# Schrödinger's Ratio

October 29, 2010 12:52 PM Subscribe

If you look around, you'll see that the ratio of 1.618:1 appears in architecture, nature, and artistic works (such as music, previously). Studied by the Greeks, the Golden Ratio is pretty much everywhere and is common accepted as aesthetically pleasing, and now it has been found to exist down into the nanoscale level, as a byproduct of investigating the Heisenberg Uncertainty Principle. We may not be able to nail down both position and speed, but it appears the macro ratio is an echo of the micro one.

The Epoch Times article makes a mistake with confusing atoms vs molecules, which makes having the full paper - Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry - behind a paywall at Science.org all the more frustrating.

The Epoch Times article makes a mistake with confusing atoms vs molecules, which makes having the full paper - Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry - behind a paywall at Science.org all the more frustrating.

As interesting as it is to learn that this ratio describes the relationship of two numbers in a large set of frequencies measured by some scientists, you may also wish to learn that The Golden Ratio appeared in my toasted cheese this morning, next to an image of the virgin mary.

posted by honest knave at 1:06 PM on October 29, 2010 [4 favorites]

posted by honest knave at 1:06 PM on October 29, 2010 [4 favorites]

Haven't we debunked the claims about the inherently aesthetic beauty of the golden ratio already? I thought it turned out that the original studies had biased the results by always placing the golden ratio rectangle in the middle of the choices and that with random placement, a different ratio turned out to be chosen more often. Sadly, I can't find where I read that anymore. Anyone know?

posted by ErWenn at 1:06 PM on October 29, 2010 [2 favorites]

posted by ErWenn at 1:06 PM on October 29, 2010 [2 favorites]

I believe this math overflow link is about this.

posted by empath at 1:08 PM on October 29, 2010 [3 favorites]

posted by empath at 1:08 PM on October 29, 2010 [3 favorites]

The full paper seems to have been published in January 2010.

Here are summaries from PhysOrg and Science News.

I lack the theoretical physics background to evaluate The Epoch Times article on its own merits, but there seems to be an informed discussion thread on MathOverflow.

posted by Prince_of_Cups at 1:09 PM on October 29, 2010 [2 favorites]

Here are summaries from PhysOrg and Science News.

I lack the theoretical physics background to evaluate The Epoch Times article on its own merits, but there seems to be an informed discussion thread on MathOverflow.

posted by Prince_of_Cups at 1:09 PM on October 29, 2010 [2 favorites]

Thank you, honest knave, but could you at least

posted by ErWenn at 1:09 PM on October 29, 2010

*pretend*that you don't have ESP?posted by ErWenn at 1:09 PM on October 29, 2010

Goes to show, you can learn something new every day, yup.

posted by VicNebulous at 1:11 PM on October 29, 2010

posted by VicNebulous at 1:11 PM on October 29, 2010

Here is the relevant graph from the Science publication.

posted by exogenous at 1:11 PM on October 29, 2010

posted by exogenous at 1:11 PM on October 29, 2010

Denn was innen, das ist außen.

So ergreifet, ohne Säumnis,

Heilig öffentlich Geheimnis.

(There is naught within, naught without:

What is within, is also without.

Do try to grasp, without delay

This open secret, deep, devout.)

--Goethe

posted by No Robots at 1:14 PM on October 29, 2010 [5 favorites]

Nope, nothing to do with supersymmetry. Supersymmetry is a kind of symmetry usually in particle physics, between fermions and bosons. (So just like we have a positron and an electron, supersymmetry imagines that there's an electron and a "selectron"-- a boson instead of a fermion.)

I do agree, leahwrenn, that the actual abstract sounds much more interesting than the annoying science "news" article. Gah, I get so frustrated when science news is so obfuscating that I can't figure out wtf they're talking about.

They are apparently talking about a system exhibiting this kind of behavior.

posted by nat at 1:21 PM on October 29, 2010

I do agree, leahwrenn, that the actual abstract sounds much more interesting than the annoying science "news" article. Gah, I get so frustrated when science news is so obfuscating that I can't figure out wtf they're talking about.

