Euler’s Identity And The Fine Structure Constant G

oh, "e^(π*i) = -1" , I have such fond memories of you from complex analysis.

What I don't understand is why you -- or others for that matter -- arbitrarily pick some combination of numbers like phi to try to get @ equal to. I mean, you could use the Riemann zeta function (at even intervals... Z(2) = pi^2/6; Z(4) = pi^4/90), or Bernoulli numbers (I don't remember them, and I cba to google it).

However, for all I've said I suppose Alpha is right. Frankie, if you find @ approximations interesting, you should keep pursuing it. After all, curiosity and innovation expands our understanding of physics. If per chance you do manage to stumble on some combination of pi, phi, gamma, e, apery's constant, etc to equal @, I'm sure physicists and mathematicians alike will congratulate you for your work.

The problem is that just sitting there trying, by hand, dozens or hundreds or thousands of little algebraic expressions, trying to get something which is a match to the experimental value of alpha tells you nothing.

Firstly, because infinitely many such expressions will exist (though most won't be nice). Secondly, you have no way of knowing if your answer is right. Thirdly, it doesn't tell you why that's the answer.

For instance, you can compute the surface area and volume of an n-sphere using geometric arguments and some complex analysis (Gamma functions etc). That will yeild a general formula for all, including non-integer, dimensional spheres. It also tells you why it's that expression and it'll work for any radius size.

If instead someone gave you the value of the volume of a 6-sphere with radius 2 and then, after a lot of trial and error, you got some simple function of pi and some other numbers, what does it tell you? Nothing. You cannot generalise it without having to repeat the same analysis for many other configurations (and even that is hit and miss) and until you come up with the correct formula (which you might never do), you'll not see why it's that formula.

Numerology is not the way to go in such things. Unfortunately all to many cranks here can't see past it.

But in this paper, α is not a number, but a measurement. All measurements have uncertainties associated with them, and latter measurements may be made to refine the estimate.

The paper actually says
1/α = 137.035999068±0.000000096
so
α = 0.0072973525701±0.0000000000051
which is consistent with the widely accepted (CODATA) value:
α = 0.0072973525680±0.0000000000240
Perhaps, one day CODATA will accept the result you quote, but probably they are going to wait for experimental and theoretical confirmation.

Plugging in your preferred value for α (0.0072973525701±0.0000000000051),
x = 0.0070327±0.0000051
which makes your form work.
0.1 × e(-φ²) + ( 1 + √3 + x) / 1000000 = α
But this increased precision of α leads to increased precision in x, which does not help your situation.

x is now over 1300 standard deviations away from zero. (The less precision value of α I chose meant x was less than 300 standard deviations from zero, so you can see how increased precision has hurt you.) Without a physical explanation for x being different from zero, this just emphasizes that you are on the wrong track. Measurements come with uncertainties, while numbers like e, φ and √3 do not. Equations relate numbers to numbers, and by explaining only most of measurement, you have that glaring left over part with no "explanation." 0.0070327 -- worse you don't really know what the last 3 digits might be and whatever you say could be entirely falsified by a later, better measurement.

So much for the bare numerology of this.

The physics is worse than the numerology, since "1000000" is clearly made up to be pleasing to humans, not the physics of the universe. As AlphaNumeric has written, you give no support why the form α = 0.1 × e(-φ²) + ( 1 + √3 + x) / 1000000 is preferred. And x being significantly non-zero means you aren't even matching that form.

I don't believe I advocate anything other than for Frankie to pursue his quest-- how exactly I'm uncertain, that's his choice. I'm not quite sure if you -- or anyone else for that matter-- construed what I said in an earlier post to advocate numerology, because that certainly was not my intent. My intent was neutral at best.

I don't want to pull a fast one or change subject here, but isn't physics an empirical study? Hell, math is an empirical study. Am I properly using the word "empirical"? I'd say empirical is "trial and error", and that's exactly what physics (imo) is.

