themselves are composed of fusions of simpler ones. The organization just isn't there, even if the raw computational power may be.

Jess Tauber

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In post 10 Don mentions values of q, the numerator in the subtracted part of his equation, and gives 1 as well as 1/(sqrt2+1) as values.

Were you aware that both of these are reciprocals of Metal Means? For the equation 1/2(N+(sqrt(Nsq+4)), where N=0, the Mean is 1. For N=1 we get Phi, 1.618..., and for N=3, we have sqrt2+1, 2.414...! What would happen if you plugged the reciprocal of Phi, 0.618... into your numerator variable??

Jess Tauber

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Hi Jess,

And thanks for all the interesting feedback.

I am of the opinion that the underpinnings of the universe are essentially

both logical and mathematical (and therefore "eternal"). In other words,

I believe that the universe may very well be, at its most fundamental level,

comprised not of "particles" or "strings", but of "self processing information".

I also agree with you that $\phi=\frac{\sqrt(5)+1}{2}=1.61803398874...$

is one of the most important numbers in all of nature.

Your recent posts gave me an idea, and I think it's a pretty interesting one.

Here it is, and after you read it, please let me know what you think about it...

There are, at present, many proposed "Grand Unified Theories" and "Theories of Everything",

which involve a very special mathematical object called $E_{6}$, whose fundamental representation

is $27$ dimensional. Thus,

$27=3^3=5^2+\frac{\sqrt(5)+1)}{2}+\left(\frac{\sqrt(5)+1}{2}\right)^{-2}=5^2+1.61803398...+\frac{1}{2.61803398...}$

may be the most "fundamental" number of all. (At this point, we just don't know.)

However, since $\frac{\sqrt(5)+1)}{2}+\left(\frac{\sqrt(5)+1}{2}\right)^{-2}=2$,

all positive integers of the forms $2N$ and $2N-1$, that is,

all positive integers including $27$ can be viewed as involving $\phi$.

Note that the above representations of the number $27$ involve the numbers $1,2,3,5,\phi,\phi^{-2},$

all of which are either Fibonacci numbers, or closely related to them.

Now, as it turns out, the most simple "super accurate" counting function for

polygonal numbers of order greater than 2 seems to be...

$\varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-\pi^{2}\right)}\right)$

where:

$\alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(27*\left(\ln\left(x\right)\right)^{-1}+1\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1}$

which results in the table:

$x$_______________________$\varpi(x)$_________________$B(x_{F\alpha})$____________Difference

10_______________________3______________________5___________________2

100______________________57_____________________60__________________3

1,000____________________622____________________628_________________6

10,000___________________6,357__________________6,364________________7

100,000__________________63,889_________________63,910_______________21

1,000,000________________639,946________________639,963______________17

10,000,000_______________6,402,325______________6,402,362_____________37

100,000,000______________64,032,121_____________64,032,273____________152

1,000,000,000____________640,349,979____________640,350,090____________111

10,000,000,000___________6,403,587,409__________6,403,587,408__________-1

100,000,000,000__________64,036,148,166_________64,036,147,621_________-545

1,000,000,000,000________640,362,343,980________640,362,340,978________-3002

10,000,000,000,000_______6,403,626,146,905______6,403,626,142,382_______-4523

100,000,000,000,000______64,036,270,046,655_____64,036,270,047,407_______752

200,000,000,000,000______128,072,542,422,652____128,072,542,423,321______669

300,000,000,000,000______192,108,815,175,881____192,108,815,179,516______3635

400,000,000,000,000______256,145,088,132,145____256,145,088,131,948_____-197

500,000,000,000,000______320,181,361,209,667____320,181,361,209,475_____-192

600,000,000,000,000______384,217,634,373,721____384,217,634,375,674______1953

700,000,000,000,000______448,253,907,613,837____448,253,907,608,937_____-4900

800,000,000,000,000______512,290,180,895,369____512,290,180,895,206_____-163

900,000,000,000,000______576,326,454,221,727____576,326,454,224,725______2998

1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,399______482

This version of my counting function is also noteworthy in that as $x$ goes into infinity,

the value of $\alpha^{-1}$ as measured at the scale of the electron mass is only approached

and never goes higher than $137.035999083778...$, thereby strongly implying that

the value of the fine structure constant at zero energy can be neither observed,

nor deduced, and will therefore always remain a mystery.

Don.

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Just toying around with my original FSC proportion here:

All the numbers together sum to exactly 600. The numerators sum to 310, the denominators sum to 290. That is, 300 + or - 10.

Then the left numbers sum to 265 and the right to 335. That is, 300 + or - 35. Finally, the right numerator minus the left denominator sums to 301 and the left numerator plus the right denominator sums to 299, or, 300 + or - 1.

This might SEEM to be a jumble of random differences, but look again: 1, 10, 35 are every other Pascal Triangle TETRAHEDRAL number, the fuller set being 1, 4, 10, 20, 35, 56, 84, 120. More specifically, 1, 10, 35 are the running sums of squares of odd integers 1, 1+9, 1+9+25... (the complement of what is found as atomic numbers in the Janet Left-Step periodic table, where we find the running sums of the squares of even integers being the atomic numbers of the right edge, s2 configuration elements).

