An Exact Value For The Fine Structure Constant.

[tetrahedron]

Also: In the squares flanking the numbers in the proportion: For the numerators the right square for both 137 and 173 is 225, 15sq. For the denominators 128 and 162 we have 0, or also 0sq, for the left square. A kind of antisymmetry?

 

For the numerators the left square varies: 49 or 7sq, and 121 or 11sq. For the denominators the right square varies: 256 or 16sq, and 324 or 18sq.

 

For the denominators if we sum 16+18 we get 34Fib, which is also the difference between 128 and 162. If we sum the squares 256+324 we get 580, which is 4Lucx5Fibx29Luc. For the numerators if we sum 7+11 we get 18, which is also Luc. Summing 49 and 121 gives 170, which is 5Fibx34Fib.

 

All this stuff coheres in frightening ways mathematically. If we knew enough about the fractional components could we do something similar? Are they, say, the sums, differences, multiples, etc. of the inverse of Fib, Luc, squares, and related numbers.

 

There is, IIRC, for the Pascal Triangle, an analogue structure which uses fractions instead of whole numbers in its diagonals. I don't remember whether anyone ever bothered to calculate what any Fibonacci or Lucas analogue would be for such a structure. Maybe all Nature is doing is mating together the positionally equivalent results from the one triangle to the other? That might explain why the fractional component of the 137 FSC doesn't seem to be easily relatable to the whole number part. Remember that for Phi, the Golden Ratio, Phi(1.618...)=phi(0.618.. and 1/Phi)+1 (among other expressions). I think that other Metal Means work similarly if not identically. Is that what this is all about?

 

Now, looking more abstractly at the situation from a bit more distance- the numerators are the low energy, and the denominators the high energy, versions (again assuming 162 is in the mix). Then the left members of the proportion involve Bohr approximations, with a point nucleus, correct? And 173 (and presumably 162) involve more realistic approximations with a nucleus that has an actual volume and cross-sectional area, among other things. So what is the relationship between the two dimensions here, or is it more complicated, say, in some kind of tetrahedral mapping instead of 2D?

 

Jess Tauber

[tetrahedron]

Now, with all this going on, I wondered whether the proton mass to electron mass ratio, a bit over 1836, might be amenable to a Pascal-type analysis. Well, clearly we don't have more numbers to play with, just the one.

 

BUT, let's see... 44sq gives 1936, which is 1836(44sq)+100(10sq). 44 is 11Lucx4Luc, and 100 is 5sq(Fib)x4Luc.

Yet 43sq is 1849, different from 1836 by 13Fib. 45sq yields 2025, which differs from 1836 by 89Fib. 46sq is 2116, different from 1836 by 180, 2x5Fibx18Luc. 47sq is 2209, different by 273 (related? to 173 of the FSC+ 100?). 273 is 3x7Lucx13Fib. And 47 by itself is Luc. How far up can we go? Dropping in the other direction, 42sq, gives 1764, differing from 1836 by 72, another half/double square (which is also 100-28, the latter part being 4Lucx7Luc). And 41sq=1681, different by 155 (100, sq10,+55Fib); 155 is also one of the numbers I mentioned earlier, being 1/4 620 related to Phi. It is also 5Fibx(13Fib+18Luc), showing that addition is important too here. Finally, for this pass, 40sq=1600, differing by 236 and 39sq=1521, differing by 315.

 

Can anyone here see the relationships in the last two?

 

While for this set we lack more basic numbers, there ARE the Muon and Tauon families- which would give their own different masses for the electrons and the hadrons composed of their particular quarks. There are THREE of these families too in the Standard Model, with the oft sought but never seen 4th family as well. Could these be the analogues of what we see in the FSC system?? The different families DO increase in mass and energy. Has anyone ever calculated what masses Mu and Tau protons would have?? The other Standard Model particle massas relative to each other will also be dimensionless constants- can they be related in this way?

 

Jess Tauber

[Don Blazys]

To pascal,

 

Quoting pascal:

All this stuff coheres in frightening ways mathematically.

 

Whenever I read something like that, I salivate like Pavlov's dog.

 

However, wading through substantial amounts of data and information from various sources

(and even from post to post) is cumbersome, tiresome, time consuming and often confusing.

 

Therefore, it would be great if you could "lay out" your ideas using easy to understand

charts, graphs, lists and tables with accompanying explanations and links to sources.

