The Efimov constant is only approximately ≅ 22.7. It would be a little surprising to have a physical constant end in exactly 0.7, since it would imply that there's something special about base 10 and bases that are a multiple of 10.
Section 3 of this paper talks about the actual transcendental equation that provides the constant.
Rational numbers tend to have easily recognizable patterns in their fractional digits in all bases (at least as long the numbers aren't too big), so it wouldn't imply there is something special about base 10. In fact, you can always choose the base as q for a rational number p/q and get rather simple fractional digits.
It's not repeating digits I'm talking about, though, it's termination. If it were exactly 22.7, that's 227/10. The question would be "Why is the denominator 10?"
It's just that it's a non-sequitur to say "oh, that looks rather simple and it has a 10 in it, there must be something special about our number system" (which is what you say in your first comment). I implicitly subsumed termination under 'simple pattern' since repeating sequences can be described in a finite way too. The only surprising property of 22.7 in this context is that it's rational.
Because you used the denominator "10." If the constant is precisely 22.7 then it can be represented in any other base with just as much precision, you'd just use a different format for the number before and after the decimal point.
227/10 will not terminate in any numerical base which isn't a multiple of ten. Combining this with the idea that no numerical base should be privileged over another, we can rephrase this to say that it's surprising for a physical constant to be a rational (but not integral) number.
It's not surprising for a physical constant to be an integer because we really believe that integers have special qualities. It's not surprising for a physical constant to be irrational because that's just the way numbers are. If it's rational but not integral, you want to justify the denominator somehow.
There are many constants out there. Sooner or later it seems natural that you'll find one that happens to dive nicely into base 10. Why does that imply anything special about base 10. It would simply be a coincidence.
It's a staggeringly unlikely coincidence; the odds of getting a rational number when selecting uniformly from any bounded interval are 0% -- not approximately zero, but exactly zero. In contrast, the odds of getting an irrational number are 100%. It's not possible to select uniformly over an unbounded interval, but rational numbers don't get any easier to find.
(Technical note: though the figures of 0% and 100% are correct and exact, they actually do not mean that selecting a rational number is impossible. For a similar idea, note that if I select a number uniformly from the interval [0,1], the prior odds of choosing whatever number I did choose were also zero, but by the construction of the problem I had to end up with something.)
This means that if you find a rational number, you should immediately suspect that it's like that for a good reason. You're much more likely to win the jackpot of every lottery you ever enter, even if you buy a ticket every day, than to encounter a rational number by coincidence.
Or using a different analogy, the world population is roughly 7 billion. It would be extremely unlikely, so unlikely that we can essentially rule it out beforehand on theoretical grounds, for any of them to be a rational number of centimeters tall, or to mass a rational number of grams. That sort of thing can't happen.
True, but due to precision limits in measurements you're never actually dealing with this scenario. You always have some number of significant digits which implies you're always going to see a rational number for your measurements.
Further there are strict levels of granularity on a number of measurements, for example you run against the plancks constant when measuring a number of things. So this argument for looking at all of [0,N] falls apart if we aren't dealing with measurements that are continuous.
Using probability theory for the definition of interesting numbers is a daring thing to do. There is a rather surprising mathematical result (I unfortunately don't remember the context) that contained the number 27/32 (or something like that). Does that mean there is something extraordinary spacial about base 32? Difficult to say.
Well, sort of by definition a mathematical result will tell you why you got the result you got.
I haven't argued that a physical constant which terminates in base 10 implies that 10 is more special than other bases. I've argued to support the idea that a physical constant terminating in base 10 is surprising. It is, superlatively so. In the words of the OP, the question you need to answer, if you think you have a constant exactly equal to 227/10, is "why is the denominator 10?". It could be that 10 is a holy number, or it could be that the 10 represents some other physical quantity that you weren't paying attention to before. But assuming it's a coincidence is a strange thing to do, and in fact it is not natural to think that if you just looked at enough numbers of course you'd eventually see a rational one. Unless you're choosing numbers by a process that biases you heavily towards getting rationals, you are not likely to see a rational until long after you've chosen an infinite number of irrationals.
The whole genesis of string theory was, first, observing that a sequence generated by a badly-understood phenomenon was the same sequence given by a mathematical result about strings; and then, second, concluding "aha! There must be strings involved in this!" Plenty of people argue that string theory isn't going anywhere, but nobody says the conclusion "there must be some strings around here somewhere" was unjustified. The "it's a coincidence" theory persuades no one.
I vaguely recall that if a circular disc with a circumference of pi (3.14 ...) is held at an angle of 22.7 degrees relative to a flat surface, it projects an ellipse that has a circumference of 3.000 Coincidence? Eh, probably. Might've also been 26.8 degrees.
