Euler’s Identity And The Fine Structure Constant G

ERROR correction:

1) I did of course mean that F(n) and L(n) diverge rather than converge...

2) phi^30/10 = 186049.8 = 99.86% c (miles/sec), not 10^2*phi^30

Apologies for the mistakes and thank you for the links, Guest. I will take a look.

Best,
Raphie

This post has been edited by Raphie Frank on Oct 18 2007, 12:08 AM

Guest00, you have not gotten to the core of my message.

α is a measured number. It's exact value rarely matters. Knowing it to a few parts per thousand is often good enough. But the claim that a certain formula "explained" α was not true because it had an extra x which was 1300 standard deviations away from zero. If the formula "explained" α it would not have the x which remains non-zero and non-explained.

Since 1983, international convention has defined the meter in terms of the second and the speed of light. So in meters per second, the speed of light has an exact value, as it does in miles per second (or even miles per hour -- don't confuse the units). So c is an exact number in all these units. I criticize the formula and the new formula in that they are 1) complicated and 2) not exact.

Numerology is no substitute for reason and physics. Raphie Frank continues to compound his error by issuing new formulas which by their newness prove the old formula was in error, despite Raphie Frank's conviction then that the method was sound. This does not bode well for eventual success.

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 + √5.202 + 8.192 × 10^-5 = 186282.39708449827+
186282 39937 / 100584 = 186282.3970512208701185079+
So there's no exact match here, either.

And Raphie Frank is correct when he says √5 + 0.1/√5 = √5.202
(√5 + 0.1/√5)² = (√5)(√5 + 0.1/√5) + (0.1/√5)(√5 + 0.1/√5) = (√5)(√5) + (√5)(0.1/√5) + (0.1/√5)(√5) + (0.1/√5)(0.1/√5) = 5 + 0.1 + 0.1 + 0.01/5 = 5.202 = (√5.202)²
Which I think is arithmetic manipulation at or below the GCSE level? I'm not a UK educator.

Raphie is also correct that I am not the first to say "shoot from hip" -- as I quoted him saying it above.

But the formula in meters is also off.

161.14 × φ^30 + √5.202 - 8192 = 299792458.000702725830638763733669595935776255+

Even if there was "deep significance here" there would be more deep significance in that Raphie Frank has a numerological expression for miles/km. Which is also wrong.
1000 * [ 5 × 10^11 × 2^(1/(φ^30)) × (2^(1/(φ^30))-1) + √5.202 + 8.192 × 10^-5 ] / [ (161.14 × φ^30 + √5.202 - 8192) ] = 0.6213711923468789+

Dear RPenner,

Here is a part of that post I wrote last night, but deemed too long too post...

===================

You make the point, RPenner, and I thank you sincerely for the time you took to respond to my earlier post, that one must understand QED in order to derive formulas of possible interest to physicists and I just can't say as I am convinced that this is the case. Such knowledge certainly improves ones odds, and I have been working at frantic pace to educate myself in that area, but is such knowledge either a necessary or sufficient condition to generating such formulas? IF, in fact, the laws of physics apply to patterns of human organization and perception, as Jung, for instance posited, and great Physicist/Thinkers from Pauli to Einstein to Penrose would seem to agree or have agreed, then it stands to reason at least to postulate that for each and every physical phenomeneon there is a mental correlative; that the mental and the material realms can be thought of as bijective mappings of one another.

IF, in fact, this is the case, then, for instance, the concept of the holographic mind, in which every part contains the imprint of the entirety, as a holographic plate divided can yet project the entire image, may not be so farfetched after all. Whether we are speaking of quarks, leptons or bosons, or physics, biology and sociology or cells from the brain, nose or posterior region, the question becomes far more one of perspective and orientation and emphasis than one of objective validity. In other words, one multi-dimensional "crystal," many facets.

And what would be one of the implications IF this were to be the case? And this, as before, is all a question, not an answer, but I believe it to be a worthwhile question to ask... Well, at least in theory, we could study social systems or the psyche or even language to generate hypotheses regarding the workings of the cosmos and vice-versa. But there's a problem, an obvious one to me, and it's something I call "Babelization." Even as knowledge and information explodes in exponential upon exponential fashion according to what Ray Kurzweil termed "The Law of Accelerating Returns" (and long gone are the days when one man or woman can know "everything"), we simply don't have the common argot to effectively communicate across disciplines even when we're speaking the same language.

