View Poll Results: Multiple poll: Check all you agree. Logarithms of lepton mass quotients should be pursued. 24 27.91% Alpha calculation from serial expansion should be pursued 22 25.58% We should look for more empirical relationships 26 30.23% Pythagorean triples approach should be pursued. 21 24.42% Quotients from distance radiuses should be investigated 16 18.60% The estimate of magnetic anomalous moment should be investigated. 26 30.23% The estimate of Weinberg angle should be investigated. 21 24.42% Jay R. Yabon theory should be investigate. 16 18.60% I support the efforts in this thread. 47 54.65% I think the effort in this thread is not worthwhile. 30 34.88% Multiple Choice Poll. Voters: 86. You may not vote on this poll

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## All the lepton masses from G, pi, e

It's OK

X = sqrt(137.03599911) = 11.7062376154766

11.7062376154766 = X
00.0854245431237 = X^-1 /(2pi)^0
00.0000992128957 = X^-3 /(2pi)^1
00.0000000183389 = X^-5 /(2pi)^3

=================================
11.7917613898350 = exp(pi^2/4)

_3.1415926536222 = pi (measured...)
_3.1415926535897 = pi (exact)

Regards, Hans

Tip: The Windows calculator is exact to 32 digits.

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Gold Member
 Quote by Hans de Vries It's OK X = 11.7062376154 11.7062376154766 = X 00.0854245431237 = X^-1 /(2pi)^0 00.0000992128957 = X^-3 /(2pi)^1 00.0000000183389 = X^-5 /(2pi)^3 ================================= 11.7917613898350 = exp(pi^2/4) _3.1415926536222 = pi (measured...) _3.1415926535897 = pi (exact) Regards, Hans
so when did you first see this?

also it is maybe not so important but what do you mean by the "measured" value of pi?

 Recognitions: Gold Member Science Advisor Rilke has a funny sonnet about mohammed being in a cave and a visitor comes and shows him something. it could be like the experience of seeing this. do you read German? I will see if I can find the poem Yes, here it is: Mohammeds Berufung Da aber als in sein Versteck der Hohe, sofort Erkennbare: der Engel, trat, aufrecht, der lautere und lichterlohe: da tat er allen Anspruch ab und bat bleiben zu dürfen der von seinen Reisen innen verwirrte Kaufmann, der er war; er hatte nie gelesen - und nun gar ein solches Wort, zu viel für einen Weisen. Der Engel aber, herrisch, wies und wies ihm, was geschrieben stand auf seinem Blatte, und gab nicht nach und wollte wieder: Lies. Da las er: so, dass sich der Engel bog. Und war schon einer, der gelesen hatte und konnte und gehorchte und vollzog.

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 Quote by marcus so when did you first see this? also it is maybe not so important but what do you mean by the "measured" value of pi?
I did checked it before I posted

"measured" just because it's derived from a measured value, (Only
to distinguish it from the exact value )

Regards, Hans

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Gold Member
 Quote by marcus so when did you first see this? ...
I mean, when did you first discover this numerical relation?
Have you known for weeks, for months? Or did you just
notice this a day or two before posting it?

I think it must have been a strange experience
so I am curious about what it was like

Recognitions:
 Quote by marcus Mohammeds Berufung Da aber als in sein Versteck der Hohe, sofort Erkennbare: der Engel, trat, aufrecht, der lautere und lichterlohe: da tat er allen Anspruch ab und bat bleiben zu dürfen der von seinen Reisen innen verwirrte Kaufmann, der er war; er hatte nie gelesen - und nun gar ein solches Wort, zu viel für einen Weisen. Der Engel aber, herrisch, wies und wies ihm, was geschrieben stand auf seinem Blatte, und gab nicht nach und wollte wieder: Lies. Da las er: so, dass sich der Engel bog. Und war schon einer, der gelesen hatte und konnte und gehorchte und vollzog.
Schön

Fun, isn't it? At least it might give an entry from another
direction to reveal something of the underlaying
physics (or geometry)

Regards, Hans

Recognitions:
 Quote by marcus I mean, when did you first discover this numerical relation? Have you known for weeks, for months? Or did you just notice this a day or two before posting it? I think it must have been a strange experience so I am curious about what it was like
I found it a week or two before posting. tried to see if it
could be further simplified but couldn't resist to post it
anymore. (Even though I'm the kind of person who will
never feel 100% sure about anything )

Regards, Hans

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Gold Member
 Quote by Hans de Vries Fun, isn't it?
I do not know what to say.

if it is a coincidence then
it is the most decimal places coincidence I have

I wish some of the others would say something

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Gold Member
 Quote by Hans de Vries I found it a week or two before posting. tried to see if it could be further simplified but couldn't resist to post it anymore...
well I am glad you posted it here
where we can see and discuss some

I suppose the next thing is two things:

1.write email to some physicists
(I would think immediately to write Lee Smolin because
it could have some connection with his CNS scheme for
iteratively generating the fundamental dimensionless constants)

2. prepare an article to post on http://arxiv.org

Alejandro knows about arXiv, has some experience.
Once it is on arXiv then it will always come up when people do
keyword searches. Then it may be of use to someone who
finds it by accident and is building a theory to explain alpha.

