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Short – but not irrelevant – exercise February 7, 2007

Posted by dorigo in mathematics, physics, science.

You are given three different positive real numbers A<B<C (C is the largest). Each of them has a uncertainty equal to a part in ten thousand or smaller. How large does the integer number N=1/Q have to be in order for  the following formula to hold:

 (A+B+C)/[(A^Q + B^Q + C^Q)^N]  – 1 < 10^(-5) ,

as a function of the ratio C/B?

The question has some relevance to the strange coincidence found by Koide on charged lepton masses. I was startled by the astounding coincidence between that ratio (save a probably well-motivated factor of 2/3 at the denominator) and unity (the most up-to-date number is 1. +- 0.000026), but then I started thinking at what really is the probability of something like that occurring by chance.

Assessing the real “significance” of such a coincidence -that is, assessing how likely it is that such a simple formula can describe the ratios of lepton masses, numbers that the Standard Model does not explain and which appear to be falling out of nowhere, is important, but probably unfeasible. In fact, one would then also need to estimate how likely it is that a formula describing “by chance” these ratios can be obtained by using eigenvectors of a quite simple 3×3 matrix.

Despite the trouble with assessing the probability of the coincidence stated above, it remains in my opinion a quite interesting open question.


1. Kea - February 7, 2007

Cool post! Of course, the 3×3 matrix is not random, either, but derived from theory. What is the probability of that? You are right to worry about this, though.

2. Bee - February 7, 2007

to make the issue even more complicated: if you’d take all the masses of the standard model particles, pick three of them, and scramble them together in a random equation, what’s the probability that there is such an equation fulfilled with the above precision? Okay, the Koide equation fulfils various symmetries, so it’s not actually random – you might want to put some constraints on the elegance of the equation.

what’s your opinion about the strange formula?

3. dorigo - February 7, 2007

Hi Kea,

yes, the 3×3 matrix is one of the simplest matrices one can think of, and this makes the matter quite hard to brush off with pseudoexperiments or other frequentist stuff like that. The part concerning the equation, though, can be computed easily, and I will do that.

Hello Bee,

I am afraid your question overflows my poor neurons. I explicitly avoided it, settling on the easier problem stated above…

My opinion on Koide’s formula ? I highly value these efforts, in a time where we seem to have lost all beacons. I find it quite stimulating, actually: it makes me feel as if we have not even started to scratch the surface of what is the stuff made of.


4. Alejandro Rivero - February 7, 2007

R. Foot gave a short statement for Koide’s: the vector built from the square roots of the masses of the charged leptons has an angle of 45 degrees with the vector (1,1,1). Or, equivalently, with the vector for the [square roots of] three equal masses.

I got puzzled because the original derivation of the formula was done in a preon/subquark model. In fact CarlB likes preons, so he has kept looking to the formula for more time. I raised my own theory of supersymmetry between composite and elementary particles to bypass the preons: if leptons where not composite but susy to composite, the original technique of Koide could be applied for the mass of the composites, and then susy invoked to get it into the leptons. Now, the current phenomenological approach is to forget about any intrinsic mechanism and to try to see what symmetries can generate the formula; some progress is done in the context of Ma A4 family symmetry. And such symmetries could also have some consequences in the quantity of Higgses.

About probabilities of formulae, I.J.Good tried to study such subject, a mix of algorithmic complexity theory and physics, and some others have followed suit, but the results have never been very interesting.

5. Carl Brannen - February 8, 2007

Since most of the sinners have already posted, I should say something.

When the Koide formula is rewritten as that 3×3 matrix, one finds another odd number, and that is \delta = 0.22222204717(48). The formula for the masses of the charged leptons is then

m_n = \mu_n (1 + \sqrt(2)\cos(\delta +2n\pi/3))

where m_n is the mass of the nth charged lepton (electron, muon and tau) and where \mu_n is an appropriate scaling factor. See this for the value of m_n and an extension to the neutrinos.

The thing is that 0.22222204718 is an “interesting” number, and that takes two degrees of freedom out of the charged lepton masses. It becomes very difficult to get that many digits of coincidence, the six of Koide’s formula plus another six here.

6. Arun - February 8, 2007

I expect the probability of a Koide formula for A,B,C will be rather small. Suppose it turns out to be so. Then what?

7. Thomas Larsson - February 8, 2007

Some other coincidences. The Lubos number 6 pi^5 = 1836.118.. is close to m_p/m_e (it was actually first discovered, and rejected, in the early 50s), and the number 4 pi^3 + pi^2 + pi = 137.0363.. is close to 1/alpha. I saw that on the arxiv, together with some motivation which linked the three terms to SU(3), SU(2) and U(1), respectively.