Biology uses one set of symbols and terms and paradigms. Chemistry another. Physics another. And these are practically sibling disciplines. Now bring in the philosophers and the psychologists and the politicians and sociologists and artists and on and on and you've got a multi-threaded system chalk full of co-evolving strands of knowledge and experience diverging in logarithmic fashion even as we are facing global dangers of unprecedented magnitude: Nuclear proliferation to rogue nations and terrorist groups, Global Warming, dwindling supplies of renewable energy, increasing concentration of wealth in violation of the "Tinbergen Norm" (Tinbergen developed the "Gravity Model of Economics) etc. etc.

An ironic thought, I know, in an age of globalization and connectedness on a scale also unprecedented, and I do not wish to sound alarmist because i have great hope and optimism for the future of humankind also, but the danger is that, as Yeats might have put it, the center cannot hold. All systems may be self-organizing, which I believe to be the case, but unfortunately such self-organization oft times takes violent form and that is something obviously I believe we can agree that most of us want to avoid.

The bottom line is this:

I believe we need to start figuring out some shortcuts to better communication across disciplines (not to mention across race, color, creed, class, etc.) so that we might more effectively "tie it all together" (social string theory, anyone?) before we blow it all apart. And in order to do that, my take is that we need better "algorithms" for understanding and communicating with one another and ought to figure out ways to pool our resources in distributed computing, parallel-processing fashion.

===================

All that said, let me make one point clearly: I do not believe that understanding can come from pure thought alone. But what I do believe is that reason, intuition, imagination, experience, observation, testing etc. are all together a part of the overall equation.

Furthermore, as pertains to my comments regarding a pooling of resources, I believe we must figure out ways to bridge disciplines. Not empiricism, not poetic social theories, but the two in tandem. Not positivism, not phenomenology, but both in tandem. "Hard" and "soft," reason and intuition in tandem. My goal is not to be a mathematician or a physicist, but to learn enough to be able to work with mathematician and physicists.

Another point I'd like to make, you mention the level of mathematics involved, and point it out with a certain amount of judgement associated, but I feel, as you feel with Guest00, that you are missing the thrust of the basic point I am trying to make (although perhaps I have not made it well...). In Vicoish, back to basics manner, I don't believe that one NECESSARILY requires higher levels of complex mathematics to intuit certain knowledge, so much as one requires complex and interdependent and relational patterns of thought.

For instance, one of the beauties of number theory is that one can let the embedded algorithm of the series "do the work" for you. As one need not understand a computer's assembly code in order to use it, one also need not understand prime numbers in order to use public key code, and so on.

Fib numbers, for instance, related obviously to Lucas numbers, and therefore to the powers of phi in compound form (phi^2n + phi^-2n = L^n; phi^2n-1 - phi^-2n-1 = L^n) can be thought, relative to one another, as a flowing series of whole number Pythagorean triples (i.e. a "moving equilibrium" in the vein of some of the premises of functionalist sociology) with each F(2n) equal to F(n-1)^2 + F^(n+1)^2.

Does any of this have application in functionalist sociology? Or with respect to string theory or bit string physics? Or IA? Perhaps, yes. Perhaps no. The answer is that I DON'T know, but, to reiterate a point I have made repeatedly, I believe the inquiry to be more than worthwhile.

We need new approaches, because the old ones are no longer sufficient as Peter Woit, I believe, made amply clear by negation in his book "Not Even Wrong," a critical response to string theory.

In closing, let me simply make reference to the words if Stanford Professor of Physics Emeritus H. Pierre Noyes...

=================

A Short Introduction to BIT-STRING PHYSICS"
http://arxiv.org/pdf/hep-th/9707020.pdf

When I first met Ted Bastin in 1972 and heard of the Combinatorial Hierarchy (hereinafter CH), my immediate reaction was that it must be dangerous nonsense. Nonsense, because the two numbers computed to reasonable accuracy — 137 ? hc/e 2 and 2127 + 136 ? hc/Gm 2p — are empirically determined, according to conventional wisdom. Dangerous, because the idea that one can gain insight into the physical world by “pure thought” without empirical input struck me then (and still strikes me) as subversive of the fundamental Enlightenment rationality which was so hard won, and which is proving to be all too fragile in the “new age” environment that the approach to the end of the millennium seems to encourage [84, 86]...

=================

I largely agree with Noyes. For whatever that is worth.

Kindest Regards,
Raphie

This post has been edited by Raphie Frank on Oct 18 2007, 01:49 AM

Ah. more ERROR CORRECTION

phi^2n + phi^-2n = L(n)
phi^2n-1 - phi^-2n-1 = L(n)

not L^n

Obviously.