Yes Alpha, you are right about expressing things like numbers. But while there are infinitely many ways to express pi [be it fractions like "22/7," a very crude approximation or "4*arctan(1)" or "(6*Zeta(2))^(1/2)"], in the end, there is only one pi.

Perhaps I have a very narrow minded view, perhaps I may be ignorant. I was a math major, so maybe I'm biased. If anything, the intent of my criticism of Frankie was geared toward what rpenner said," The physics is worse than the numerology, since "1000000" is clearly made up to be pleasing to humans, not the physics of the universe. As AlphaNumeric has written, you give no support why the form α = 0.1 × e(-φ²) + ( 1 + √3 + x) / 1000000 is preferred. And x being significantly non-zero means you aren't even matching that form.".

What do you mean. It's called RESEARCH. First you accuse everyone of no math, then of too much arithmetic.

http://www.vorpublishing.com/fine-structure-137.html
http://www.geocities.com/Area51/Nebula/3735/fine.html
http://www.people.vcu.edu/~chenry/Jerry%20Iuliano.htm

I just wish someone would relate it to the Feigenbaum Constant.
oooooh, they already have

http://www.greatdreams.com/numbers/jerry/feignbm.htm

How about to my birthday?

Alpha,

I wrote a quite long post, but will not post it. It was simply too long. Instead, I will simply make note of the fact that "numerology" is not always without relevance to the physical realm. Penrose tiling, for instance, played a key role in the discovery of quasi-crystals, once deemed a physical impossibility. Prime numbers play a role in signal processing and cryptography . The Fibonacci Series has been employed in the construction of search algorithms. Perfect Numbers and Mersenne Primes find application in string theory classifications. And so on...

By the way, please don't make the assumption that I just scribble out equations upon a piece of paper, okay? Without a little bit of an underlying cosmology, you don't just "shoot from the hip" out formulas such as this one...

((((2^(phi^(-30))) - 1) * (2^((phi^(-30)) - 1)) * 10^12) + (sqrt 5 + .1/sqrt 5) = c*m/s = 186,282.397

If you're familiar with the construction of perfect numbers or know the number of edges of a dodecahedron or icosahedron, that might be interesting to you. If not, perhaps not. But one interesting information "chunk" I made mental note of today in relation to that equation up above there, thanks to Guest00, is that, -1/30 effectively forms what one might term a "double loci" at index positions 4 and 8 of the Bernoulli numbers. http://www.bernoulli.org/ ).

Believe me, there is a method behind the madness. In the most general way, I start with the concept of balance and "compound" symmetry and "relative zeroes". As above below and all that vedic nonsense and I've been basing much of my thought and educated guesses upon a form of math I've been developing privately, something I call "particle" math or "Kolo" math (Kolo is slavic for "circle") in which pi is effectively set to zero, meaning, for instance C=2r or 2R and A=r^2 or R^2 where r = 1/100 and R = 100 ... but that's too much to go into herein and I'll save it for another time...

Cheers,
Raphie

Were those things developed by just plugging in hundreds of different numbers and someone spotting that the ones with the right properties followed a pattern or where they derived in a more rigorous, analytic manner?

Things involving 2^2n (like Fermat primes) come up in string theory due to the sizes of supersymmetry algebras or the number of basis elements in an exterior algebra. It's nothing to do with numerology, the expressions are derived in a rigorous manner.

That isn't numerology though, it's due to how you work in modulo arithmetic systems. If you read up about how thr RSA method works, it's based around the problem of factorising prime numbers. It's the result of a lot of deep number theory, not someone plugging in numbers repeatedly and spotting a pattern.

See, to me that's nothing but numerology. You have taken a bunch of numbers and gone "Wow, look, it comes out as the speed of light". Yes, the speed of light in our totally arbitrary system of units. For a start, you give the units as m/s, when 186,000 is the speed of light in miles per hour. See how unimpressive it is? If you pick different units, your result is meaningless. If we all worked in 'feet per minute', you'd be thinking how pointless that formula is.