8. riqie arneberg - February 8, 2007

Stick to physics, leave numerology to the whackos who also study astrology and crop circles.

9. agm - February 8, 2007

From a physical perspective, or from the perspective of approximating zero in some computer code running an analysis?

10. Alejandro Rivero - February 8, 2007

Well, most of this is already described in physicsforums and it is worthwhile to repeat it only because the audience of this blog does not coincide exactly with the audience of physicsforums (seems to be a superset). Lubos number is actually published in Phys. Rev. 82, 554 (1951), being the shorter PhysRev article I am aware of.

The question about the probability of simple formulae was addressed recently by David H. Bailey and H. R. P. Ferguson, for instance in Mathematics of Computation, vol. 53, no. 188 (Oct 1989)

They also isolate Lubos number as the most singular case in an algorithmic scan, and point to a work of IJ Good in 1957. The article physics/0405094, which gives the right historical references, tries to justify this number in terms of extra dimensional volumes. The number surfaced again in a series of correspondence in Nature during 1983-85, motivated by Ganzfeld: Nature 304, 11 (07 Jul 1983) Nature 305, 672 Nature 306, 530 NATURE 308 (5962): 776 and Nature 313, 524.

11. island - February 8, 2007

Stick to physics, leave numerology to the whackos who also study astrology and crop circles.

No offense, but I live for the day when people like this have to eat their arrogant certainty… along with their meaningless UNcertainty… 😉

It’s a number provided by nature and we should expect that a theory will some day provide a reason for it.
-Paul Dirac

12. Tony Smith - February 8, 2007

Tommaso asks “… what really is the probability … a simple formula can describe … numbers that the Standard Model does not explain occurring …”.

In a letter to Physics Today (November 1971 page 9), in the context of Wyler’s calculation of the fine structure constant alpha, Asher Peres wrote (I have corrected a typo where Peres wrote 3 for 5 in one equation):

“… The question simply is how closely we can approximate alpha^(-1/4) by playing with integral powers of 2, 3, 5, and pi. In other words, we have to find integers x, y, z and t such that

( 1 – delta ) alpha^(-1)
[is less than]
( 2^x 3^y 5^z pi^t )^(1/4)
[is less than]
(1 + delta ) alpha^(-1)

where delta = 1.5 x 10^(-6) [… for accuracy +/- 1.5 ppm …].

This can be written as

-4 (logalpha + delta)
[is less than]
x log2 + y log3 + z log5 + t logpi
[is less than]
-4 (logalpha – delta)

This formula has a very simple geometrical significance: The integers x, y, z, t, form a unit lattice in a four-dimensional space.
The expression x log2 + y log3 + z log5 + t logpi represents a three-dimensional surface in that space, and the distance between two limiting surfaces is

8 delta {(log2)^2 + (log3)^2 + (log5)^2 + (logpi)^2}^(-1/2) = 5.4 x 10^(-6)

by elementary geometry. Therefore, we expect to find, on the average, one lattice point inside the slab, within any three-dimensional area of size 185,000. This is the volume of a sphere of radius 35.
It follows that one could be suprised if he finds a solution for xyzt whose distance from the origin (or any given point) is much smaller than 35.
In Wyler’s formula however, the distance is
(19^2 + 7^2 + 1^2 + 11^2)^(1/2) = 23
a number quite comparable to the radius of the sphere expected to contain one lattice point. Thus, we cannot dismiss the possibility that Wyler’s result is a mere numerical coincidence …”.

The Physics Today editor commented immdiately following Peres’s letter:

“Some theorists have told us they feel the above findings weaken the interest in Wyler’s calculations. Wyler himself feels that the difference is that his formula is derived from a theoretical formalism which is related to the physical world – the conformal group O(4.2) which is the natural invariance group of Maxwell equations.”

It might be interesting to do a similar analysis for the Koide equation.

Tony Smith

13. Carl Brannen - February 9, 2007

With square root lepton mass treated as eigenvalues of a 3×3 matrix, the Koide relation relates the sum of the eigenvalues of that matrix with the sum of the eigenvalues of the square of that matrix. I.e. it’s a relation between the square of the trace of a matrix and the trace of the square of that matrix. That’s pretty simple and it makes it difficult to parameterize as random chance.

A mass formula has to have the right units. One might try:

(a+b+c)^n = (p/q)(a^n+b^n+c^n)

for random a, b, c, and positive integers p,q,n. That one gets a hit good to 5 digit accuracy with n,q = 2, p = 3 is a bit surprising.

Circulant matrices have been used before in modeling lepton generations. This has to do with the mixing angles, not the masses, so having the masses show up as eigenvalues of a simple circulant matrix is interesting. For example, see:


Harrison, me, and Koide spoke about these funny coincidences at the DPF06 meeting in Hawaii. (Let’s see if that link works, where is the “preview???”)