You mentioned, RPenner that an expression for miles/km would be, possibly, of greater significance. Would you mind expanding on that thought? I believe it would be useful, not just for me, but for others who might come this way.

In spite of my dispute with what I view as your over-emphasis on empirical measurement to the exclusion of pure mathematics (of whatever stripe), empirical measurements which often must be updated once new evidence is "uncovered" or better measurements are made, I nevertheless have great respect for your knowledge and the depth of thinking you clearly bring to any discussion you are a part of.

Best,
Raphie

This post has been edited by Raphie Frank on Oct 18 2007, 02:13 AM

Pardon?

[Quote] And Raphie Frank is correct when he says √5 + 0.1/√5 = √5.202
(√5 + 0.1/√5)² = (√5)(√5 + 0.1/√5) + (0.1/√5)(√5 + 0.1/√5) = (√5)(√5) + (√5)(0.1/√5) + (0.1/√5)(√5) + (0.1/√5)(0.1/√5) = 5 + 0.1 + 0.1 + 0.01/5 = 5.202 = (√5.202)²
Which I think is arithmetic manipulation at or below the GCSE level? I'm not a UK educator.[/Quote]

Perhaps I may have misinterpreted your post as an attack at my understanding of mathematics-- if, however, it is an attack...

Firstly, the intent of my previous post(s) was to get across that no matter what, there are ambiguities that arise in certain numbers and/or how we calculate them, period. I used arbitrary, "stuff", if you will, to illustrate this. Second, I am a yank. I am not a welsh, a liverpudlian, or a brit (I say this in jest). So please don't count on me taking the A-levels, but maybe you'll take a paltry mcats score of 41r as a substitute.


[Quote]Since 1983, international convention has defined the meter in terms of the second and the speed of light. So in meters per second, the speed of light has an exact value, as it does in miles per second (or even miles per hour -- don't confuse the units). So c is an exact number in all these units. I criticize the formula and the new formula in that they are 1) complicated and 2) not exact./Quote]

[Quote]The easiest way to deal with the speed of light is to declare the use of natural units, where c=1.[/Quote]

I agree with you on both points; however, I would like to note our definition of certain things, like the seasons, are in constant change. How we measure the second now is different from how we measured it 100 years ago... the definitions continuously evolve. The fact that they do shows that there will always be a sense of ambiguity (with regards to certain things), which is my entire point. Frankly, I don't give a piss if a caucus or committee or bozo the clown for that matter says that an inch is equal to ... (you fill in the blank). If memory serves, I believe the "foot" will serve as a good example... it was originally a measurement dependent on the size of the king's foot. Not to be philosophical, but the fact that we are humans is proof enough that we are fallible and prone to errors/mistakes.


[Quote]α is a measured number. It's exact value rarely matters. Knowing it to a few parts per thousand is often good enough. But the claim that a certain formula "explained" α was not true because it had an extra x which was 1300 standard deviations away from zero. If the formula "explained" α it would not have the x which remains non-zero and non-explained.[/Quote]
This however, I disagree on. Perhaps it's because I look at things from the point of view of the mathematician. Exactness IS important. To illustrate my point, the area of a whole unit circle is pi, the end, period. Whether those "units" are in pink tutus, kangaroos, or cute koalas doesn't matter (don't take what I've just said word for word, if you are you are missing the point). Approximations, like pi = 3.14 is not acceptable.

Let us assume arguendo that this wikipedia entry is correct where it says, "practical science and engineering will rarely require more than 10 decimal places. As an example, computing the circumference of the Earth's equator from its radius using only 10 decimal places of pi yields an error of less than 0.2 millimeters."
Accuracy to one decimal place or a billion, it matters not. Pi is pi and that's that.
But having said that, I admit that sometimes, we mathematicians are guilty of hoping and/or praying that in the end, things will work out (i.e. canceling out pis and es and squares of numbers).

The physicist and mathematician are always arguing, even if our disciplines are similar. Frankie is doing something similar to throwing a dart at a dartboard a million miles away blindfolded-- that I think we can all agree on. If per chance I failed to communicate my point I will attempt to rectify things. I suppose however that's a problem for the linguist, not physicist.

I don't think that is my point. I think I'm of the opinion that in order to say anything meaningful about the use of α, you need to know QED. In order to suggest physical reasons why α is approximately 1/137, you would need to develop a physical theory that explains more than QED and the whole rest of the Standard Model of Particle Physics.