Infact, given ANY number, you can invent a set of units which has the speed of light as that speed in your new units. For instance, natural units have c=1. Not a particularly impressive formula for the speed of light there.

No, you mean pi is effectively set to 1.

You cannot do that geometrically without doing a considerable amount of differential geometry. I remember having a discussion with a Cambridge postdoc about this once and while you can construct discrete geometries which has pi not equal to 3.14... but some other constant everywhere, you cannot construct a continous geometry which has pi equal to a constant, not equal to 3.14... The value of pi in terms of circles will vary over the space and the size of circles.

Not enough understanding it seems. You shot from the hip and missed the target.

φ = (1+√5)/2
r = (2^(1/(φ^30))-1) × 10^6 = 0.37256+
s = 2^(1/(φ^30) -1) × 10^6 = 2^(1/(φ^30)) × 10^6 / 2= 500000.18628+
rs + √5 + 0.1/√5 = 186282.397166+

Since there are exactly 1609.344 meters per mile, the speed of light = 299,792,458,000/1,609,344 = 18737028625/100584 = 186282.39705+ miles/second
or exactly 186282 miles 2096 feet and 5 21/127 inches per second.
or about 186282 27/68 miles/second.

So the speed of light is exactly

186282 39937/100584 miles per second.

Unless your caclulation works out to this exact value, you are polluting the truth, not contributing to it.

This post has been edited by rpenner on Oct 17 2007, 06:14 PM

Slight amendum to my last post. I corrected him with the incorrect units. I said miles per hour when obviously I ment miles per second. In that case my correction isn't required (I've never seen anyone use m/s for 'miles per second' before).

My point about pointless numerology stands though.

This post has been edited by AlphaNumeric on Oct 17 2007, 06:24 PM

I was going to write about transcendental functions and algebraic numbers and the impossibility of computing a rational number from this form. But, since the entire direction is wrong headed, there's little point to go into the math reasons why it can't work.

Basically what science is about is generalization. Discover by progressive experiment what's the most basic rules a system obeys. Numerology is about ascribing (without support) meaning to relations between numbers and has nothing to do with the rules of the system being studied. Numerology is magic -- the practitioner expects to learn about the universe by doing mere sums, not by actually modeling or predicting anything.

The easiest way to deal with the speed of light is to declare the use of natural units, where c=1. Problem solved with exactly the appropriate amount of effort. It is unforgivable that the huge expression given doesn't exactly match the speed of light in miles per second which is what was claimed.

The easiest way to deal with the fine structure constant is to keep it as a parameter of the theory. Then you never need to know its digits except when comparing experiment and theory. This is the very approach used in QED, including the paper cited by the numerologist.

rpenner

You're very good at mathematics. Do you work in this field? I will never have a mathematical grasp yourself.

rpenner>''So the speed of light is exactly

186282 39937/100584 miles per second.

Unless your caclulation works out to this exact value, you are polluting the truth, not contributing to it. ''

Yeh bub. Ain't it 186, 356 mps. I thought it was a fraction over.

Neo, I don't mean to acerbically criticize you-- in fact, it's meant to be my neutral opinion--, but I believe rpenner's intention was that there is always some degree of inaccuracy or uncertainty when expressing c. That is to say, despite c being constant, say 186,282 miles per second, the moment we realize that it's really 186,281.918765324* (this is an arbitrary value which I have made up for the purposes of explaining inaccuracy; in no way am I claiming this to be the true value of c), all of a sudden we realize how inaccurate our original value(s) are/were.

Another example I'd like to use is this: Suppose we can express c, the speed of light as a function: a^2*b + d^3*e^3 - 4 = c (again, this is some arbitrary function I have made up for use in this example). If we were to say, plug in sqrt(2)* for a, in the best case scenario, we'd use sqrt(2) instead of 1.414 or 1.4142 or 1.41421356....

Not to say who's right or wrong, but I think rpenner was wise when saying something along the lines that the important thing here is not to look at these things as numbers or values, but measurements and the context in which they appear in.