 

After all, a picture is worth a thousand words and the "LaTex" in this forum is both powerful and user friendly.

This is important because most folks are not as well versed in particle physics as you are, least of all myself.

 

Speaking of lists, can we somehow aquire a complete list of particles and their associated FSC values?

 

So far, all we have is this:

 

Massless: [math]\alpha^{-1}=137.03604(11)[/math]

Electron: [math]\alpha^{-1}=137.035999084(51)[/math]

W-boson: [math]\alpha^{-1}=128.08(42)[/math]

Z-boson: [math]\alpha^{-1}=127.08(42)[/math]

 

which is rather measley.

 

If we can't find one, then perhaps we should make one.

 

Don.

[CraigD]

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 …

I wholeheartedly agree on the serious and important business of fun with math. I’m only partly joking about fun being important, because if at least some people didn’t find math pleasant – many, I think, like me, viscerally so – I don’t think enough people would have done it to ever establish math, or even representational language, as an individual and societal human behavior. Without that thrill akin to love that comes from intuiting patterns and formally teasing/proving them out, humans would have remained as we appear to have been for hundreds of thousands of years before the beginning of history, living much like any other animal in our ecological niche. Something closely akin to fun with math birthed human culture and history, and remains, as I see it, what critically distinguishes us humans mentally from our very close primate species relatives. Washoe and Koko showed the glimmerings of it in their affection for rhyme and wordplay, but to the best of my knowledge, no non-human animal has ever shown the least spark of interest in number theory.

 

... 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.

...

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

137 is cool! Since reading your post, Jess, I’ve been playing happily with it and other “repeating digit palendromes”.

 

Numerology, though, isn’t the same as number theory (“numberology”, if such a word were in common use), any more than a numeral is the same as a number, the map the same as the territory, or the written word “rock” the same as a rock.

 

Numerology is a special kind of analysis of the digits of numeral representing numbers. As such, it depends strongly on the numeration scheme used to represent numbers. Most numerology, being popular mostly among mystics with naïve, considering only base 10 Arabic numeral schemes.

 

Consider, [imath]n = \frac{1}{137} = 0.0\overline{07299270}_{10}[/imath]

 

It is an unusual reciprocal of an integer, because not only does it have palindromic repeating digits – that is, [imath]\frac{n}{10^{-5}} = \frac{n}{10^{-6}} \pmod{10}[/imath], [imath]\frac{n}{10^{-4}} = \frac{n}{10^{-7}} \pmod{10}[/imath] ... [imath]\frac{n}{10^{-2}} = \frac{n}{10^{-9}} \pmod{10}[/imath] – but it has more digits of its base (4 out of 10) than any other reciprocal of a small integer (I checked through [imath]\frac{1}{50000}[/imath]). In bases other than 10, however, checking through base 500000, it doesn’t have any palindromes.

 

Of all the small integers in all bases, I found [imath]n = \frac{1}{898} = 0.00\overline{00321444412300}_5[/imath] to have the most impressive palindrome. Checking through [imath]n = \frac{1}{10000}[/imath], it’s the largest base to have a palindrome containing all the digits in its base, for bases up to 100. All base 2 palindromes contain both of its 2 numerals, as do, it appears, about 22% of base 3 palendromes, but after base 5, there don’t seem to be any for small (<10000) integers.

 

The numerological specialness of numeral representation of [imath]\left( \frac{1}{137} \right) _{10}[/imath] and [imath]\left( \frac{1}{898} \right) _5[/imath] depends as much on the number 10 and 5 as they do on the numbers 137 and 898. Since the choice of base for a given numeral system, or even the choice of regular-exponent-of-a-base numeration systems, is arbitrary and cultural (in our case, almost certainly due to us having 10 fingers), the numerological specialness of numbers like these has as much to do with our culture as with fundamental qualities of nature or numbers.

[Don Blazys]

Quoting CraigD,

Without that thrill akin to love that comes from intuiting patterns and formally teasing/proving them out....

Yeah! If there is no challenge, then life quickly becomes very boring.

This is true even when it comes to movies and other forms of entertainment.

Imagine a James Bond flick in which .007 finally retires and the entire film is about

him just relaxing by the pool and drinking vodka martinis "shaken, not stirred". :cocktail:

Heck, we would all be fast asleep! It would be a complete and utter flop! No one would go see it!

 

Don.