For example, to use π to compute the surface area of a sphere you only need to know a formula in geometry. But lets start with a number, x, that's the sum of the squares of the reciprocals of all the odd positive integers. A simple algebraic relation between π and x exists and is easy to write down, but proving it requires a lot more mechanics than just the simple high school geometry.

But even if your sums did take well-explained numbers that total to important physical constants (and this has not happened yet) the ability to compute a sum that totals to a goal does not explain the goal.

Does 3 + 7 or 4 + 6 explain 10 better? Does 1 + 2 + 2 + 3 + 4 + 5 or 13 + 4 explain 17 better? Neither is an explanation -- these are just totals.

If string theory is true, it is necessary, not sufficient.

I disagree because this would suggest that intuition is more valid than reason in dealing with the world, which is a hypothesis which has been soundly rejected over hundreds of years. It would suggest language skills, math skills and science skills come at least as easily to the uneducated and illiterate as those who make education a large percentage of their life's effort. For this reason too, it is experimentally rejected.

Tens of thousands of highly educated people designed and reasoned about every part the computer you read this article on. If you reject the supremacy of reason over pure intuition, ideologically forces should compel to build an Internet chat device uncontaminated by reason.

This analogy means little to me, since I know too much about the physics of actual crystals.

I disagree. Many multidisciplinary scientific journals exists because such common languages do exist. In English, the top two are Nature and Science.

I disagree, but even in High School I was well-versed in all three of those. They have a large amount of commonality which becomes more clear the more you know. So I think of those words in the sense you are using them are more three starting points than three disciplines. Each of these starting points are gateways to sub-disciplines of Science which have many sub-disciplines themselves. Tribology, Ballistics and High Energy Physics are all physics sub-disciplines with their own specialized journals. But they overlap with each other because each of them is a science of the observable universe. Likewise Biology, Chemistry and Physics (as disciplines) overlap.

The "softer" sciences like sociology and economics are "soft" in the sense that they are less reliable for predictions, because it's harder to do pure experiments on people. But to the extent that they are sciences more than wishful thinking, they represent advances of human knowledge.

Economics and the Law of Conservation of Energy suggest that finite resources will always be an issue, even for a universal dictator.

You left out education without which none of us, as Newton wrote it, could "stand on the shoulders of giants."

I disagree. How can two things be yoked in tandem if they pull in opposite directions. Communism was a "poetic social theory" which empirically generated authoritarian horrors.

Are you using Kant's or Comte's positivism?

Since mathematics is how science communicates found relationships, I disagree.

Which shows the difference between using technology and creating technology. Even the flint-tipped spear requires complex skill and understanding to build, but relatively little to use.

You left out Bode's Law.

If your formula's for the speed of light in miles per second and meters per second were exactly correct, then for free you get an expression for miles in terms of meters. I was pointing out for the sake of ridicule that your expression here is many times more complicated that the by-treaty established numerical values and is incorrect.

Perhaps I may have misinterpreted your post as an attack at my understanding of mathematics-- if, however, it is an attack...

I was simply making an observation on the basis of RF's post. RF who is in the UK, I assumed knows a GCSE.

It was only after our ability to measure the speed of light got better than our ability to use the old definition of the meter that the need to change the definition of meter was required. In 1983, the empirically demonstrated stability of the speed of light (which matches theoretical prediction from Maxwell and Einstein) resulted in the adoption of this definition of meter.

This however, I disagree on. Perhaps it's because I look at things from the point of view of the mathematician. Exactness IS important. To illustrate my point, the area of a whole unit circle is pi, the end, period. Whether those "units" are in pink tutus, kangaroos, or cute koalas doesn't matter (don't take what I've just said word for word, if you are you are missing the point). Approximations, like pi = 3.14 is not acceptable.

But we aren't talking about approximations, but measurements, formulas and "explanation." The x in my rewriting of RF's formula doesn't have an explanation attached to it. And since zero is the only thing in RF's world that doesn't need an "explanation" it would mean RF met his goal if the formula fit α (up to measurement error) with x = 0. Instead x is statistically significantly different than zero. (Which is to say, x is (much) more than 3 times the scaled uncertainty in α.)

Why would anyone think that the fact that I was born on the 137th day of the year is not significant in this argument? The connections between my birthday expressed in seconds after christ, pi, the fine structure constant and the 911 hijackings are extremely significant, to better than 10 places in base 16. (expecially if I can use natural logarithms). That can't just be a coincidence.

Hello Guest00,

Other than the fact that "all agree" must subtract 1, namely myself, thus "all minus one" perchance would have been the proper phraseology, why on this good earth did you choose 1 million as the distance of the target I am shooting at? Why not, say 999,9999?