Jess Tauber

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Hmm- first let me apologize for misquoting my Metal Mean value N=3 >> (sqrt2+1). I meant to write N=2 there.

As for 27, it does seem to have special properties. Long ago, while working on some problems in representing the Standard Model particles geometrically, I decided to figure out how to map a cube so that it possessed an axis going through two opposing vertices through the center of the cube, said axis itself within a plane that could rotate along the axis. The cube was then rotated against the plane until the lengths of projections dropped from the vertices and perpendicular to the plane had lengths that were in 0:1:2:3 proportion (the idea being that these represented N/3 fermion particle charges for neutrinos, down quarks, up quarks, and electrons. There being 8 vertices in toto the two sides of the plane also could capture regular and antimatter as well).

Anyway, the angle of rotation to produce this turned out to be arctan(sqrt27), and had a value of a bit more than 79 degrees.

Now, with regard to the FSC value of @1/137, it is of high interest to me that the exact value of 1/137 is very special as well. It is a repetitive palindrome .0072992700729927. Not too many of those around, at least up to 1/500 (which is as far as I bothered to look). Not only this, but its complement, 136/137= .99270072992700... also uses the same 4 numerals: 0, 2, 7, 9. Most such complements don't. So this fractional sequence is self-complementary (so related to fractals?). Also note that 7-0=9-2. And the end pairs 72 and 27 sum to the central pair 99.

It is curious from this perspective that the first three nonzero values after zeroes, 729, is the square of 27 (also since 729 is 9x81, it is 3 squared x 9 squared). The part made from the last three nonzero terms, 927, isn't so lucky. It is 9x103. 103 doesn't ring any bells for me, maybe I'm not looking hard enough? Interestingly 9+2+7+7+2+9= 36. That keeps popping up too (remember that 173-137=36). If we add 103 to 81 we get 184. If we subtract 81 from 103 we get 22. NOW, if we subtract 22 from 184 we find 162, with 173 as their mean. Ain't numerology er, hmmph, I mean 'number theory', grand?

I wonder whether anything similar could be done with the fractional values of 128, 173, and 162?

Jess Tauber

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Now, with regard to the FSC value of @1/137, it is of high interest to me that the exact value of 1/137 is very special as well. It is a repetitive palindrome .0072992700729927. Not too many of those around, at least up to 1/500 (which is as far as I bothered to look). Not only this, but its complement, 136/137= .99270072992700... also uses the same 4 numerals: 0, 2, 7, 9. Most such complements don't.

Here are a few more that have equally interesting properties (some of them using fewer than 4 numerals):

1/101 = 0.00990099009900990099...; 100/101 = 0.9900990099009900...

1/110 = 0.00909090909090909090...; 109/110 = 0.9909090909090909...

1/111 = 0.00900900900900900900...; 110/111 = 0.9909909909909900...

1/273 = 0.00366300366300366300...; 272/273 = 0.99633699633699699633...

1/303 = 0.00330033003300330033...; 302/303 = 0.99669966996699669966...

1/1001= 0.00099900099900099900...; 1000/1001 = 0.999000999000999000999...

Note: 273 is the boiling point of water in degrees Kelvin!

Note: 273/2 = 136.5, almost 1/FSC!

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Ah, but your decimal expansions lack the 2 in mine of 1/137! This makes all the difference :lol:

Jess Tauber

Here are a few more that have equally interesting properties (some of them using fewer than 4 numerals):

1/101 = 0.00990099009900990099...; 100/101 = 0.9900990099009900...

1/110 = 0.00909090909090909090...; 109/110 = 0.9909090909090909...

1/111 = 0.00900900900900900900...; 110/111 = 0.9909909909909900...

1/273 = 0.00366300366300366300...; 272/273 = 0.99633699633699699633...

1/303 = 0.00330033003300330033...; 302/303 = 0.99669966996699669966...

1/1001= 0.00099900099900099900...; 1000/1001 = 0.999000999000999000999...

Note: 273 is the boiling point of water in degrees Kelvin!

Note: 273/2 = 136.5, almost 1/FSC!

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Quoting pascal:

Ain't numerology er, hmmph, I mean 'number theory', grand?

Above all, math is supposed to be fun, and I happen to enjoy reading about curious, unusual or

otherwise interesting properties of numbers, even if they have nothing to do with actual mathematics.

Recreation, amusement, joy and laughter are definitely elements in the set of things that make life worth living,

and even presidents of the United States find it necessary to "lighten up" from time to time.

Take for instance, Bill Clinton.

He was a "popular" president because he never took himself too seriously,

was able to laugh at himself and never lost his sense of hummer.

We should do the same.

Don.