I think anyone here would be glad to help. It is quite interesting
that a coincidence (let us call it that) should go out 10 decimal places

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 Quote by marcus 2. prepare an article to post on http://arxiv.org Alejandro knows about arXiv, has some experience. Once it is on arXiv then it will always come up when people do keyword searches. Then it may be of use to someone who finds it by accident and is building a theory to explain alpha.
Yep, I know from the ArXiV , and I can see it is not easy to fit there, as the goal of the preprint distributions is to keep researchers up-to-date about advancements in *their* area. It is clearly not a hep-th/ , as it does not have a theory under, and probably not a math-ph/ . It could be focused as sort of "state-of-art memotecnics" or something in a line able to fit as physics/ (think about the American Journal of Physics kind of articles).

 Recognitions: Science Advisor I found an interesting one: The idea is to see if Nature's limitation to three gene- rations might have some relation with the identity: $$3^2 + 4^2 = 5^2$$ For the three consecutive integers 3,4 and 5, leading to only 3 states rather than a whole series. We write for the three lepton masses: . . $$3^2 \ ln(m_e) \ \ \ + \ \ \ (4^2+n)\ ln(m_\tau) \ \ \ = \ \ \ (5^2+n)\ ln(m_\mu)$$ . . We then look at the value of n. We do find $n \approx 1.00086$ using the following Codata values for the mass ratio’s: __3477.48 ____ (57)__ tau/electron ___206.7682838 (54)__ muon/electron ____16.8183 ___(27)__ tau/muon The least well known value, $$ln(m_\tau)$$ has to be exact to within 0.005% to get a result so close to 1.0000. This is less than the current experimental uncertainty which means that the exact value of n=1.0000 is still within the experimental uncertainty. Regards, Hans . . . PS. Please be aware that the chance for coincidences might be bigger than you think! PPS: The formula is invariant under the transformation $$\{ m_e, m_\mu, m_\tau \} \rightarrow \{ xm^y_e, \ xm^y_\mu, \ xm^y_\tau \}$$ where x and y are arbitrary numbers.

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Actually there are two "pythagorean sequences", the other one is (-1,0,1), having as square the 1,0,1 sequence. Intriguingly, in your combination there is a twist, because the -1 is not going along with the 3, but with the 4.

 Quote by Hans de Vries PS. Please be aware that the chance for coincidences might be bigger than you think!
I can see your conflict; your first set of formulae and this third one have different assignations for the quotient of logarithms,
$${\ln m_\mu/m_\tau \over \ln m_e/m_\mu}={\pi^2-1 \over 2\pi^2-3}; \, {\ln m_\mu/m_\tau \over \ln m_e/m_\mu}={3^2\over 4^2+1}$$
so surely at least one if them is a random coincidence.
(EDITED: Or, perhaps, there is some happy circunstance where an expression happens during some series expansion of the another.
(EDITED again: in fact, using the expansion $$\pi^2/6=\zeta (2)=\sum_n 1/n^2$$, its first term is $$10/18$$, ie $$9+1 \over 17+1$$ so both expresions are no so far away: the second formula can be seen, at least, as the first term of the expansion of $${\pi^2-1-1/2 \over 2\pi^2-3-1/2}$$. Good enough if one takes into account that both HdV formulae are approximate guesses of an hypothetical exact formula.
Said this, both relations have an intense geometrical flavour. We are exploring two avenues of discretization of mechanics which could, in the long run, provide some justification: either to parametrise the ambiguity in taking two derivatives of the position to get Newton law, or to parametrise the ambiguity in the two sequencial derivatives of Lie Bracket.

