## Tuesday, July 02, 2019

### Innocence of beauty in Feynman lectures on physics

Massimo Pigliucci chose Richard Feynman as a champion and user of beauty in physics – and therefore a natural target of a hit piece. While Pigliucci's knowledge of Feynman or physics is basically non-existent, he made a good choice: Feynman did indeed like to refer to beautiful laws, derivations, and pictures in physics.

In this extensive, 56-kilobyte-long blog post (almost exactly 10 times Pigliucci's rant), I decided to review references to the words beautiful, beauty, and pretty in the Feynman Lectures on Physics (1963, online). If you click at the hyperlink, you may get 71 hits. But when you try to see all of them, the number gets reduced to 65. Moreover, a dozen or two dozens of hits use the word "pretty" as a synonym of "rather" ("pretty soon") and whenever it's so, I automatically omitted the references.

Some of the hits also lead to comments by others. The remainder provides us with a rather comprehensive picture where and why Feynman referred to "beauty in physics" and in related discussions. The words "beauty" and "pretty" are given several meanings that may be partially distinguished from each other. I will try to do so, to comment on each of them, and to decide whether each usage of the "beauty" in physics may be justified.

I think that all sane readers will agree that there was nothing objectionable in his usage of the aesthetic terminology. "Beauty" has often referred to Maxwell's equations, quantum mechanics, its simplicity, simplicity of derivations, simplicity of equations expressing laws, a girl on a beach, actual pictures encoding a physical situation, architecture, design of eyes built by Mother Nature, and more.

The lectures have 3 volumes, each of them has many chapters, and I will sort them from the first volume and from the early chapters. The errata only contain texts from individual chapters that don't have to be discussed separately.

First, in the Nobel Lecture that is also available on that website, Feynman described a story about his discoveries as well as unsuccessful research dreams:
And, so I dreamed that if I were clever, I would find a formula for the amplitude of a path that was beautiful and simple for three dimensions of space and one of time, which would be equivalent to the Dirac equation, and for which the four components, matrices, and all those other mathematical funny things would come out as a simple consequence—I have never succeeded in that either. But, I did want to mention some of the unsuccessful things on which I spent almost as much effort as on the things that did work.
If I understand it, he wanted a first-quantized path integral description of a Dirac fermion. I have actually struggled with that problem as well – and independently – and I also found nothing. A Dirac fermion has some discrete degree of freedom, the spin and/or the sign of energy (there are four components), and it's questionable whether the evolution of discrete or finite-dimensional Hilbert spaces may be captured by any path integral as well.

Concerning beauty, you see that Feynman just wanted a formula for an amplitude that would be "beautiful". He didn't find it so he couldn't tell us exactly what was beautiful about it. ;-) But it would be a short, simple, concise rule that could be written on a T-shirt, I guess. The word "beauty" partly plays a motivating role. A person may be attracted to the members of the opposite (or the same) sex and it's analogous with physics discoveries. One of the things that attracts us is some generalized "beauty". To ban it sounds as silly as to ban beautiful women when it comes to sex.

Now, the chapters. The numerical labeling I/3 describes Volume/Chapter.

I/3: Other sciences, nuclear processes in stars

Feynman discusses how other sciences are affected or owned by physics.
In this chapter we shall try to explain what the fundamental problems in the other sciences are, but of course it is impossible in so small a space really to deal with the complex, subtle, beautiful matters in these other fields.
In this sentence, he talks about beauty in other sciences. What does it mean? He clearly implicitly claims to have some aesthetic sense for the other sciences, too. Whatever excites Feynman aesthetically is "beautiful". You can't really eliminate the "joy of beauty" because it's an important motivation that drove his – and other scientists' – curiosity.
Astronomy: In this rapid-fire explanation of the whole world, we must now turn to astronomy. Astronomy is older than physics. In fact, it got physics started by showing the beautiful simplicity of the motion of the stars and planets, the understanding of which was the beginning of physics.
The planetary orbits are beautiful ellipses etc., I will discuss it later. A story about a nuclear physicist's girlfriend and a footnote there say:
One of the most impressive discoveries was the origin of the energy of the stars, that makes them continue to burn. One of the men who discovered this was out with his girlfriend the night after he realized that nuclear reactions must be going on in the stars in order to make them shine. She said “Look at how pretty the stars shine!” He said “Yes, and right now I am the only man in the world who knows why they shine.” She merely laughed at him. She was not impressed with being out with the only man who, at that moment, knew why stars shine. Well, it is sad to be alone, but that is the way it is in this world.