Dear Alphanumeric,

Thank you for correcting my error regarding setting pi to zero. Indeed, I set pi equal to one which I view as a "relative" zero in the sense that once one goes on to the y-axis, defining the y-axis (conceived of as a plane on edge) in this example as exponential, not linear, in nature, based on "z" units to the nth power), division effectively becomes subtraction and multiplication becomes addition (i.e. x^a * x^b = x^a+b ala the exponential function)

And perhaps that is as good a point as any to follow up on, to explain what I mean by the concept of the "relative zero," not unrelated to the concept of "renormalization" and "measurement" as a relative construct, in order that I may do what I can to disabuse you of this notion that I am simply "plugging" in numbers haphazardly. Although there is admittedly a process of trial and error that goes into what I term "dialectical math" aka the "math of approximation," a methodological counterpart to "particle" or "Kolo" math, and although I admittedly have been making use of a technique I call "semiotic bridging" which makes liberal use of symbolic logic), I do believe I am applying rather consistently on a conceptual level pretty standard and even accepted mathematical and physical concepts, from fractals to Fourier transforms to modular arithmetic and basic premises of number theory, etc. even if I I am yet learning the specific mathematical/physical argot (i.e symbols and nomenclature - feedback always welcome and patience requested...).

So, what do I mean when I state that "division effectively becomes subtraction and multiplication becomes addition" when one segues from the x to the y axis? Perhaps the concept is obvious to you, but nevertheless I will explore it herewith if only for others who may come along and read this.

Let's start by looking at the number pair of (2, -2) and let's place that number pair on the X-axis relative to the point of intersection of the x and y axes. What is the midpoint? The total distance between the two points is 4 and the average distance from the mean is |2|, so the answer is clearly 0.

But now let's take the average of z^2 + z^-2 where the first term moves "up" the y-axis and the second term moves "down" the y-axis. What is the midpoint of these two terms? z^2 is two "exponential units" from the mean and so too is z^-2. The "distance" between the two units is equal to z^4 (z^2*z^2) and the average distance from the mean is equal to z^|2| meaning the midpoint, aka the "equilibrium" point is equal to z^0 = 1. To get back to what we think of as "0" from the perspective of the x-axis, we would have to subtract one. Thus, z^|2|/ z^|-2| - 1 = 0, (not unrelated in an inverse construction manner, by the way,, to the construction of Mersenne numbers, which follow the form 2^n - 1)

So where is the point of "equilibrium" at the theoretical "zero point." Is it z^|2|/z^|-2| - 1 = 0 or z^|2|/z^|-2| = 1? I would argue that the "equilibrium point" is BOTH, in twin loci elliptical fashion, not this OR that, but this AND that because one (I believe) ought to include BOTH frames of reference x and y (assuming for now just the x and y axis) in the calculation, yielding an "average" of .5 (i.e. z^0 + 0)/2 much as many mathematicians have argued that the infinite series 1+1-1+1-1... equals .5, given that .5 is the "average" of 1 and 0 and the answer depends on where you take a metaphysical "knife" and "slice" the series in order to "measure" it. The limit of n slices and measurements of the "zero point" from a random sampling from x or y axis (S(n)/n) slices would converge to .5, even if in practice it might never get there. But let's not go there for now. For now, I simply wish to make the point that there is a displacement of 1 that happens the moment one switches from x to y axis and this displacement has to be incorporated into any complex analysis whether of quantitative or qualitative nature.

To sum up, with regards to the point of intersection of x and y axes, 1 can be 0 and 0 can be 1 depending on one's frame of reference, Furthermore, from the perspective of an imaginary observer standing on the y-axis, and given that the rate of change is constant, the distance between z^1 and z^2 will appear to be the same as the distance between z^2 and z^3. If one simply thinks in terms of "uniform acceleration" then the exponential rate of change will "feel" as if it is linear and, indeed, it will be the unit increments upon the x-axis that would appear logarithmic in nature and would be mathematically defined in accordance with whatever inverse "looking glass" function of z would be appropriate (not unrelated to Fourier transforms, something I am in the process of learning the mathematical language for...).

