## Finding pretty patterns in scientific graphs

May 13, 2010 by dberenstein

Here is a graph I produced today as I was studying some problems I’m interested in. The important thing is not what the graph represents physically: it’s just a bunch of trajectories in a Hamiltonian system. What is interesting is that the patterns look pretty and seem to have meaning.

Trajectories in a dynamical system

So what do I mean? You look at it and you can not help yourself from finding a visual pattern in it. To me it looks like a sword and it has an aesthetic of its own. You might have seen a lot of ‘calculations’ turn to art before. For example, the Mandelbrot set and other fractals.

When I write papers and I put plots in them, there is always some choices made to make the plot pretty and relevant. There is this undeniable satisfaction in making the graphs look good.

A not so pretty plot can convey the same information, but it causes a lot less impact on a potential audience. I believe this is one reason why Astronomy is so popular: there are experts in false-coloring graphs taken on invisible light to make stunning pictures. I think that if Astronomy was just a bunch of dots without this image manipulation, people would not feel such a close connection to it.

Please let me know what the graph above means to you. And if you have a science picture that you really love and would like to share it, let me know about it as well in the comments.

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on May 13, 2010 at 11:01 pmMosheHey David, Eric Heller visited us recently, you may enjoy his gallery:

http://www.ericjhellergallery.com/

on May 14, 2010 at 6:18 pmdberensteinThanks for the link Moshe. I really enjoyed them.

on May 14, 2010 at 5:22 amLuboš MotlI was attending a vernissage of modern art last night, so the question “what does it mean?” was pretty frequent. 😉

Of course, pretty diagrams would attract new people to reading the papers, and to the field, but do you really want them? 🙂

on May 14, 2010 at 6:39 pmdberensteinHI Lubos:

As an educator, I certainly want people to be interested in what we do. Having nice graphs also helps in presentations to larger audiences (Colloquia, etc), so it pays to make a little effort there.

If a field looks `cool’, smart people will go into it and I definitely want smart people working in my field (even if they are the competition and end up being way smarter than I am).

on May 15, 2010 at 6:03 amLuboš MotlOh, I see. Surely, I agree on colloquia and the general comments. I thought that this was created in a series of technical graphs in a technical paper.

on May 15, 2010 at 6:09 amLuboš MotlLet me take the agreement back a bit.

I actually don’t agree that you attract smarter people if you increase the proportion of the people who were attracted because they liked some superficial visual portion of the diagrams such as nice colors and lines of the right robustness.

This way of “casting” adds mostly noise to the competition. It’s great to show likable pictures, and I love them myself, but it’s still a different thing from liking the internal patterns – and especially those that wait to be uncovered.

In fact, I do think that there’s a correlation but it’s a negative one. If you wanted to get the smartest people, you should present the things in the least visually likable way, boring way, leave the people to deal with the things, give them some tests, offer them big salaries, and choose the best ones. 😉

on May 14, 2010 at 11:41 amGiotisIt definitely looks like the kind of lances knights used in Medieval jousting.

on May 14, 2010 at 3:31 pmparesh shahAesthetically beautiful graphs are definitely an attraction. I fully agree with you that ‘creating’ them gives us a lot of satisfaction.

Unfortunately now that you have primed us to see a sword, that is what I see.

At the same time it is critical that the graph must show the data/numbers clearly and accurately and not distract the readers from the message.

Incidentally my background is in business charts – but we can all learn by interacting with persons in other fields – surprising insights can somehow emerge.

on May 14, 2010 at 4:12 pmdberensteinHi Paresh:

I’m not sure that I primed the sword. It could also be Dr. Mad’s focusing ray beam or some such.

I agree that the graph has to convey meaningful information.

on May 15, 2010 at 4:16 pmRobertIt reminds me of my PhD thesis where in one chapter I studied the classical dynamics of a two dimensional system with Hamiltonian H= p_x^2 + p_y ^2 +(xy)^2 which at that time was a toy model for (M)atrix theory but is somewhat generic for what happens when two branches in moduli space meet (eg as some particle becomes massless or two branes coincide). This came up again in the study of moduli trapping by Silverstein and friends.

on May 15, 2010 at 5:00 pmdberensteinHi Robert:

What a coincidence! That’s exactly the Hamiltonian I was studying. And what a coincidence! I am studying it because of matrix theory and brane trapping phenomena.

See, even out of context it fires up the right neurons. Now if you point me to the right reference, you might have already done something I need to make some estimates, so I would be very grateful for the pointers.

on May 16, 2010 at 6:37 amLuboš MotlOh, that’s right. You should have actually assigned the reverse engineering task as a contest for your readers, David.