I am making a point here, of course, by way of example, regarding the reciprocity between human nature and physical nature.

People love round numbers. My Great-Uncle Bill (RIP), a doctor, used to call it "the tyranny of round numbers." Others, such as the once-upon-a-Pythagorean, might have called it the "harmony" of round numbers.

As far as the harmony and tyranny of precision goes, however, I will cede to the good RPenner for now and give him a nice "round" EXACT number, although it kind of misses the point to a certain degree since I am far more interested in the error than in the precision and am trying to "bridge" units of measure in dimensionless manner... but anyway, it's always nice to have a zero point as a basis of reference, so here goes...

Let...

Q = 16,114 = 2F(n) + F(n-2); n = 20 -------- (F = Fibonacci Number)
or..
Q = F(n) + L(n-1); n=20 -------- (L = Lucas Number = F(n+1) + F(n-1))
or...
Q = G(n); n=18 -------- (G = "Golden Scale" = Fib series 2,5)
==============================
L(30) = phi^n + phi^-n; n = 30 --------- (L = Lucas Number = F(n+1) + F(n-1))
==============================
B = (2^n); n = 13 = 8192 = 5th Mersenne prime + 1
==============================
b = (2^n); n = 7 = 128 = 4th Mersenne prime + 1
==============================
p = 100 (== "radius" derived from Greek "rho" = p = "sperical radius")
==============================
c' = 299,792,458 * s/m

Then...

Q*L(30) - p(B - 1) + b = pc'

= G(18)*L(30) - 100*(2^13 - 1) + (2^7) = 100*c'

= 16114*(phi^30 + phi^-30) - 100*(8191) + (128) = 100*299,792,458

As an FYI...

If we were to translate via "Kolo" (aka "Particle") math 100 c' (i.e. 100*speed of light*s/m) into 2*pi*r by expanding pi from "point particle" (where pi is set to =1) status back to three dimensional space, then 100 c' would be equal to 2*pi*r; r = 100 = 200*pi, or, to phrase it in Greek terms... "sigma*pi" (sigma = 200)

Ah, if only there were another sigma in there somewhere. Then we'd have sigma*pi*sigma, the National Physics Honor Society...

Best,
Raphie

P.S. RPenner, I just noted that you posted while I was writing my post. Will review and respond tomorrow. Many thanks for the reply.

This post has been edited by Raphie Frank on Oct 18 2007, 10:27 AM

I will go a little astray from your post but with a reason i guess.

Basically you are talking about the "initial conditions of description"-meaning you unavoidably construct every possible future of a system by the very axioms you choose it to have. For example - making a reset for every cycle, choosing the revolution of earth as one cycle which resets to 1 unavoidably results lots of people having birthday on same day an many such similarities.
Any kind of a closed (cyclic to me) system has large number but finite of possible states and this implies things will unavoidably repeat somewhere. This is what one would find as not being a coincidence but it is basically coming out because of a chosen metrics. If every moment in universe is different from any
other for example then one couldnt say that earth has ever been in the same relative to everything position more than once. And this would basically imply infinity of states which means that they never repeat
exactly the same. Never is an absolute but in this context it works for me. Basically it comes out very simply that this universe will never again repeat as it is. And that is beyond reason comprehension. Even multiverse theory should have problems with this. It is a known question though. Is universe cyclic or
not. And if it is not, not much comments are needed. This never happened before and will never happen again. Which one do you prefer and how does that make a difference of how you see your life and everything for that matter?

Dear RPenner,

Other than to make "crystal" clear :-) that in no manner whatsoever do I believe intuition can replace reason, but rather that I believe the two go hand in hand, I am going to have to take a time out on replying to your comments specifically as I will have to do a little learning and research first in order to reply properly and am on a "tantextual" (i.e. a "contextual tangent") riff, thinking about Fibonacci "half-steps," as I call them, related to my my earlier comments regarding Pythagorean triples, the golden ratio, and moving equilibriums. (i.e. F(n-1)^2 + F(n)^2 = F(2n -1))

For instance, a known relation between phi, pi and e is...

2*cos (pi/5) = phi

... but since we're in a Eulerish frame of reference here on this message thread, an alternate expression incorporating the complex plane can be defined as...

2*e^(i*pi/5) = phi + i*(sqrt 5 - 5)/2
(where sqrt 5 - 5/2 = ~ 1.1755705)

equivalent in trigonometric form to...

