Here is an example of the breaking of rotational symmetry into a discrete symmetry - you might have seen it in your own sink! These are photos of water flowing down vertically and being distorted into a horizontal direction. The first photo shows rotational symmetry. The following photo shows a distortion by which the water is spread out into 14 discrete beams.

These photos have been published in

*Hancock, M.J. and Bush, J.W.M., 2002. Fluid pipes, J. Fluid Mech., 466, 285-304.*

See also Fluid Polygons and Polyhedra.

Ooh, those are so cool! It seems sometimes the whole point of life is this 'dance' of broken symmetries. Sorry, I don't mean to be sound like a downer. Just a little frustrated...

ReplyDeletePhysics seeks fundamental models with maximal symmetries. It then inserts explicit, anomalous, and spontaneous symmetry breakings to match predictions vs. observations. The most elegant studies may miserably fail (e.g., quantized gravitation) because reality is fundamentally asymmetric.

ReplyDeleteOne can easily test for this, but why bother? 420+ years of pertinent observations have exactly nulled. What fool would examine the impertinent case? A $billion of theory saves us $100 in consummables for the clarifying experiment.

Hey Bee (and any other women), I've made a post specifically for women to come and commiserate about men or whatever else is bothering them. Feel free to come and rant.

ReplyDeletewow! the second picture reminds me of the diagram by which one draws the mechanism of string fragmentation... You know, when the string stretches, with an area proportional to the potential energy, and then breaks up, creating two additional quarks.

ReplyDeleteI wonder if the analogy goes further.

Cheers,

T.

That's pretty cool. But there's no reason that the (continuous)cylindrical symmetry is more fundamental than the discrete symmetry. Let's call your top figure as state "A" and bottom figure as "B". The thing with symmetry breaking is that, as I understand, if one increases the strength of the mass flux of the water flow, "A" becomes unstable and the unstable mode evolves into "B", which is stable. But a discussion of this sequence of "A evolves into B" (i.e., the process of "symmetry breaking") has implicitly assumed that "A" is the god-given initial condition of our universe. If we instead consider the situation when "B" is already the god-given I.C. of the universe, then "B" is always there (and is stable, therefore doesn't evolve further) and will always be there for time immortal. Then, the entire history of the universe is described by "B". One does not need to invoke "A" whatsoever to describe the universe. From that point of view, the concept of symmetry breaking is not necessarily fundamental.

ReplyDelete(Just post this comment to see if my account works. Ha ha.)

hey breeder, seems your account works :-) what was first, the chicken or the egg?

ReplyDeleteI don't get your point though. B is a subgroup of A but not vice versa. Yes, you could of course say there might be no A at all, but I hope the above egg - I mean, photo - convinces you.

Best,

B.

My first guess for the theory of these shapes should be to relate to the theory of overtones and standing waves of a drum.

ReplyDeleteI can't put my finger on it, but if without the shape from which it flows, then the patterns would not be there?

ReplyDeleteThis is a subtle relationship in which one may look at the universe, and how it came forth?

I can almost agree with plato. One can ask if there is a matching condition between the vertical flow and the horizontal one.

ReplyDeleteThere is a calculation of circular modes in http://www.kettering.edu/~drussell/Demos/MembraneCircle/Circle.html and a javas application in http://www.falstad.com/circosc/

The matching condition should tune with an overtone corresponding to a normal mode with a 14-fold simmetry. It seems sort of exceptional, does it?

ReplyDeletePlato: I can't put my finger on it, but if without the shape from which it flows, then the patterns would not be there?arivero: I can almost agree with plato. One can ask if there is a matching condition between the vertical flow and the horizontal one.Not sure what you two agree on, but I'll give it a try. I am reasonably sure whatever the water flows down at is not perfectly symmetric. However, I think what actually gives rise to the discrete beams is an interplay between pressure and surface tension. Surface tension will be able to get over small disturbances, but if they grow too large the spherical symmetry gets interrupted. (Maybe I should have read the paper, but I couldn't access it). There is for sure some matching condition for the water that gets deviated, but I've never been very good in hydrodynamics.

Anyway, to come back to the issue of symmetry breaking, do you mean to say that the direction of the beams is not fixed by the symmetry? I.e. you can rotate it by an arbitrary angle? It's like: if you put the pencil on the top and it drops, it will break the symmetry by dropping to one side. But it could have fallen to either side. So, if you'd take a sample of one million pencils, they would fall equally to all sides. From a large enough distance then, symmetry is not broken?