I think it would be easy to solve it if one knew that it has a doable solution. After all, they’re quasi-sinusoidal oscillations whose amplitude gets much bigger near x=0, so it’s an x-modulated harmonic oscillator, and the frequency – whose square is in front of y^2 – clearly goes like x. 😉

on May 16, 2010 at 8:07 amRobertBeyond eikonal scattering in M(atrix)-theory.

Robert Helling, (Potsdam, Max Planck Inst.) . AEI-2000-048, Sep 2000. 19pp.

e-Print: hep-th/0009134

Have fun!

on May 17, 2010 at 8:34 pmCodyMark Stock’s graphics around fluid dynamics are just incredible:

http://markjstock.org/

Luboš, might you be overlooking the influence it has on children? And would you disagree that there is a benefit to simply having more minds working on a problem (whether those minds are more talented or not)? I think I disagree with suggestion that presenting things in a boring, visually mundane way would attract the smartest people. Though I suspect that was at least in part a joke.

on May 18, 2010 at 5:53 amLuboš MotlDear Cody,

yes, I believe my comment applies to children, and maybe especially to children, and yes, I believe that significantly less talented people are less likely to help with physics progress in the future. In fact, I think that this is a kind of tautology.

These predispositions can still be compensated by good luck, patience, and a lot of work (among other things) but it would be dishonest to pretend that less talented people are running against some universal dynamics.

I feel that this simple and self-evident observation is kind of “politically incorrect” – it goes against the omnipotent populism and its carefully propagated myths and misconceptions – but I still think it’s very important for it to be articulated, appreciated, and taken into account in rational decisions by anybody.

Also, one can always fool himself into thinking that a person or kid XY likes physics or maths and can do important things – just by attaching cute pictures or beautiful sparsely dressed women with the exposition of the physics ideas, pictures that the person or kid will like.

But it’s still just fooling oneself. There comes a moment when the person learns that maths and physics are not about the nude women or colorful pictures and a disappointment for the person – and other problems for his or her environment – follow.

It’s great to paint pretty pictures and to be aesthetically excited by them and it’s great if they help the public to be more friendly towards science but it’s still a different type of excitement than the intellectual excitement from the patterns underlying the laws of Nature and mathematics.

Genuine maths and genuine physics *is* unattractive for a vast percentage of people, and if one does something that it’s not, it’s just no genuine maths and physics.

Best wishes

Lubos

on May 18, 2010 at 1:09 pmSophia“There comes a moment when the person learns that maths and physics are not about the nude women or colorful pictures and a disappointment for the person – and other problems for his or her environment – follow.”

Yes, we can all relate to the devastating moment when we realized that Physics is not related to pornography.

Seriously though, beautiful graphs and colorfully presented data have a role to play in attracting the attention of the public, which in turn decides indirectly where funding is directed.

on May 20, 2010 at 12:40 amJust LearningI think that the beauty in a graph is only found when one knows the equation that generated it. For me it simply isn’t enough to look at squiggly lines and see something aesthetically pleasing in them. It is far more pleasing when one sees a generated image and can identify what generated it. What is always extremely useful is being able to identify very simple curves: exponential curves, power law curves, and logarithmic curves. Or being able to spot a curve generated by a simple linear function.

What I always think about is how different equations can have different meaning to people, whenever I see physics equations these days, I always find myself thinking in predator-prey type models, such as:

http://math.bd.psu.edu/faculty/jprevite/dump/070417.pdf

So the toy model for h = x-dot^2 + y-dot^2 + x^2y^2 now becomes a relationship in population dynamics where we look at 3 orthogonal axis, one for change in pop x, one for change in pop y and one for the predation of x on y.

So using a simple relationship of:

x-dot = x – xy

y-dot = -y + xy

and

x-dot^2 = x^2 – 2x^2y + x^2y^2

y-dot^2 = y^2 – 2xy^2 +x^2y^2

h = x^2 + y^2 + 3x^2y^2 – 2x^2y – 2xy^2

This is now a polynomial equation and using wolfram alpha it has solutions:

x = 1/3 (1+i sqrt(2)), y = 1/2 (2 i sqrt(2) h+h+1)

x = 1/3 (1-i sqrt(2)), y = 1/2 (-2 i sqrt(2) h+h+1)

3 x^2-2 x+1!=0, y = (x^2-sqrt(3 h x^2-2 h x+h-2 x^4+2 x^3-x^2))/(3 x^2-2 x+1)

3 x^2-2 x+1!=0, y = (sqrt(3 h x^2-2 h x+h-2 x^4+2 x^3-x^2)+x^2)/(3 x^2-2 x+1)