2*cis (pi/5)

(where cis = cos x + i*sin x; x = pi/5)

But since I am also trying to "triangulate" to expressions related to Pythagorean triples, however, it's worth noting that we can also express 2*e^(i*pi/5) in the following manner...

limit n @ infinity of 2* [F(n)/F(n-1)] + i*F(2n - 1)/F(n)] = phi + i*(sqrt 5 - 5)/2 = 2*e^(i*pi/5)

where...
phi = F(n)/F(n-1)
i*(sqrt 5 - 5)/2 = i*F(2n - 1)/F(n)

Now, given the following identity...

F(n-1)^2 + F(n)^2 = F(2n -1)

... where F(n) equals nth indexed Fib number...

... then one could make a bijective mapping to the Pythagorean theorem...

a^ + b^2 = c^2

where...

a = F(n-1)
b = F(n)
c = sqrt F(2n -1)

Then by "dimensionless" substitution...

2*e^(i*pi/5) = 2*cis (pi/5) == 2* b/a + (i*c)/b

Looking glass thinker that I am, I will use that basic "structure" (i.e. the dimensionless expression) to see what I can learn about Bode's law, crystals and such via "Mind of God" research otherwise known as Google.

But for starters, the value of c = [sqrt F(2n-1)], it so happens, falls between F(n) and F(n+1), a term I equate with that "Fibonacci half-step" I referred to at the beginning of this post.

From F(n) to (sqrt F(2n-1)) @ infinity
= F (n) * (sqrt 5 - 5)/2
= F (n) * 1.1755705...
= sqrt F(2n-1)

From sqrt F(2n-1) to F(n+1)
= sqrt F(2n-1) * 1/(tan 36 degrees)
= sqrt F(2n-1) * 1.37638192
= F (n+1)

That 10^2/tan 36 degrees, by the way, is equal to 99.5624812% of @^-1 is something I note within the context of this thread, since we're discussing Euler and the fine structure constant, but don't place undue weight upon.

Far more interesting to me is the fact that...

(sqrt 5 - 5)/2 = 1.1755705

... is a term that showed up both in this expression...

2*e^(i*pi/5) = phi + i*(sqrt 5 - 5)/2

and this one...

F (n) * (sqrt 5 - 5)/2 = sqrt F(2n -1)

... but in the first expression it's the imaginary component, while in the second expression it's on the "real" number line. Furthermore, in the first instance, it's additive, while in the second case multiplicative, consistent with my earlier post regarding addition becoming multiplication on the complex plane.

Now, by reference to the aforementioned "Mind of God"... and all this might be old hat to you, but it's not to me and I'm trying to demonstrate "tantextual" learning in action, we find, in close to real time, that 1.1755705 = the length of the side of a Pentagon... Thus...

2*e^(i*pi/5) = Golden ratio + one edge of an imaginary Pentagon inscribed in a circle of unit length 1

or...

10*e^(i*pi/5) = 5*Golden ratios + the perimeter of an imaginary Pentagon inscribed in a circle of unit length 1

As I mentioned before, I'll have to see what this all might have to do with crystals, specifically quasi-crystals, but, in the meantime, please do feel free to "school" me.

Kindest Regards,
Raphie

This post has been edited by Raphie Frank on Oct 19 2007, 07:18 AM

ANOTHER ERROR CORRECTION:

sqrt ((5 - sqrt 5)/2)
= 1.1755705...
= the unit length of the edge of a pentagon inscribed inside a unit circle

NOT

((sqrt(5) - 5) / 2)...

which equals 1.38196601... = 1 + (1/phi)^2 or (3 - phi) depending on from which one direction one is making the "approach."

I wasn't joking Guest00 when I said I make many typos :-). One of the downsides of the relational approach, I guess. It makes one a bit dyslexic...

It's easy to play games with arithmetic, but that's not a way to extract information from numbers. Numbers can be composed in infinite numbers of ways, and have far less information that the equations of physics which are physical relations between quantities.

φ = ( 1 + √5 ) / 2
3 = number of space dimensions
4 = number of space-time dimensions
3 + 4 = 7 = 2 + 5
11 = number of dimensions in M-theory
3² = 9
4² = 16
5! / 2! = 60
11² = 121
4² + 11² = 137
2 × 5³ = 250
2^7 = 128

( (4² × (4² + 11²) × 2 × 5³ + 3 × 11² + (4² + 11²)² + 5! / 2!) × φ + 3² + (4² + 11²) × 2 × 5³ ) / ( (2 × 5³ × 16 + 2^7 + 11) × φ + 2 × 5³ ) =
( (16 × 137 × 250 + 3 × 11 × 11 + 137 × 137 + 60) × φ + 9 + 137 × 250 ) / ( (250 × 16 + 128 + 11) × φ + 250 ) = ( 567192 * φ + 34259 ) / ( 4139 * φ + 250 ) = 1/α ????