One common association of those digits with PI is the handy approximation of 22/7 accurate to three digits, off by just over .0004.
I stress that this is an approximation only! I had to explain about PI being irrational rather than exactly equal to the shortcut approximation to a couple co-workers, explaining some poor behaviors in their rotation function.
“What you see is three groups, in three different countries, reporting these multiple Efimov states all within about one month,” said Chin, who led one of the groups. “It’s totally amazing.”
One time, Feynman was giving a lecture to some middle-school kids. He used the date of some battle as an example, but he got it a bit wrong and the kids immediately corrected him. He laughed and said "Hey! 3 significant digits is pretty good for a theoretical physicist!"
A teacher asks a group of students to prove that all odd numbers above 1 are prime numbers. After a short while the students begin arguing about the best proof.
Mathematician: "We should use proof by induction, I can show 3 is a prime number by listing all of it's factors, and then from there I can show that 2n + 1 is prime ..."
Scientist: "What? that doesn't make any sense. What we need is experimental data. Let's collect enough points, 3, 5, 7, 9, 11, 13. Six points should do, now look, the general trend is that all of them are prime. We have one anomaly, but it's probably just caused by a random fluctuation in..."
Engineer: "Nah, all this theory is pointless, we only need to know if they are prime in the real world. Let's just take a few and see if they work for our purposes. 3 - prime, 5 - prime, 7 - prime, 9 - nearly prime, 11 - prime, surely that's good enough."
Computer scientist: "While you lot have been wasting your time arguing I have been writing a program that will do all the work for us"
All of the students gather around the CS's computer and watch as he presses go.
Computer:
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
3 is prime
This whole thread is messed up because it
has never even mentioned the whole other
half of the subject as in the famous
monograph, A Short Table of Even Primes.
I wonder if this principle can also be applied to nucleons. Protons and neutrons are each made up of 3 quarks. And there are some troubles in pinning down the radius of the proton. If Efimov scales apply to nucleons, this would imply that there are multiple scales in which quarks can arrange depending on the outside potential. This could explain the differences of measured proton radius depending on the used experimental setup. But I'm just wildly speculating here.
It applies (in very rough approximation) to groups of nucleons, which is where it was first discovered. People observed that the nucleon/nucleon scattering length was surprisingly long for what they considered natural nuclear scales (or equivalently, that the deuteron was very weakly bound). The circumstance where the Efimov effect is mathematically exact is when the scattering length is infinite. Fermions in this circumstance are described as unitary.
Sadly, the Efimov effect has nothing to do with the proton size puzzle.
At any rate, these depths of physics are quite complex and not very intuitive, and this kind of discovery shows it is very easy to overlook things. A giant discovery could be looming in the horizon...
More than 40 years after a Soviet nuclear physicist proposed an outlandish theory that trios of particles can arrange themselves in an infinite nesting-doll configuration, experimentalists have reported strong evidence that this bizarre state of matter is real.
I think it's quite awesome that Vitaly Efimov lived to see his theory being proved (or, at least, to see strong evidence in its favor).
It always makes me sad when I read about scientists who made great discoveries just to be laughed at until their days end, and then their theories are proved after they are dead.
A classic example is Alfred Wegener, who postulated continental drift just to be ridiculed., but of course there are many, many others.
Another, even more tragic story is that of Ignaz Semmelweis, who not only pioneered antiseptic procedures in hospitals, but, to my mind, proved their effectiveness beyond any reasonable doubt, was rejected, and died tragically.
I am pretty sure this is not related to local realism. I would naively imagine the situation as follows (not having read anything related in detail and not being a physicist). The forces between particles become very weak with growing separation but never become zero. Now imagine throwing two ping-pong balls into a pool making ripples on the surface similar to the force field or the probability of presence of the particles. There will be a interference pattern and you can probably find a valley where you can place a third ping-pong ball and it will sit there more or less stable. Now finally imagine every pair of ping-pong balls forming the valley for the third ping-pong ball. This may of course be incredible far off from what really happens - just making up images.
I like the ping pong balls over the surface of a pool, that's a neat analogy. However, I am also not a physicist, but my impression is that this is something closer to how a http://en.wikipedia.org/wiki/Lagrangian_point (in particular, the L4 or L5 lagrange point) works. In fact, I think the size difference thing they describe in the article is probably very similar to how gravity, and relative object positions work out based on their mass ratios. Kind of an equilateral force balance, with a self stable state that things will naturally gravitate towards if the background energy level is low enough. Course, then that makes me wonder if there's a 4-body version with a stable tetrahedron formation?
Section 3 of this paper talks about the actual transcendental equation that provides the constant.
http://rsta.royalsocietypublishing.org/content/369/1946/2679...