PS: A collection of fine structure calculations, including Hans’:


14. dorigo - February 9, 2007

Hi all,

I am happy to have posted on this subject, because I now have a lot of material to study. Thanks to all who contributed, particularly Carl, Tony, and Alejandro.

I think I want to try and give my own answer to Arun’s question: “I expect the probability of a Koide formula for A,B,C will be rather small. Suppose it turns out to be so. Then what?

I believe -as some of those contributing to the discussion above- that any number that is not in some way inherited from the geometrical structure of space-time has no place in a “final” theory. I can easily digest pi factors, I can accept a h/bar, maybe even an alpha_em. But I can’t take tens of different constants without an explanation.

To me, finding a coincidence like Koide’s has the potential value of a E=hv hypothesis to fit a blackbody spectrum. And those who fail to appreciate this potential value are erring as Planck himself, who disregarded his own invention and turned to finding more “natural” explanation for Lummer and Pringsheim’s experimental data.

Now, Arun’s question is how much does the numerical coincidences have to scream at you before you take them seriously. I think physicists have a very good, agreed-upon convention for that: a 5-sigma effect is to be taken seriously. I do believe it is some sort of a threshold that usually random results or fluctuations or chance do not cross.

So, the problem is, as has been noted above, how to correctly estimate that probability, taking correctly into account the peculiarities of the observation.

Not an easy task at all. But not meaningless, either… Numerology is not necessarily a bad thing. It might mean we are shooting in the dark. No wonder, the last 30 years have produced little hope to go beyond the Standard Model in a rational, economical way…


15. Markk - February 9, 2007

Not quite understanding the actual physics (if there is any here), but just looking at the formula you gave, to me that is just playing with binomial coefficients, mixing the A, B, C values. For example, for N=2, the key functional value would be
2*(AB^1/2 +AC^1/2 + BC^1/2) and so on for N=3, 4 etc. What kind of algebraic structure that is a generalization of, I don’t know. If C >> B >> A then it limits down to a couple terms at most.

16. Arun - February 10, 2007

Sorry, I am too often telegraphic and thus end up having asked in effect the wrong question.

If the Koide formula is improbable for A,B,C, then I agree wholeheartedly with anyone who says this is indicative of some hidden structure or pattern. The problem is that the pattern doesn’t seem to hold any clue as to its origin. E.g., is it a meaningful question to ask if this mass relationship is a fixed point of the renormalization group?

17. dorigo - February 10, 2007

Hi Arun,
to me, the fact that the formula can be related to the eigenvectors of a very particular 3×3 matrix is *very* indicative of some underlying structure. But I am biased by years of dogged studies of particle physics…. We need seers here, as Smolin advocates.

Anyways, I sort of understand your question but I certainly am not able to answer it, so I turn it to Carl or Alejandro…


18. Arun - February 10, 2007
19. Arun - February 12, 2007

A pity that quark masses are so uncertain.

For the U,C,T generation, the “Koide constant” ( (square of the sum of the squareroots of the masses)/ sum of the masses) ) ranges from 1.166 to 1.185. The nearby small integer ratio is
7/6 = 1.1666…

For the D,S,B generation, the “Koide constant” ranges from 1.311 to 1.424. The nearby small integer ratios are:
4/3 = 1.333…
7/5 = 1.4

I slightly incline towards the idea that it might be difficult not to fall on a Koide curve :). Then of course, I remember the precision of the lepton numbers.

20. dorigo - February 12, 2007

Hi Arun,

I agree, there is a (numerable) infinity of curves in the graph you posted in your site. That is why, IMO, it becomes crucial to do the exercise I propose, that is see how much the proposed relation holds upon increasing the precision on the tau mass… Unfortunately, I do not see much happening in the next few years. Not at the level at which we would stop calling the Koide formula “a coincidence” and start considering it a fundamental law.


21. Carl Brannen - March 9, 2007


In the last few days I started trying to figure out how to modify the Koide formula for use on the quarks. I think that the quark masses as given in the PDG are highly dependent on model and are not accurate.

But I’ve got some beautiful results in the mesons, baryons and resonances. It turns out that the neutron and proton masses are three times the scaling constant of the eigenvector form of the Koide relation, and the difference between the neutron and proton masses are also closely related to these constants. I suspect that you will find these results very suggestive. You can see these things in LaTex here which is a useful place to discuss them.

22. dorigo - March 9, 2007

Wow Carl, thank you for letting me know! I am jumping at the link now!