Does this tell you something new about α that you didn't know before? No.

Most certainly. As I'm sure you, Rpenner, are all too aware (I'm saying it for other's benefit), given a number X and a 'margin of error' Y (say X=α and Y = experimental error ~ 10E-9), there are infininitely many rational, algebraic and trancendental numbers in the range [X-Y,X+Y].

Ralph, giving a list of a few of those numbers doesn't mean you have progressing physics, it means you've spent some time mucking around with a calculator.

I used to do that in school, as a 9 year old. A problem a teacher once gave the class involved finding two numbers, A and B, which added to 10 and multiplied to give 20. The class then spent 30 minutes getting an answer to about 3 decimal places by playing about the calculators. It told us nothing of how to generalise such a problem. It was only years later when I came across quadratic equations and saw how they are constructed from their roots that I realised the generality of the system. Two numbers which sum to give 10 and which multiply to give 20 will satisfy x²-10x+20=0. Finding the roots of a general quadratic is much richer and interesting than solving each quadratic by hand (and except for a very small amount of cases, you never get the exact answer from your calculator).

See the difference? One is pure numerology, the other gets to the structure underneath. You waste your time just coming up with rough approximations.

Dear Dr. Penner,

Tell you what, because I see your point and actually AGREE with you, by and large. Will you PLEASE just keep an open mind, rather than assuming there are no guidelines or "triangulation" points of reference behind what I am doing?

I am not in school YET, and need the peer (or professorial, as the case may be... I would welcome your advice regarding a text book on the Standard Model) review in order to refine and discern that which I don't know and need to learn, particularly in terms of nomenclature and semiotic notation, although I suspect we will never see eye to eye because my approach is structural and prescriptive in nature and begins with QUALITATIVE observation and those observations guide the language in which I learn to express those observations, as does the audience to whom I am expressing it, as does the feedback I receive in dialectical manner.

As for your constant reference to the infinite number of ways we can express any numeric quantity, that's a lemma, and even if it were an opinion, it's one I agree with, but that's PRECISELY my point. In an infinite universe with an infinite number of ways to express anything, or in an infinite system of infinite order ala Ramsey theory, order that MUST BE IMPOSED BY THE OBSERVER, there are then infinite ways to CONSTRUCT our conceptions of "reality," based upon whatever points of references we agree to culturally.

But here's the thing, once you get started, there's a path dependent component, just as the computer keypad yet has the vowels placed on the outside against all common sense, Italian society resisted the Arabic (actually Indian in origin) numeric system for quite some time when Fibonacci attempted to introduce it. They were fine with all those letters because it was a common language and commerce depended upon it, meaning there was RESISTANCE, economic and otherwise, to an idea we now generally agree was the better idea.

As for me, to contextualize this, my reference is lifelong observation and thought and reflection, and self-education borne of a liberal arts background, and a constant challenging of pre-convention and societal norms leading to "data points" of experience most don't have. And I have offered up those challenges because I have been committed from young adulthood, not to a "good enough" world, but to a "better" one, and it's a choice motivated and guided by personal experience, experience I will not delve into herein, but suffice it to say that I understand the value of belief in oneself and others more than most as "factors of production."

I am interested in the hidden variable, the potential energy that we leave untapped and never find because we are not looking for it, and I speak socially here, not mathematically. I am interested in "hidden populations" and "statistical discrimination." It's what drives me because I see far, far too many good people for whom the current "system" does not work, many of whom are driven to self-abuse (I speak specifically of artists I know).

Why do we have the "No Child Left Behind Act" Dr, Penner? Does this not imply a consensus social conviction that many children were left behind before? What are we doing for those people, now adults? What heuristics or theoretical paradigms do we have in place to identifty, for instance, the "adult prodigy," because, no, Dr. Penner, it may be sufficient, but it is certainly not necessary that giftedness manifests itself at an early age.

Our definitions "rule" us, and limit the measure of the man (and woman) far too often and for far too many of us.

Necessity is the mother of invention. I do what I do for people like Helen Buyninski, booted from Columbia with straight A's and 1600 SAT's because she was a little too different (name used by permission) and, in a general sense, for those so desperate in neglected corners of our global society that they graffiti the walls of buildings with the word WATER.