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This particular factoid I posted yesterday has me stumped. Assuming the whole-number parts are correct (and exactly what are we supposed to do with the remainders if any?), what could possibly make this work in the real, natural world? It's like a kind of Sudoku puzzle, with all the parts balanced so as to show differences that parallel tetrahedral numbers based on summing squares of odd integers. There are only a couple of natural phenomena that I'm aware of that come close to this sort of thing, with complementation all around- color perception and categorization, and music, and these are both neurological, biological. No hint of anything like that at the level of the number values for the FSC, UNLESS we're talking about some kind of 'role of the observer' effect, with an Anthropic bias. I've often wondered in the past couple of years, as I explored the periodic system and the meshwork of numbers behind it where the computations were supposed to be implemented. Its almost as if complex systems are conspiring so that higher levels alter the lower ones to better suit their needs. We see something like this in the periodic system with regard to the ground state electronic configurations of copper and silver, where the positional misplacement relative to the Lucas (atomic) number mapping trend gets 'fixed' by altering the configurations to fit the trend despite the misplacement. This would imply that TODAY'S periodic system isn't the same as yesterday's. It also puts a monkey wrench into any sort of Design argument, making reality more of an active work in progress by processes that really don't care about what goes on underneath the hood so long as the thing runs.

And the fact that the FSC numbers aren't whole integers might mean that later hierarchical level additions have had their effect here as well. Were the numbers tweaked somehow?

Jess Tauber

All the numbers together sum to exactly 600. The numerators sum to 310, the denominators sum to 290. That is, 300 + or - 10.

Then the left numbers sum to 265 and the right to 335. That is, 300 + or - 35. Finally, the right numerator minus the left denominator sums to 301 and the left numerator plus the right denominator sums to 299, or, 300 + or - 1.

This might SEEM to be a jumble of random differences, but look again: 1, 10, 35 are every other Pascal Triangle TETRAHEDRAL number, the fuller set being 1, 4, 10, 20, 35, 56, 84, 120. More specifically, 1, 10, 35 are the running sums of squares of odd integers 1, 1+9, 1+9+25...

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These 4 FSC numbers (if one includes 162) are also all midway between squares: 128 between 0sq and 16sq, 162 between 0sq and 18sq, but also 137 between 7sq and 15sq, and 173 between 11sq and 15sq. Note that 7 and 11 are both Lucas numbers. 15 is the product of a Fibonacci number 5, and 3 which belongs ambivalently to Fib and Luc. Moreover, the difference values between the FSC numerators and flanking squares involve both Fib and Luc numbers as well: for 137 we have 88, which is 8Fibx11Luc (the total distance between the squares being 176, which is 100(sq10)+76Luc), and for 173 we have 52, which is 13Fibx4Luc. (the total distance between the squares being 104, which is 49(sq7)+55Fib). Is there a pattern to this set of mappings? 15-7=8Fib, twice that of 15-11=4Luc. And 128-88=40, which is 4Lucx10 and 5Fibx8Fib, while 162-52=110, which is 11Lucx10 and 55Fibx2(ambiv).

Jess Tauber

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To: pascal,

Lots of food for thought there.

Your ideas have resulted in some breakthroughs that are truly profound.

I now have to find the time to think them through and sort them out.

Hopefully, I will be able to post them by the middle of next week.

Cheers,

Don.

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Thanks, Don, for your supporting words. Unlike yourself I hardly ever get any feedback whatsoever, as if eyes cross when people read what I write on various blogs (even my own!).

Since last night I came up with a few more observations about the four numbers in the FSC proportion.

Given that the numerators are flanked by equidistant squares, it is curious that for the group around 137, where we have 7sq and 15sq, that 15+7=22, which is 2x11, a Luc number. Then for the group around 173 we have 11sq and 15sq, and 15+11=26, which is 2x13, a Fib number.

If we subtract the flanking square differences for the numerators 137 and 173, that is, 88-52, we get 36, which is identical to 173-137 (Is that necessary?? Haven't worked that out yet). But 88+52=140, which is 7Lucx4Lucx5Fib.

Now, if we switch the differences between the flanking squares, so that we have 137+52, we get 189, which is 9x21Fib, but also equidistant by 100(10sq) between 89Fib, and 289, which is 17sq. And 17 is 34Fib/2. 137-52=85, being 17(half Fib)x5Fiband remember that the right flanking square for both 137 and 173 is 15sq. And we know that the denominator is 128, which is half 16sq. So we have references then for: 15, 16, and 17 in the functions for 137/128 and the squares flanking.

If we also switch for the right side of the FSC proportion, we get 173+88=261, which is 9x29Luc, but is also 361(19sq)-100(10sq), and 173-88=85, being 17(half Fib)x5Fib. The denominator is 162, which is half 18sq. So here we have references then for: 17, 18, and 19 in the functions for 173/162 and squares flanking.

So recapping: 15,16 17 on the left, and 17,18,19 on the right, with the center numbers in each triplet being related to the denominators, and the right and left members of the triplets related to the numerators. AND 137-52=85=173-88. This latter sort of same differences effect is also found among the double tetrahedral numbers within the atomic nuclear counts!

Still, this is an odd set, but there are other dimensionless constants in Nature- could they be using numbers below 15 and above 19??

Jess Tauber

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