[tetrahedron]

Now you've wowed me out of my socks, CraigD! That's quite an effort, with very very interesting results. I've wondered about other bases, but had no way of accessing them. For 1/898(base5)...3214444123... summing end-pairs 32+23=55, 14+41=55, 44=44, together 154, 11Lucx7Lucx2(ambiv). For 1/137(base10), ...729927... the sum of the three pairs is 198, or 18Lucx11Lucx1(ambiv).

 

Is this part of a larger pattern? Just for a lark I added (all in base 10) 898+137 yielding 1035, which is 5x9x23

 

These three numbers also just happen to be in the sequence after the Lucas numbers, so 3,1,4,(5),(9),14,(23),37,60,97...

 

137 is by itself, digitally, part of the Lucas sequence 2,(1),(3),4,(7),11,18... Now note that for both we have the first two numbers, miss the third, and the have the fourth in each sequence. 1035 almost looks as if it is in the Fibonacci sequence, with 2 missing and replaced by 0, but with reversal of order of what's missing. 1035 is also the sequence of two Pascal tetrahedral numbers based on squares of odd integers, 10 and 35 (remember what I wrote in an earlier posting about the FSC numbers and how they deviate from 300 when summed pairwise).

 

898-137=761. Here I can't see any obvious connections except perhaps 1000-761=239, and then we have again for the series after Lucas ...(9),14,(23)... but here reversed, with 14 again missing. In earlier posts we saw the generation of 261, which is 19sq-10sq. Here 761 would be 19sq+20sq. Can 1035 be gotten by combining 2 squares?

 

Jess Tauber

[LBg]

Can 1035 be gotten by combining 2 squares?

 

Not by adding two squares, but by subtracting:

34^2 - 11^2, 42^2 - 27^2, 62^2 - 53^2, 106^2 - 101^2, 174^2 - 171^2, 518^2 - 517^2

[tetrahedron]

34Fib, 11Luc, 42=2x21Fib; 62 as per the Phi based set, 53=100-47Luc, 106= 2x the last entry. I've seen 101 elsewhere, dunno what it implies (related perhaps to 199Luc?), 174 is 6x29Luc, 171 is 200-29Luc. Don't know what to do about 518, 517.

 

Some time back I imagined calling all these numerical interconnections the 'Golden Tapestry'- how does that sound to you folks (assuming someone else isn't already using it)?

 

Jess Tauber

 

 

Not by adding two squares, but by subtracting:

34^2 - 11^2, 42^2 - 27^2, 62^2 - 53^2, 106^2 - 101^2, 174^2 - 171^2, 518^2 - 517^2

[LBg]

137 is a prime number

 

137 is a prime of the form 8n+1

137 is a prime of the form 3n-1

137 is a prime of the form 6n-1

137 is a prime of the form 2n+3

137 is a prime of the form 30n-13

137 is a prime of the form x^2+101y^2 (x=6, y=1)

 

The sum of digits of 137 is a prime (namely 11)

 

137 is the lesser of a pair of prime twins

 

137 is a prime p such that 3p-2 is prime

137 is a prime p such that 2p+1 is composite

137 is a number n such that (10+n!)/10 is prime

137 is a number n such that 6n-1, 6n+1 are twin primes

137 is a number n such that (13^n - 1)/12 is prime.

 

137 remains prime if any digit is deleted

 

137 is not the sum of 2 primes

 

137 is the number of primes between 2^10 and 2^11

 

Fib(137) is a prime number

 

137 is odd but not divisible by 5

 

137 is the sum of 4 positive cubes in one or more ways

 

137 is both the sum of two nonzero squares and the difference of two nonzero squares

 

137 = 4^2 + 11^2 and 4/137=0.0291970802919708..., 11/137=0.080291970802919708... same digits!

 

137 occurs in the pythagorean triples (105, 88, 137) and (137, 9384, 9385) and no other

 

137 is a number of the form x^2 + xy + 2y^2, (x=1, y=8, and 1+8 = 9 = 3^2)

137 is a number of the form x^2 + 2*y^2, (x=3, y=8)

 

137/60 = 1/1 + 1/2 + 1/3 + 1/4 + 1/5

 

137 is the number of restricted hexagonal polyominoes with 5 cells

 

137 is a number n such that Mordell's equation y^2 = x^3 + n has no integral solutions

[Turtle]

137 is a prime number

...snip...