So, again, which is which? The answer for me is obvious. IT DEPENDS. For the y-axis folk "standing" on the exponential y-axis 1 "appears" as if it were 0 although we x-axis folk might laugh and say "Ha! That's not zero! That's 1! Don't you know anything?!?" And meanwhile, all the y-axis folk are scoffing at the x-aixs folk calling -1 zero. And so on.

And now, just for fun let's turn our (unlabeled) x-y polar coordinate graph at a right angle and assume our reference point to be the horizon (a reference point constructed in the mind of the observer) and now x will "seem" as if it were y and y will "seem" as if it were x, but because we have the horizon as our reference point, and saw the rotation, we still know that x is still x and y is still y, even if displaced by 90 degrees.

Meaning that now we have another point of reference. Not just our knowledge that x represents linear growth and y represents exponential growth and not just the horizon, but also a mental component THROUGH TIME in which we have observed and remember the rotation (i.e. the "change" or "progression") that took place. Dispense with any of these ponts of reference and we would be right back at a point where we don't know what is what, where up could be down or left or right or future or past or whatever...

For instance, instead of removing an unobserved rotation by removing the time component, let's remove the triangulated reference point of the horizon, once again a CONSTRUCTED point of reference based on observer POV, and imagine our x-y polar coordinate axis hurtling through space on the level of galaxy or elementary particle or whatever. Where is x and where is y? Which is which? And what is our scalar reference? The answer is "We simply don't know" As noted before, from the perspective of our y axis, the x axis "seems" logarithmic and vice-versa, meaning that we have to make a CHOICE about which is which, and all else stems from there in terms of measurement.

The most apt metaphor that comes to mind for me is "Gulliver's Travels" in which Gulliver travels both to the land of the Lilliputians and to the land of the Brobdignagians (Giants). In Lilliput, Gulliver is non-plused by the war over how best to eat a hard-bolied egg, just as the Brobdignagians are beyond confused when Gulliver explains the arcane partisan politics of the U.K. Gulliver's "relative zero" point is his own frame of reference, just as the Lilliputians and Brobdignagians frames of reference are their "relative zeroes."

This is all related to the issue of measurement in miles versus meters, and, in anthropomorphized terms to issues of TRANSLATION and TRANSFORMATION, but I will save the bulk of that for another post. In the meantime, let it suffice to say that I "discovered" the fine structure constant while researching how the current mile became the mile, my interest being that I have a working HYPOTHESIS that the laws of physics apply on all levels, including human, from the individual psyche to the family to the collective to the collective of collectives (I plan to return to school for an advanced degree in Sociology, FYI, and what I am doing is related to Functionalism and Symbolic Interactionism both. My approach is critical realist in nature while my motivation is an interest in social justice...)

My anthropomorphized starting point?

e=mc^2

translated to...

Z = PI^2
Zightgeist (sic) = Power * speed of Idea^2

which has now morphed a bit into the still qualitative, but less specifically anthropomorphized, equation...

z = pc^2
Equilibrium ("Zero") Point = progression * connectivity^2

which, as i conceive of it, is basically the Nash Equilibrium + expansion/contraction, but again for another time...

In any case, in an attempt to see if there might be a correlation between phi and the speed of light (phi being correlated in my mind with both progression and connectivity), I plugged a number in to Google and got just one result and that result was a web page, long since forgotten, related to alpha. That piqued my interest, especially because of the reason I was doing the search. As I mentioned above, I was interested in issues of TRANSLATION and TRANSFORMATION, and the reason I was doing the search was that in researching the possible correlation between phi and c, I noted the following numerical relation:

10^2*(phi^31) = 301,034,900 = 99.58% c *s/m = 299,792,458 (meters/sec)

and

10^2*(phi^30) = 186,049,800 = 99.86% c * s/m = 186,292.397 (miles/sec)

I thought, Wow! What a neat coincidence! And then I thought "Wow! What a neat coincidence! Is that a MEANINGFUL coincidence?"