I should elucidate a little bit more on what I find may be troubling.

ReplyDeleteI had to start from some position?

While my url and I are not perfect, there are somethings that seem so to me?

As the sphere, and thus from this, all things "as geometrics" in explanation?

I have this problem with information such as this, that it all must be inclusive with what you are saying Bee.

B:an interplay between pressure and surface tension.I am trying. I will go on with more shortly at my own site.

Speaking of interesting patterns in water, you may have already come across this:

ReplyDeleteResearchers in Japan "have developed a device that uses waves to draw text and pictures on the surface of water."Turns out that Bessel functions (the same ones that come up in elementary differential equations classes when solving for vibrations of a circular drumhead) were key in the design. (There used to be a great video of this on YouTube, but it got taken down for copyright reasons).

Bee - Glad to know that my account works for posting. I cannot deny that egg exists :)

ReplyDelete"B is a subgroup of A but not vice versa" - That's true. (I'm not good at group theory but can understand that A is invariant under a rotation of any angle, while B, the 12-pointed star, is invariant only under a rotation of N*pi/6) But these symmetric properties are mathematical; don't necessarily imply that there's any profound physical principle (that makes A more special than B) behind them.

Also, although A --> B can be called a process of symmetry breaking, it is not a case of the more interesting "spontaneous" symmetry breaking. The thing is, we need to turn up the volume of the water jet input to force A to change into B. A is a (stable) steady state response to a weaker jet, while B is the steady state response to a stronger jet. So, it's a rather boring kind of symmetry breaking.

(Then again, I could be wrong. I'm not a theoretical physicist, just surf the web for fun.)

- arivero: "theory of overtones and standing waves of a drum"

ReplyDelete- sujit: "draw text on the surface of water"

A somewhat related point here: Although the polygonal pattern in the lower panel looks exotic (compared to the "water bell" in the upper panel), it is mathematically possible to create such a pattern just by a superposition of simple sinusoidal functions.

For example, this superposition

f(x,y) = cos[A(px+y)/2]+cos[A(px-y)/2]+cos[Ay],

where p = sqrt(3) and A is arbitrary, gives rise to repeated hexagonal patterns in the x-y plane. (This is well-known for those who study hexagonal cells of thermal convection.) Moreover, the above f(x,y) is an eigenfunction of the Laplacian operator. So, if we build a drum with the right shape and beat it the right way, we can produce exotic polygonal patterns in the membrane.

By the same token, writing exotic "text" on the surface of water shouldn't be too difficult. It just requires a more sophisticated set of superpositions.

hi breeder,

ReplyDeletehard to say what kind of symmetry breaking you find interesting, but yes, there are different kinds of transitions you can have. in most cases it's some kind of a temperature drop that leads to a transition into a less symmetric state. I didn't mean to give an example for the Higgs-field if that's what you're aiming at?

it is mathematically possible to create such a pattern just by a superposition of simple sinusoidal functions.As long as you're in the linear regime.

Best,

B.

for enjoyment of the general audience, it can be worthy to note that the web has a lot of info about

ReplyDeleteChladni Figures, including some youtube videos:http://www.youtube.com/watch?v=EprMFajNzfQ

http://www.physics.brown.edu/physics/demopages/Demo/waves/demo/3d4030.htm

http://www.physics.ucla.edu/demoweb/demomanual/acoustics/effects_of_sound/chladni_plate.html

http://fusionanomaly.net/ernstchladni.html

http://media.www.jhunewsletter.com/media/storage/paper932/news/2004/10/01/Science/Profs.Study.Vibrations.Earn.Grant-2244347.shtml

The last link points out that some ancient seers could rely on Chaldni patterns!

A question to be ask is which the contour conditions (Dirichlet, Poisson, etc) are in the case of the scattered flow. Looking at the pictures, it seems that the flow colapses when the speed is unable to sustain it against the force coming from the surface tension (strain?) of the fluid.

ReplyDeletean interplay between pressure and surface tensionYep, surely this interplay is the adequate alternative to the boundary condition of a vibrating drum. We have on one side the matching conditions with the vertical flow, on the another the boundary condition for the flow to colapse.

do you mean to say that the direction of the beams is not fixed by the symmetry?Hmm some pattern experiments select a given node by putting the finger in a point of the plaque. So perhaps there is some equivalent of the finger hidden somewhere in the matching conditions.