137 is a number n such that Mordell's equation y^2 = x^3 + n has no integral solutions

 

:thumbs_up

 

137 is the 30^th prime, the 27^th chen prime, & the 53^rd non-polygonal number. ;)

[tetrahedron]

LBg ti:kama:nude: >137 = 4^2 + 11^2 and 4/137=0.0291970802919708..., 11/137=0.080291970802919708... same digits!

 

And 4 and 7 are Lucas numbers!

 

LBg....>137 occurs in the pythagorean triples (105, 88, 137) and (137, 9384, 9385) and no other

 

And 88 is the interval between 49 (7sq) and 137, and between 137 and 225(15sq)

137+105 is 242, twice 121 (11sq), 11Luc. 137-105=32, 32 being half and double square. 105 is also 21x5, both Fib numbers, and remember that 88 is 11Lucx5Fib. In 9384 we have 4x21Fib for the last two digits (or 3x4x7 all Luc) and the first two are 3x31, where 31 relates to Phi based 62, and is also 13Fib+18Luc. In 9385 the last two digits are themselves both Fib (8,5), and together as 85 are 13x5, again both Fib.

 

LBg....>137/60 = 1/1 + 1/2 + 1/3 + 1/4 + 1/5

 

This is rather interesting- but what does it mean?

120=1x2x3x4x5, and may be the end of the periodic table atomic-number-wise (we'll know in a few years). Remember that 137+128+173+162=600. Isn't there a periodicity of some sort along the lines of 60 units in the Fibonacci sequence or some other (can't remember). In any case this makes me think of other motivations for the FSC.

 

LBg....>The sum of digits of 137 is a prime (namely 11)

 

Again with the Lucas numbers!

 

Turtle....>137 is the.....53rd non-polygonal number

 

47+53=100. And Lbg- you missed this?!? My faith in you is crushed. How can I ever open my heart to you again? So sad.....

 

Jess Tauber

[Turtle]

...

Turtle....>137 is the.....53rd non-polygonal number

 

47+53=100. And Lbg- you missed this?!? My faith in you is crushed. How can I ever open my heart to you again? So sad.....

 

Jess Tauber

 

i don't want to speak out of line, but i don't think Lbg is up on the non-polygonals. [ EDIT: checking my data files i find that all Lucas numbers >= 7 up to 370248451 are non-polygonal. ∆, i posit that all Lucas numbers are non-polygonal. Hold on!! i was looking only at Lucas primes when i wrote that....will do some more checking..... ](we have proven that all primes are non-polygonal btw, so all Lucas primes must be non-polygonal.) another btw, not all Fibonacci numbers are non-polygonal.

[Turtle]

i don't want to speak out of line, but i don't think Lbg is up on the non-polygonals. [ EDIT: checking my data files i find that all Lucas numbers >= 7 up to 370248451 are non-polygonal. ∆, i posit that all Lucas numbers are non-polygonal. Hold on!! i was looking only at Lucas primes when i wrote that....will do some more checking..... ](we have proven that all primes are non-polygonal btw, so all Lucas primes must be non-polygonal.) another btw, not all Fibonacci numbers are non-polygonal.

 

ok; i was mistaken on the strike-through bit. my bad! :doh: while all Lucas Primes are by default non-polygonal, not all Lucas Numbers are non-polygonal. for example, 322 is polygonal twice. >> 322 2 [4, 55, 7, 17] (4^th 55-sided number & 7^th 17-sided number)

[tetrahedron]

Heptagonal with 18 units would be Lucas- does it count? Actually sequence ..7Luc, 18Luc, 34Fib, 55Fib is interesting too.

Tetradecagonal with 76 units Lucas.

 

Actually the table at http://en.wikipedia.org/wiki/Polygonal_number shows that the numbers are constructed across of triangular numbers, and then down with triangular number increments added. This is very similar in flavor to the table of Fibonacci-like numbers, where one can put the first Fib sequence across, then repeat is shifted up 1 move (because of the sequence 1,1), then downward entries have Fib number increments added. If one can do this with the triangulars, and the Fib-like series, then one should also be able to do it with higher-dimensioned Pascal diagonals (since each diagonal associates with a dimensionality). Now, if one can do this for the primary Pascal numbers, what about for reverse-engineered matrices for other sequences that give the other Metal Means?

