The Rule of Three Proved By Physicists

An anonymous reader writes "In 1970, Russian physicist Vitaly Efimov developed mathematical proof (PDF) that any three-particle substance, referred to as a trimer, will scale up or down in size by a factor of 22.7 and that if the particles are not all of the same type, 'the scaling factor of 22.7 decreases according to the particles' relative masses.' In 2006, physicists in Austria proved that Efimov's trimers can be created in laser-cooled environments. And now, in 2014, physicists in Austria, Germany, and the U.S. have physical proof that Efimov's trimers do indeed scale by a factor of 22.7 if they are comprised of the same particles or a lower ratio relative to their particles' masses if they are comprised of a mixture of different particles (abstract 1, abstract 2, abstract 3). 22.7 — a.k.a., the rule of three — now appears to be as significant as pi."

Comments

Sometimes it's annoying when there is

[fisted]

left out

infinite amounts? well yeah, if 3 is about infini

[gl4ss]

"Each 22.7-times larger Efimov state is also 22.7-squared times weaker, requiring the optical trap to be cooled even further to allow the new state to form. Grimmâ(TM)s group perfected its techniques and detected the state at the very edge of experimental limits."

"Meanwhile, the two other groups managed to observe three consecutive Efimov states by taking advantage of a footnote in the theory: When a trimer is built from a mixture of different particles rather than an identical set, the scaling factor of 22.7 decreases according to the particlesâ(TM) relative masses. In other words, nesting dolls made of atomic mixtures become closer in size, enabling more of them to be observed within the experimental window."

cool sure, but ""The law is a geometric progression of evermore-enormous trios of particles, spanning in a theoretically infinite sequence from the quantum scale to (if the particles were cold enough) the size of the universe and beyond. âoeAlthough we didnâ(TM)t see an infinite number of them, thereâ(TM)s pretty strong evidence when you see three in a row,â Chin said.""

yeah sure, universe size yeah why not if we have 3 smallest and every sequence is weaker...

Re:Even more amazing

[Sique]

The volume of a pizza of radius z and height a is pi*z*z*a.

Re:Pi?

[mrbester]

Not much of a leap on my keyboard, / being next to . so an easy typo to make...

Re: Pi?

[orangesquid]

22.7 is 2*10^1+2*10^0+7*10^-1.
If you express the same quantity in a bar other than ten, the .7 is no longer going to be a 7 after the decimal point, 7*10^-1. Note that you multiply it by TEN to get the 7 in the denominator of 22/7. In another base, you'd be multiplying by something other than 10, so you wouldn't get 7.

Re:Pi?

[Anonymous Coward]

http://rsta.royalsocietypublishing.org/content/369/1946/2679.full

If you check section three, you'll be able to compute that the number in question is 22.6943825963666951931....

Clueless

[JimSadler]

Maybe I'm as dumb as a rock but this article doesn't make much sense to me. Maybe a bit more info about trimers would have helped.

Re:Clueless

[hansraj]

I found the summary confusing but the article made more sense.

The theory was that there exist configurations of three particles that is stable in a strange sort of way. The strange part is that if a certain configuration was stable then putting the particles in the same configuration but the distances blown up by a certain factor (22.7 if the three particles were the same) gives another stable configuration. So you can keep blowing up the distances in multiples of 22.7 and would get an infinite sequence of stable configurations. These configurations are necessarily quantum and not classical since the distances involved would be much larger than the range of the forces between the particles. (Although even the initial distances are large too, if i understood correctly, you would agree that they _will_ get pretty large at some point).

Now some independent groups have shown the existence of such states with the required blowup. Since similar-particle setup required cooling things down to the limit of present day technology, only _one_ configuration was observed initially. Someone used a system of different particles resulting in a blowup factor less than 22.7 allowing them to observe _three_ of these configurations, essentially validating the theory.

Hope that made sense (IANAP).

Re:Clueless

[azav]

Soooooo, this might explain stability of atoms in certain structures and certain distances?

Or how certain subatomic particles make stable structures and become atoms?

Am I reading into this right?

Re:Clueless

[hansraj]

I don't think so.

Stability of ordinary matter is well explained by other more traditional theories (strong/weak forces for nucleus, electromagnetic for atoms and molecules, gravity for even larger structures). This theory described stable states that initially no one believed existed.

Morever, these configurations are stable but quite fragile.

Re:Clueless

[jovius]

What I gathered is that the configuration is like mirrors facing each other. The observer of the particle configuration is somewhere in the middle of the reflections, which go from an infinitesimal small scale to the infinitely large, in a sort of continuum. The distance of one mirrored reflection from another is successively 22.7 times more (or less) than the distance from the previous reflection, depending which direction is observed.

The number 22.7 is not necessarily a constant, and it depends on the properties of the particles. In the article it is said that the scaling factor decreases according to the particles’ relative masses, for example.