23. Carl Brannen - March 13, 2007

Okay, my write up was horrible with a coupld serious errors. The actual calculation was not I wrote (which was a simplification, the idea of the calculation), but instead a much more accurate method. What can I say, I was very excited and couldn’t wait. Alejandro Rivero complained, and I redid it. He now agrees with the result.

Since then, things are moving quickly. I’ve got tree diagrams for the fine structure constant now, and some very suggestive coincidences in the coupling constants of the weak and gravitational forces. The details are in the above link.

The short form description of the method is that it replaces the complex numbers of QM with collections of Feynman diagrams for the particles that are bound together. It allows one to solve the “recursion problem”, that is, if QM allows a complex particle like a quark to be treated as a point particle, there needs to be a formalism that covers the transition of free states to bound states in such a way that the properties of the composite state can be predicted from the properties of the free states that make it up.

it allows one to produce free states from a collection of bound states, and then to bind that composite state with others.

24. dorigo - March 13, 2007

Hi Carl,

I think I will make a separate post of this later today


25. Carl Brannen - March 22, 2007


Things are making wonderful progress in both theoretical and phenomenological areas and they are starting to converge. I now have a very good idea where the mysterious number in the eigenmatrix form of the Koide relation, 0.22222204717(48), arises from. It has to do with the fact that in a massless wave equation one can replace \psi with e^{i\psi}. This is done by the Bohmian mechanics and they have the right idea, it turns a quantum thing that is normalized to probability into a classical thing that is normalized to energy.

The basic idea behind the new perturbation technique is to solve exactly the color binding force problem, and then treat all the other coupling constants as perturbations. But to do this, you have to solve the color binding force which requies nonlinear theory.

To give a hint on how this is done, here is a derivation of the spectrum of weak hypercharge:

Consider permuations on three elements, R, G, and B. Label the identity 0, label the 3-cycles 1 and 2, and label the odd permutations that preserve one of the three elements R, G, and B. The group law under composition is:


Next, one replaces the six permutations with complex numbers and rewrites them as six quadratic equations:

0 = 00 + 12 + 21 + RR + GG + BB,
1 = 01 + 10 + 22 + RG + GB + BR,
2 = 02 + 11 + 20 + RB + GR + BG
R = 0R + 1G + 2B + R0 + G2 + B1
G = 0G + 1B + 2R + R1 + G0 + B2
B = 0B + 1R + 2G + R2 + G1 + B0

For examle, the equation for R has a factor G0 and a factor B2 because G0 = B2 = R. (Please pardon the reuse of the integers as variables, they should be subscripts.)

Solve these six quadratic equations in six variables. You will find that the spectrum for the 0 operator is the same as the rational numbers that appear in the weak hypercharge spectrum. That is, one has 0, 1/6, 1/3, 1/2, 2/3, and 1. Note that the +/- signs are arbitrary, as are the assignment of a given state as a “particle” as opposed to an “antiparticle”. The tradition is to define the things that make up normal matter as “particle”, but this is arbitrary. Furthermore, it breaks down for bound states like pions. With the elementary fermions redefined as composites, there is no consistent way to pick out which are particles and which are antiparticles.

An important subgroup of the permutation group on three elements is the even subgroup, {0,1,2}. Of the six solutions, four are nonzero only in the even subgroup and correspond to weak isospin singlets. The other two have odd components and correspond to the weak hypercharge numbers for the doublets.

Basically, the idea is that the elementary fermions are composites of three preons that can swap places, where “place” is defined by color. The exponential function converts a classical wave function into its quantum analog in a manner which reminds me of how Bohmian mechanics works but is closer to the density function theory of the condensed matter theorists.


26. Composite models in the making « A Quantum Diaries Survivor - March 22, 2007

[…] dorigo in mathematics, Blogroll, physics, science, internet. trackback Carl Brannen posted today a long comment in a recent post I wrote about the Koide mass formula. He is working at composite models of […]

27. Bruce Schechter - June 10, 2008

This is probably irrelevant, but Koide’s formula reminded me of Descarte’s famous theorem relating the radius of curvatures of four mutually tangent circles. If one of these circles is replaced by a straight line, then Descarte’s theorem states that the sum of the radius of curvatures of the three remaining circles squared equals twice the sum of the squares of the radius of curvatures. If the factor of 2 were somehow 2/3 you’d have Koide’s formula. As I said, this is probably totally useless but maybe it will spark some thought in your far more erudite brains…

28. dorigo - June 10, 2008

Hi Bruce,

interesting, I was unaware of the theorem. Well, not useless; a coincidence between math and physics is usually to be taken seriously. Geometrical analogues such as the one you mention also are often helpful in furthering our understanding. However, my brain is not so erudite – maybe that of some other contributors to this thread may be stimulated though.


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