 

Jess Tauber

 

 

i don't want to speak out of line, but i don't think Lbg is up on the non-polygonals. [ EDIT: checking my data files i find that all Lucas numbers >= 7 up to 370248451 are non-polygonal. ∆, i posit that all Lucas numbers are non-polygonal. Hold on!! i was looking only at Lucas primes when i wrote that....will do some more checking..... ](we have proven that all primes are non-polygonal btw, so all Lucas primes must be non-polygonal.) another btw, not all Fibonacci numbers are non-polygonal.

[Turtle]

Heptagonal with 18 units would be Lucas- does it count?

 

no; it doesn't count. off by 1 is the bane of number theory.

 

Actually sequence ..7Luc, 18Luc, 34Fib, 55Fib is interesting too.

Tetradecagonal with 76 units Lucas.

 

Actually the table at http://en.wikipedia.org/wiki/Polygonal_number shows that the numbers are constructed across of triangular numbers, and then down with triangular number increments added. This is very similar in flavor to the table of Fibonacci-like numbers, where one can put the first Fib sequence across, then repeat is shifted up 1 move (because of the sequence 1,1), then downward entries have Fib number increments added. If one can do this with the triangulars, and the Fib-like series, then one should also be able to do it with higher-dimensioned Pascal diagonals (since each diagonal associates with a dimensionality). Now, if one can do this for the primary Pascal numbers, what about for reverse-engineered matrices for other sequences that give the other Metal Means?

 

Jess Tauber

 

i am familiar with that wiki page and while accurate insofar as it goes it is woefully incomplete. i should say that i find much of this interesting, however i put no stock in don's associating polygonal number density to the fine structure constant. the equations he derives have become ever more complex over several years and require tweaking every time someone calculates polygonal number density to a higher value. fwiw i dare say one could do similar juggling with any strictly increasing function.

[Don Blazys]

Quoting Turtle:

i should say that i find much of this interesting

So do I. :)

 

Quoting Turtle:

however i put no stock in don's associating polygonal number density to the fine structure constant.

That's perfectly understandable. Ever since I proposed the problem of developing a counting function for

"regular figurative numbers" or "polygonal numbers of order greater than 2" as they are otherwise known,

I too have found it very hard to believe that the fine structure constant should in any way be involved.

 

That said, as a mathematician, I must investigate all possible constructs, and as it turns out,

the constructs that do involve the fine structure constant are infinitely more accurate than

the constructs that don't. That's just a fact! I never, ever, expected it! It's not my fault!

 

Quoting Turtle:

the equations he derives have become ever more complex over several years and require tweaking

every time someone calculates polygonal number density to a higher value. fwiw i dare say one could do

similar juggling with any strictly increasing function.

 

That is not true. The function:

 

[math] \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) [/math]

 

has remained unchanged ever since I introduced it. Approximation functions for erratic sequences require error terms

which are then adjusted as new data becomes available, and if you look at the six possibilities in posts #7, #12 and #25,

then you will find that in every case, the only thing that ever gets "tweaked" is indeed the error term and nothing else!

 

My derivation of the fine structure constant is another story. When Lars Blomberg first e-mailed me that

he would be able to calculate [math]\varpi(10^{15})[/math], I knew that I would then be able to use that data to derive:

 

[math] \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} [/math]

 

where [math]A=2.566543832171388844467529...[/math] is the "Blazys prime number generating constant", and which,

in this particular case, approaches the "Gabrielese value" [math]\alpha^{-1}=137.035999084(51)[/math] as [math]x[/math] goes to infinity.

You see, the above derivation of [math]\alpha[/math] is simply a consequence of the available data, and since the data is still inadequate to

derive a final conclusion, it too contains an error term which will be determined as higher values of [math]\varpi(x)[/math] become available.

 

Like I already pointed out, polygonal numbers are among the most studied numbers in the entire history of mathematics,

and Google searching "polygonal numbers" finds this counting function on the very first page! That's how important this is!

 

We are the very first to even attempt a counting function for this most important sequence of numbers! That's a historical fact.

Thus, we are indeed making history, and future mathematical historians will definitely judge us as to how we cooperated with

and treated each other.

 

...May they have mercy on us!

 

Don

[tetrahedron]

Don, exactly what is it that you think is generating the FSC physically, if one assumes that your equations work. For me, in my own work, I've hypothesized that it is something woven into the fabric of space-time-energy-mass itself, perhaps relatable (there's that word again...) to the Planck units.

 

In any case the numbers are unlikely to just pop out of nowhere. What is making them go?

 

Jess Tauber