They are apparently talking about a system exhibiting this kind of behavior.

posted by nat at 1:21 PM on October 29, 2010

erwenn: you're welcome! Also:

empath: thanks for that MathOverflow link. It's fantastic in general, and the level of metatalk in the following comment is especially wonderful:

posted by honest knave at 1:22 PM on October 29, 2010

empath: thanks for that MathOverflow link. It's fantastic in general, and the level of metatalk in the following comment is especially wonderful:

*It seems to me that "being a Fields medalist" doesn't carry any information one way or the other about the quality of the question, which should be the primary reason that people choose their votes.*posted by honest knave at 1:22 PM on October 29, 2010

Is there an uncanny valley for the golden ratio?

posted by doublehappy at 1:28 PM on October 29, 2010

posted by doublehappy at 1:28 PM on October 29, 2010

doublehappy: I don't think so. 3-by-5 cards don't look ugly. (legal-sized (8.5-by-14) paper does, to me, but I figure that's just because it looks like normal paper that somehow grew extra-long.)

posted by madcaptenor at 1:47 PM on October 29, 2010

posted by madcaptenor at 1:47 PM on October 29, 2010

The

posted by overeducated_alligator at 2:00 PM on October 29, 2010

*Epoch Times*has close ties to Falun Dafa, a group with some shall-we-say "interesting" ideas about cosmology/God/the universe/space-Buddhas. (Though I would call them mostly harmless and completely undeserving of their intense suppression by the PRC government).posted by overeducated_alligator at 2:00 PM on October 29, 2010

The Richard Borcherds who asked the question on MathOverflow does, in fact, appear to be

posted by Frobenius Twist at 2:25 PM on October 29, 2010

*the*Richard Borcherds, well-known mathematician. I've been curious about this sort of thing myself, because the group E8 occurs in my research, and it is mind-boggling to me that it could ever have anything at all to do with the real world; E8 is unwieldy and strange!posted by Frobenius Twist at 2:25 PM on October 29, 2010

The abstract reads like it was spit out by a Markov generator.

posted by five fresh fish at 2:27 PM on October 29, 2010

posted by five fresh fish at 2:27 PM on October 29, 2010

*I've been curious about this sort of thing myself, because the group E8 occurs in my research, and it is mind-boggling to me that it could ever have anything at all to do with the real world; E8 is unwieldy and strange!*

Can you talk more about that?

posted by empath at 2:28 PM on October 29, 2010

E8 is a particular affine algebraic group, which is a fancy way of saying that it's a matrix group (i.e., a subgroup of a group of matrices) and it also has a nice topological structure. It's a special kind of algebraic group called semisimple; the semisimple groups are in some sense the most interesting algebraic groups.

The semisimple algebraic groups have been completely classified, a feat which was a big achievement in 20th century mathematics. In this classification, there are some "exceptional" groups, ones that have a different sort of structure than most of the others on the list. The structure of E8 is convoluted (to me, at least) for the following reason. To get most of the semisimple affine algebraic groups, you impose a simple condition. To get the groups of type A, for example, you just take matrices of determinant 1. Getting E8, though, is quite complicated (check out the Wikipedia article). Its existence wasn't even known for a while! Its structure is complicated in other ways that I won't bore you with that have to do with the fine structure of semisimple groups (structures called root systems and Weyl groups, for example).

These groups can be seen as symmetry groups, so the fact that affine algebraic groups occur in physics isn't so surprising to me. However, there seems to be a lot of heat around E8 in particular, which is like picking a strange foreign film as the best bet to win the Academy awards. (That is a tortured metaphor!). That's the part I find confusing. (My own research is related to the geometric and representation-theoretic structure of these groups).