Re:Cue the flood of numerologists ...

[hawkinspeter]

Yes, and 22+7 is exactly equal to the number of days in a month (+ or - 2 days).

Re:Even more amazing

[LifesABeach]

Damit, Coffee, all, over, my, keyboard! :p

22.71415926535...?

[Ted Stoner]

Is it really 22.70000000 or is there some interesting splatter of digits beyond? I didn't see any more specific number in the article.

As someone not schooled in this discipline

[azav]

Why is this relevant?

What does it teach us that a normal person could understand?

Please explain to us why someone should care about this.

Re:No good

[azav]

Mexico has us beat.

We were #1. We were #1.

22.7 and pi

[iguana]

"22.7 — a.k.a., the rule of three — now appears to be as significant as pi."

22/7 approximates pi.

Iiiinteresting...

Re:22.7 and pi

[Gibgezr]

It is pretty interesting!

I always use 355/113 as my "super quickie fractional representation" of pi. It is accurate out to 6 decimal places, which makes it useful enough for most purposes. 22/7 only gets us to 2 decimal places, unfortunately.

Re:Even more amazing

[Gr8Apes]

Thanks for the funniest thing I'll probably read all day. :)

Re:Planescape

[Will.Woodhull]

And I was sure they were talking about The Mathemagician's Rule of Three:

• "Just the place for a Snark! I have said it twice:
• That alone should encourage the crew.
• Just the place for a Snark! I have said it thrice:
• What I tell you three times is true."

--The hunting of the snark, Lewis Carrol. Fit the First, second stanza.

50's monster movies vindicated!

[Thud457]

So a 200-foot praying mantis is still a possibility as long as it's scaled up some power of 22.7. Got it.

3.14159265358979323846264

[pezpunk]

yay! i'm going to memorize as many digits as i can of the Rule of Three!

Re:Pi?

[BronsCon]

To truncate with rounding, yes. Lazy truncation, however...

And 22/7...

[GPS Pilot]

is a good approximation for pi.

Re:And 22/7...

[ghmh]

Raised to the power of grayskull.

Can someone shed some light

[monkeyFuzz]

.. on why this is significant?

Re:infinite amounts? well yeah, if 3 is about infi

[K. S. Kyosuke]

I thought the proof was verified decades ago. Perhaps you wanted to say "evidence" instead?

Re:Even more amazing

[Richy_T]

This needs to be a t-shirt.

And by posting this, I guarantee that it already is.

Otherwise known as the Rule of 2 (plus one)

[0xdeadbeef]

There must always be a master, an apprentice, and the apprentice's secret apprentice.

The Exact Value...

[Sigfried]

is most likely transcendental. According to Efimov's original paper, the magic value "22.7" (we shall M) is given exactly by e^(pi / |s0|), where s0 is a pure imaginary solution (very close to i) of an equation (9) he derives earlier and is related to the three-body problem (there are an infinite number of real solutions s1,s2...). If you define s0 = i x where x is real, then (by converting from trig to hyperbolic trig) it can be shown that the number M is given by e^(pi / x), where x is the positive real solution to:

0 = (8 / sqrt(3)) sinh( pi * x / 6) - x * cosh( pi * x / 2)

If you copy/paste the right-hand side into the WolframAlpha website, you will see that the curve has exactly two non-zero solutions, approximately (+-) 1.00624. You can ask for more digits, and it will give you x=1.0062378251027814891..., which means to eight digits, M = 22.694382595... This equation above is a transcendental equation whose non-zero solutions are neither rational nor algebraic, and very likely M itself is also transcendental. Proving these sorts of things, however is very difficult. The best we can hope for is that the number can be expressed as an infinite expansion whose terms have a nice form which converses rapidly. A few more clicks on Wolfram Alpha and I'm sure someone will work it out.

Re:Even more amazing

[yo303]

Further, if the density of the pizza is p, the Earth's mass times the gravitational constant is u, and the pizza's distance to the center of the Earth is l, the force on that pizza is p*i*z*z*a*p*u / l*l.

Rule of Two is more important

[Gothmolly]

I'm more concerned about the Rule of Two. When Obama goes, who is there to take his place?

Smaller scales

[justthinkit]

Our knowledge tends to be high in the middle of the spectrum.

When we try to grasp the entire Universe, we end up looking at pixels of info. We know little and speculate much.

Similarly, at the smallest scales we have to resort to atom smashing to learn a bit and speculate a lot.

This theory suggests that life is LEGO-like after all. Once we know this underlying pattern, we can look for it, and we can use it to build better models of stuff very large and very small.

I would add that these results are consistent with my own theory.

Re:Even more amazing

[Richy_T]

Indeed, there appears to be something of a selection at least one of which appears to miss the joke.