As I said, it's a question, not an answer, but I believe it to be a question worth asking.

FYI... the full expression for that miles/per hour equation is as follows...

(((2^(phi^(-30))) - 1) * (2^((phi^(-30)) - 1)) * (10^12))
+ sqrt(5.20200)
- (8192 / (10^8))
= c*s/m + u
= 186 282.3970512 + u
(u=-6.09 * 10^-8)

The hypothesized correspondence in meters/sec?

(16114*phi^30)*10^-2
+ sqrt(5.20200)
- 8192
= c*s/m + u
= 299, 792,457 + u
(u=-7.027 * 10^-4)

where...

16114 is the 18th "node" of the Golden Scale (Fib series beginning with 2,5 = F(n+2) + L^(n)) where n = 20, F(n+1) + L^(n-1)) where n = 19 and F(n) + L^(n-2)) where n = 18

F(n) = n-indexed Fib number
L(n) = n-indexed Lucas number
(note: at infinity F(n) and L(n) converge at the rate of sqrt 5 = phi + 1/phi)

8192 = M13 = (2^13 -1) + 1 where M13 equals the 13 Mersenne number (2^13 - 1) and 5th Mersenne prime (8191), while one "completes the loop" insofar as it connects the beginning back to the end, and I believe it may be related to the "cancellation of one loop divergences" in (SO) 8192 lattice group...

(sqrt 5 + .1/sqrt 5) = sqrt 5.202 (will post the real number derivation at another point in time)...

And so on...

Both equations follow the generalized form ax(1) + bx(2) - dx(3) = c (+u), and I associate this structurally with the Minkowski metric (Minkowski normalized the speed of light to 2^31-1), with b corresponding to a "spherical loop" correction and d corresponding to a "linear loop" correction. The various x's I view as a scalar adjustment.

At the very least can a I get a few points for creativity? :-)

Kindest Regards,
Raphie

P.S. RPenner, you are not the first to suggest "shooting from hip" syndrome. I don't believe this to be the case. That said, I am struggling to ground these ideas to ground and as per my first posts some months back onthese boards, I happen to believe there is more than one way to skin a cat, Shrodinger's or otherwise...

P.P.S. Updated to correct a few typos. I am sure there are more...

This post has been edited by Raphie Frank on Oct 17 2007, 11:11 PM

Frankie, with regards to the infinite series sum, I S[(-1)^n ; {0,infinity}] (Grandi's series), it's important to note that it's divergent. I believe any 2nd semester calculus student should be able to realize this with little trouble. I'll leave the proofs with the assumption that the knowledge is a priori.

Yet, I could show II S[(-1)^n ; {0, j}] to equal (-1)^j * (j+1). Ah, but they aren't the same, are they? I and II are different as I will shortly explain.

The important thing here is that we sometimes commit huge errors by treating infinity as a number, when it is in fact a concept. Euler himself committed this error on several occasions (which is debatable) -- I'd recommend Dunham's "Euler: The Master Of Us All" as an excellent read to novices, amateurs, and experts alike.
Infinity and infinities alike must be taken care of very carefully.

I digress, being born of latin decent-- though English is my native tongue-- has allowed me to read some of Opera Omnia, albeit with difficulty. I believe Ed Sandifer's articles (which are free!!!) are also excellent reads. I think they'll aid you in your quest and clear things up which I cannot do atm, though I may post with a follow up later tonight.

(a) http://www.maa.org/editorial/euler/How%20E...nt%20series.pdf
(b)
http://www.maa.org/editorial/euler/How%20E...%20calculus.pdf

I'm a bit pressed on time, but another excellent read is Euler's "Calculus on the infinitely small". You can google it if you're interested.


I would like to express my apologies to the board and moderators for posting links when I shouldn't be (I think it said in the rules you can't post links till you reach a certain number of posts, but I cba to read the rules atm). I also don't like promoting other peoples books, because that is not my intent (i.e. not here to sell stuff).