Footnote: Now I’m rushing through this! How much each sentence in this brief story contains. “The stars are made of the same atoms as the earth.” I usually pick one small topic like this to give a lecture on. Poets say science takes away from the beauty of the stars—mere globs of gas atoms. Nothing is “mere.” I too can see the stars on a desert night, and feel them. But do I see less or more?
The footnote is clearly equivalent to the Ode to the Beauty on a Flower. Feynman simply expresses the view that with the physics understanding, he sees some additional aspects of the beauty of stars or flowers which make him similarly pleased as the "regular" visual beauty pleases most people. But one can't have these new types of perceptions without the background. On the other hand, his being a physicist doesn't rob him of the conventional, laymen's perceptions of beauty.

Yes, it's emotional but these emotions are real, too. Even if a person excited by the new, scientific kind of beauty hiding in our description of flowers were a "part-time artist" or "a partly emotional man", you just shouldn't ban them. If artists and emotional men are allowed, the half-artists are half-emotional men must be allowed, too!

The story was about the nuclear physicist named Fritz Houtermans. Sadly, the intellectually limited girlfriend who was unimpressed by his knowing what no one else knew wasn't quite an ordinary girl. She was his later wife Charlotte Riefenstahl who was herself employed as a physicist. Riefenstahl and Houtermans have actually married twice, in 1930 and 1953, and divorced twice, too (Fritz has married four times in total). Pauli was a witness on both weddings. Houtermans did the thermonuclear stellar work in 1929 (with Robert Atkinson), one year before the first wedding with Charlotte. Well, most women – and probably even most women employed as physicists – just don't get it. (Some do!) They are not really "excited" by physics and its beauty.

It's possible that these days, the Feynman Lectures on Physics could be banned just because of this story that points out that a female physicist was lacking something that her husband possessed. But that's how it often works. I would bet that he knew all the data from the previous paragraph and beyond – and he discussed the story deliberately because some affirmative action was already getting started and he hated it.

I/4: Gravitational potential energy
...so long as we are not too far from the earth (the force weakens as we go higher) is$E_{\text{potential gravitational}} = {\rm weight} \cdot {\rm height}$ It is a very beautiful line of reasoning. The only problem is that perhaps it is not true. (After all, nature does not have to go along with our reasoning.) For example, perhaps perpetual motion is, in fact, possible. Some of the assumptions may be wrong, or we may have made a mistake in reasoning, so it is always necessary to check. It turns out experimentally, in fact, to be true...
In the previous text, Feynman is heuristically deriving the proportionality of the potential energy in Earth's gravitational field to the weight and to the height. The derivation as well as the result are beautiful because they're simple, clever, and seemingly unassailable. The unknowns are almost being eliminated.

You know, there could be longer, less transparent, more foggy, and more questionable derivation which we could call "ugly". The ugliness would be related with many suspicious pieces of the derivation. Clearly, Feynman chooses the word "beauty" for something that is more likely to be true because it has fewer places where it may go wrong.

I/7: Kepler's laws

Here are the promised Kepler's laws.
He made voluminous tables, which were then studied by the mathematician Kepler, after Tycho’s death. Kepler discovered from the data some very beautiful and remarkable, but simple, laws regarding planetary motion...

Six years later a new measurement of the size of the earth showed that the astronomers had been using an incorrect distance to the moon. When Newton heard of this, he made the calculation again, with the corrected figures, and obtained beautiful agreement...