The butterfly effect, Dr. Penner. That's what I am getting at here. I BELIEVE the laws of physics can be applied on an anthropomorphized level. I believe we are all connected, all of us, of that I am quite convinced -- it's all a matter of degree -- and it seems to me that studying fractals and patterns of information and material/immaterial correspondences (ala Actor-Network Theory) that propagate and spider out across all disciplines, as well as the relationships between them, are as good a place as any to start figuring out how to throw out some metaphorical life preservers to those amongst us lacking "access."

Because those lacking access simply cannot do it alone. They are (mixed metaphorical analogy here continued...) cast adrift from the ocean liner of state and society, and let me tell you, it's a whole lot easier to climb up the side of a ship when there's a ladder, meaning those with financial, intellectual and cultural capital to spare have, IN MY VIEW, a moral obligation to make sure those ladders are offered down the side of the ship. But ala the "just world hypothesis," we look down at those in the water and think "Ah they deserve to be there! Let 'em drown..."

The bottom line is this: It's not so much if I am right or wrong. The issue is "Would we be better off as a world society if I were more right than wrong regarding some of my more unconventional approaches and ideas (many of which can be found online, and many of which I will yet present...)? Obviously I believe so or I wouldn't be doing this, setting myself up knowingly in such manner and risking the appendage of "crackpot" to my name in people's minds.

In time, I hope I can convince you that at least my approach is worthwhile. I HOPE I can, but deem it unlikely. Barring that, I simply request the open mind and a little patience as I learn an unfamiliar language, because your opinion carries much "weight" in these here parts, and to be quite frank (no pun intended), it is rather terrifying to disagree with you in such vigorous manner.

Kindest Regards and great respect,
Raphie Frank

Raphie Frank: Self Portrait of a Poet of the Possible
http://politinotions.blogspot.com/2007/01/...of-poet-of.html

This is very much true. For instance, 1/1² + 1/2² + 1/3² + .... = pi²/6. This result was discovered centuries ago, but it took many years before anyone could prove it. It wasn't until Euler came along and published 5 (yes, 5) distinct ways of showing that the sum of 1/n² is equal to pi²/6 that people saw why it equalled that.

It was only noticed to be equal to pi²/6 because of the fantastic aptitude for series summation mathematicians used to have. When you don't have computer or calculator, your mental arithmetic and algebraic manipulation skills certainly sharpen up a bit!

Leaving your name dropping of Ramsey theory aside, I don't think you've got Rpenners point. I elaborated on it in my last post. Since we don't know alpha exactly, only to less than a dozen digits, you are free to construct numbers which have a certain 12 digits or so and then have anything else after tha and chalk it up to 'experimental uncertainty'.

What does this tell you? Nothing, because there's an infinite number of possible numbers you can construct which 'match' the experimental value of alpha.

Now take my example of 1/1² + 1/2² + 1/3² + .... = pi²/6. pi²/6 is an exact number, no experimental unknowns involved. The proof of that identity involved a great deal of mathematical machinary and knowledge of number theory and complex analysis. No trial and error, pure creative application of logic. In arriving at that answer, a lot of revealing work in complex analysis was done. There was no 'constructing our conceptions of reality', it was showing that a certain sequence of numbers exactly equalled another in a very elegant but rigorous way.

All you are doing is just playing about with any numbers which take your fancy. If you came up with a series of numbers which sum to alpha which were constructed from a QED method, it would at least give some viable explaination as to why thos numbers should be thought to sum to alpha and not just some number which looks like alpha for a few digits but isn't.

Euler proved 1/1² + 1/2² + 1/3² + .... = pi²/6 and not just 1/1² + 1/2² + 1/3² + .... approximately looks like pi²/6 to 20 decimal places (which was all mathematicians before him could do). He proved it despite not knowing more than about 15 decimal places for pi (infact, he didn't need to know any!). He could see it was a summation problem so thought about how he could connect infinite summations to such a number. For those people familiar with complex analysis, the use of residues within the complex plane is well known for such applications. There was specific logic structure behind the result, not just approximations.

You are doing nothing but approximations. There's no reason to think any of your answers are exactly any particular quantity you're trying to model. If, to the best experimental level, alpha = 137.035999068 and you come up with two randomly constructed sums which gives 137.0359990686738482.. and 137.03599906821904828..., which one is better? You have no way of deciding, because there's no logic behind your sums. If you'd constructed it from some processes or parts of QED, then you'd have justification behind your answer and you'd have one answer, not a series of progressive curve fits to an experimental value.