Anyhow, I hope that made at least a bit of sense . . . .

posted by Frobenius Twist at 2:59 PM on October 29, 2010 [2 favorites]

The semisimple algebraic groups have been completely classified, a feat which was a big achievement in 20th century mathematics. In this classification, there are some "exceptional" groups, ones that have a different sort of structure than most of the others on the list. The structure of E8 is convoluted (to me, at least) for the following reason. To get most of the semisimple affine algebraic groups, you impose a simple condition. To get the groups of type A, for example, you just take matrices of determinant 1. Getting E8, though, is quite complicated (check out the Wikipedia article). Its existence wasn't even known for a while! Its structure is complicated in other ways that I won't bore you with that have to do with the fine structure of semisimple groups (structures called root systems and Weyl groups, for example).

These groups can be seen as symmetry groups, so the fact that affine algebraic groups occur in physics isn't so surprising to me. However, there seems to be a lot of heat around E8 in particular, which is like picking a strange foreign film as the best bet to win the Academy awards. (That is a tortured metaphor!). That's the part I find confusing. (My own research is related to the geometric and representation-theoretic structure of these groups).

Anyhow, I hope that made at least a bit of sense . . . .

posted by Frobenius Twist at 2:59 PM on October 29, 2010 [2 favorites]

It didn't but that's okay. I have never understood a single thing i've read about it.

posted by empath at 3:02 PM on October 29, 2010

posted by empath at 3:02 PM on October 29, 2010

Perhaps a more direct way of describing it is the following: some symmetry groups are easy to understand, like the symmetries of a square (you can rotate 90 degrees in either direction or you can flip the square over, and you still have a square). E8, on the other hand, is hilariously complex as a symmetry group (take some complicated 248-dimensional object and then consider its symmetries). I'll be completely honest (and this may not be entirely fair -- I freely admit I know very little physics) but I sometimes wonder if physicists have picked E8 partly

posted by Frobenius Twist at 3:26 PM on October 29, 2010

*because*it is so fancy and complex.posted by Frobenius Twist at 3:26 PM on October 29, 2010

*I sometimes wonder if physicists have picked E8 partly because it is so fancy and complex.*

In this particular case, I think not. The theoretical model for the physics being discussed here is the 2D Ising model in a magnetic field. It is a pretty simple and natural setup, and has been studied for a long time (invented by Lenz in 1920, if Wikipedia is to be trusted here.) The discovery that this model has something to do with E8 was made much later -- it was known by 1986, but probably not much earlier.

posted by em at 3:56 PM on October 29, 2010 [1 favorite]

*The Epoch Times? Seriously?*

Because nobody else at the time had a link to the paper in question. There are better sources, yes, but I did not find them when I put the post together; if I had, I would have put them in/used them instead.

posted by Old'n'Busted at 4:21 PM on October 29, 2010

Because its continued fraction expansion is all 1's, the golden ratio is 'the most irrational number', which its fans are determined to live up to, I guess.

Also can be written as an infinite nested square root consisting of all 1's.

posted by jamjam at 11:31 PM on October 29, 2010 [1 favorite]

Also can be written as an infinite nested square root consisting of all 1's.

posted by jamjam at 11:31 PM on October 29, 2010 [1 favorite]

*The Epoch Times? Seriously?*

I gotta say: I tend to get less angry with the writing in E.T. than with other papers.

posted by not_that_epiphanius at 12:31 PM on October 30, 2010

I haven't trusted physicists with math ever since I learned about normalization with delta functions.

posted by Twang at 5:46 PM on October 30, 2010

posted by Twang at 5:46 PM on October 30, 2010

*If you look around, you'll see that the ratio of 1.618:1 appears in architecture,*

**nature**, and artistic works (such as music, previously).Well, actually, that's mostly confirmation bias. A ratio of about 1.4:1 to 1.8:1 (or ratios that freely vary in that approximate range) are readily labeled as "following the Golden Rule" by the sort of oversimplifying "science" writers that want easy answers, but most of the usual examples - nautilus shells, sunflower seed packing, etc. - are only approximately like the Golden Rule.

posted by IAmBroom at 7:15 PM on October 30, 2010

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Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi–one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.

posted by leahwrenn at 1:00 PM on October 29, 2010