If a law does not work even in one place where it ought to, it is just wrong. But the reason for this discrepancy was very simple and beautiful: it takes a little while to see the moons of Jupiter because of the time it takes light to travel from Jupiter to the earth...

There we see a beautiful ellipse, the measures starting in 1862 and going all the way around to 1904...

This figure shows one of the most beautiful things in the sky—a globular star cluster...

It is hard to exaggerate the importance of the effect on the history of science produced by this great success of the theory of gravitation. Compare the confusion, the lack of confidence, the incomplete knowledge that prevailed in the earlier ages, when there were endless debates and paradoxes, with the clarity and simplicity of this law—this fact that all the moons and planets and stars have such a simple rule to govern them, and further that man could understand it and deduce how the planets should move! This is the reason for the success of the sciences in following years, for it gave hope that the other phenomena of the world might also have such beautifully simple laws.
Great, we have six beautiful things here. One is "beautiful" in the conventional way understandable to the laymen, a globular star cluster. Many cosmic photographs simply look beautiful to many just like many beautiful paintings. If someone imposed a ban for physicists on talking about beauty, they would probably get unable to talk about the regular beauty of such pictures, too.

Another beautiful thing is an agreement that Newton got when he corrected a mistake concerning the Moon's characteristics. An agreement is "beautiful" because things fit together. The word "beauty" isn't really necessary, it's a different type of "beauty" than others. An agreement simply makes a scientist happy. It's a good thing. It's normal for people to call good and pleasing things "beautiful".

The remaining three references say that Kepler's ellipses and the laws were beautiful. Ellipses are beautiful because they're not just some "generic curves resembling circles but different from circles". An ellipse may be obtained from a circle by a squeezing. An ellipse may be constructed in many other ways, too. The multiplicity of simple prescriptions that give you an ellipse – and its focal point – is a part of the beauty. And Feynman explains why the understanding of the planetary motion that suddenly emerged was "beautiful" by itself. When we suddenly understand something and the law may be written down, it's "beautiful". The knowledge is more beautiful than the previous ignorance in the same sense in which a flower that we see is prettier than the fog in which the flower was hiding! Fog and ignorance isn't as beautiful as the understanding of something based on clear statements and shapes.

I/9: Pendulum's motion
If we watch the dynamics of this machine, we see a rather beautiful motion—up, down, up, down, … The question is, will Newton’s equations correctly describe this motion?
What is beautiful here is the simple oscillating motion. You know, it's beautiful because, as Feynman's friend-artist said in another (magnetic) context, it's just like fudging. ;-) But the oscillating motion is beautiful, and so is the sine function. It's as aesthetically perfect as a circle. It's a different way to represent the beauty of the circular motion. And it's beautiful when real physical systems actually reproduce this idealized mathematical function.

Would you find it sane if physicists were prevented from expressing the sentiment that finding a pure sinusoidal pattern somewhere in Nature is beautiful?

I/12: Definitions of a force
That is what Newton’s laws say, so the most precise and beautiful definition of force imaginable might simply be to say that force is the mass of an object times the acceleration.
Feynman just discusses the interpretation of $$F=ma$$ as a possible definition of a force. It's beautiful because it's simple and it could aesthetically satisfy someone. He points out that without extra laws, like the inverse square law for gravity, this $$F=ma$$ would be useless. So in this sense, he is fighting against some excessive excitement that someone could have because of his feelings of beauty!

I/15: Verification of Einstein's formula
$$\Delta E = \Delta (mc^2):$$ This theory of equivalence of mass and energy has been beautifully verified by experiments in which matter is annihilated—converted totally to energy: An electron and a positron come together at rest, each with a rest mass $$m_0$$.
The experiments find a precise match in a nontrivial equation which is said to be "beautiful". Obviously, a scientist is supposed to be happy about such a precise confirmation. A question could be why it's called "beauty". But again, it's "beautiful" when you hit the target. It's like an arrow that goes through some nice hole inside an angel. Such a precise picture looks more beautiful than if you hit the angel's flesh somewhere and there's a lot of stinky blood and gore around!