 Blog Entries: 6 Recognitions: Gold Member Perhaps it is physics after all; I have just revised (ah, google!) a point that I should have noticed instantaneously: that the Pythagorean condition appears naturally in the definition of Cayley spinors. A guy called Andrzej Trautman has worked out the diophantine case SL(2,Z). If interested in this line, check http://math.ucr.edu/home/baez/week196.html and references therein.
 Recognitions: Science Advisor Alejandro, I'm trying to go back to more physics but first this. The two lepton ratio formula's may also be combined into the following: $$\ln {m_\tau \over m_\mu} = {\pi-{1 \over \pi}}$$_______ $$\ln {m_\mu \over m_e} = {4^2+1 \over 3^2} (\pi-{1 \over \pi})$$_______$$\ln {m_\tau \over m_e} = {5^2+1 \over 3^2} (\pi-{1 \over \pi})$$ Now if and only if the term $\pi-1/\pi$ was exact then all three formulae would be within experimental range. Now it isn't but the $\pi-1/\pi$ term is the only thing that I could connect to some real physics up to now. It's inspired on the way how you rewrote: $$\ln {m_\tau \over m_\mu} = {\pi-{1 \over \pi}}\ \ \ \ \ \ \equiv \ \ \ \ \ \ \ln {m_\tau \over m_\mu} = 2\sinh(\ln \pi)\ \ \ \ \ \ \equiv \ \ \ \ \ \ {m_\tau \over m_\mu} = |\exp( \sinh(\ln\pi) )|^2$$ the sinh() gives us something in the space-time domain or in momentum space if we consider it to be a boost like in: $$\sinh \xi=\frac{ v/c}{\sqrt{1-v^2/c^2}}$$____$$\cosh \xi=\frac{1}{\sqrt{1-v^2/c^2}}$$ with: $$\tanh \xi=v/c$$ ____$$\exp \xi=Doppler Ratio$$ The Doppler Ratio now becomes $\pi$ interestingly enough (for blueshift) and $1/\pi$ (for redshift) corresponding with a speed v/c (of rotation?) The term $\pi-1/\pi$ could for instance correspond with the imbalance in momentum change when absorbing a photon from the back and a photon from the front. The term $|\exp( )|^2$ may possibly be associated with going from phase space (defined in x and ct) to probability space. if the masses were to be defined as "mass density probabilities" (mass * wave function) then the imaginary part of the argument would define phase while the real part would lead to the mass ratio at each point of the wave function. Another, although numerological, reason to become interested in this approach is that the most exact equations I got up to now are found in the "boost domain", that is: $$\ln {m_\mu \over m_e} \ = \ 2\sinh(a) * 1.000000093 ,$$____$$\ln {m_\tau \over m_\mu} \ = \ 2\sinh(b) * 1.0000047$$ with: $$a = 1 +\sqrt{1/2} \ \ \ and \ \ \ ab^2 = \sqrt 5$$ I don't know if a and b are really a pair but at least there is a simple relationship. That's the best I can do for sofar in the hope to get some physical meaning out of it. Regards. Hans
 Blog Entries: 6 Recognitions: Gold Member Yep, I intended a connection with relativity when casting sinh here, but I am still far to understand how it goes. There is some dinamical analogy between the tetrad of elementary particles, $$\nu, e, u, d$$ and the tetrad of relativistic coordinates $$t, \rho, \theta, \phi$$. On one side quarks are unobservable, so they feel very much an angular coordinate -which lacks of metric scale-. On another neutrinos are sort of special, so they feel as the trivial time coordinate. But we are very far from getting real meaning of this analogy; my own research program is in the backburner fire already for more than five years (see hep-th/9804169 and "cited by" there) with no remarkable milestones. Hans, are you using some symbolic algebra program to search for your equations? I ask because even if the results are not publishable as a physics article, it could then to be a very valid article for computational journals.

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 Quote by arivero TL attributes the second one to Lubos, but I have read it attributed to Feynman himself, just as an example of the kind we are discussing, random relationships. It could be interesting to know the origin of the first one to alpha, as it is a variant of HdV idea.
Let me to update on this. While Lubos Motl seems to have proposed the 6 pi^5 equation in an independent way, some previous postings in the net give it as folklorical knowledge, sometimes related to a couple of footnote-quoted papers by Armand Wiler:
Wyler,A., 'On the Conformal Groups in the Theory of Relativity and their Unitary Representations', Arch. Rat. Mech. and Anal.,31:35-50, 1968
Wyler,A., L'espace symetrique du groupe des equations de Maxwell' C. R. Acad. Sc. Paris,269:743 745
Wyler,A., 'Les groupes des potentiels de Coulomb et de ¥ukava', C..R. Acad. Sc. Paris,271:186 188
According F. D. T. Smith, this work of Wyler was noted in the Physics Today Aug and Sept 1971 (vol 24, pg 17-19 according M. Ibison).

Regretly old comptes rendues are not in the internet so I can not verify these papers. The Physics Today comment attracted some discussion in the Physical Review Letters,
http://prola.aps.org/abstract/PRL/v27/i22/p1545_1
http://prola.aps.org/abstract/PRL/v28/i7/p462_1
http://prola.aps.org/abstract/PRD/v15/i12/p3727_1
And well, there they refer to a "Wyler equation for alpha",
$$\alpha=(9/8 \pi^4)(\pi^5/2^4 5!)^{1/4}=1/137.03608$$
not to an equation for proton mass. So we are still in doubt about the priority of the proton/electron quotient.

PS: please do not believe the Journal-Ref of Tony Smith, he is always exchanging trickeries with the arxiv (and please do not comment about it in this thread!!!). But Tony is always a good starting point to remember exotic, sometimes forgotten, research.

 Quote by arivero Perhaps it is physics after all; I have just revised (ah, google!) a point that I should have noticed instantaneously: that the Pythagorean condition appears naturally in the definition of Cayley spinors. A guy called Andrzej Trautman has worked out the diophantine case SL(2,Z). If interested in this line, check http://math.ucr.edu/home/baez/week196.html and references therein.
By the way, Trautman (I guess it's the same guy) considered
applying categories to gravity a long time ago. He presented
an article on it at Dirac's 70th birthday party.

The HdV work is cool. I'm tired of arguing with a string
phenomenologist that I know, who believes that it's in
principle impossible to calculate (from a fundamental theory)
the mass ratios.

Cheers
Kea

 Tags koide formula, lepton masses