I/20: Gyroscope, Dirac equation, and non-trivial need for mathematics

You know, in this paragraph, Feynman discussed gyroscopes:
This is a strange characteristic, and as we get into more and more advanced work there are circumstances in which mathematics will produce results which no one has really been able to understand in any direct fashion. An example is the Dirac equation, which appears in a very simple and beautiful form, but whose consequences are hard to understand.
The Dirac equation is beautiful, according to Feynman, and the seemingly complex values of the components of the Dirac matrices or epsilon symbols – also found in the cross-product analyses of gyroscopes – have some beauty. He expressed the insight that mathematics is actually needed to get some predictions out of a beautiful theory. You often need to get your hands dirty.

Some physicists who have really done very little "real science work" in their life, if any, may talk about the beauty but they insist that the beauty of an equation is also perfectly preserved while solving a problem. Most of the deluded talk about the "background independence" is an example of that. But that's nonsense. The equation may be beautiful but the calculation of some application of that equation may be rather messy. You may need to substitute numbers and obtain complex and convoluted algebraic expression, you may need to spontaneously break the symmetries that exist at the level of the equations, and so on.

The messiness of the solutions or applications don't diminish the beauty – or probability of being correct – of the theory itself. On the contrary, a powerful beautiful theory also has implications for general messy enough situations!

I/22: Beauty of math in physics, addition and multiplication
On the other hand, in theoretical physics we discover that all our laws can be written in mathematical form; and that this has a certain simplicity and beauty about it.
Mathematics' role in physics is a beautiful fact in general. One reason is that operations such as addition and subtraction, which he discussed in that section, reappear at many places. To some extent, repetition of nice things is pretty. Think about two breasts or something like that.

I/23: Resonance and vacuous insights

Feynman explained resonances in Nature beautifully. The word beauty appeared here:
This is an example of very poor science. From two numbers we obtain two numbers, and from those two numbers we draw a beautiful curve, which of course goes through the very point that determined the curve! It is of no use unless we can measure something else, and in the case of geophysics that is often very difficult.
Here, one drew a curve that was determined by one point – and it was beautiful by construction, because it was being chosen from a set of beautiful curves of some simple kind that may be associated with points. But here, like elsewhere, Feynman ends up saying that the beauty may be deceptive. If one just translates the input data to another form so that no new insight can be made, it's too bad.

I/26: Helping a girl, refraction, and least time

In this chapter, there are three beautiful things:
In Fig. 26–4, our problem is again to go from A to B in the shortest time. To illustrate that the best thing to do is not just to go in a straight line, let us imagine that a beautiful girl has fallen out of a boat, and she is screaming for help in the water at point B. The line marked $$x$$ is the shoreline. We are at point A on land, and we see the accident, and we can run and can also swim.
Well, the first beautiful thing was a girl who fell out of a boat. Feynman was going to save her. To do so, he presented a very creative or beautiful "analog system" for refraction. He needs to get to her location partly through water, partly on the shore, the speeds are different etc., and the question is where you enter the water. It's like light moving in two environments.

These days, California would probably prevent Feynman from saving a beautiful girl during the lectures at Caltech, too.
[O]ur first inclination might be to say, “Well, that is very pretty; it is delightful; but the question is, does it help at all in understanding the physics?” Someone may say, “Yes, look at how many things we can now understand!” Another says, “Very well, but I can understand mirrors, too..." ...

Evidently the statement of least time and the statement that angles are equal on reflection, and that the sines of the angles are proportional on refraction, are the same. So is it merely a philosophical question, or one of beauty? There can be arguments on both sides. However, the importance of a powerful principle is that it predicts new things.
Girl's beauty isn't too dependent on physics but the real physical discussions about beauty touch the principle of least time – or action. Feynman clearly believes that the principle of least action is beautiful and explains that the beauty is equated with the predictive power. The principle isn't just some useless babbling. It may be seen in many contexts and applied. That's beautiful. You see some "beautiful mental shapes" resembling the principle in various contexts, e.g. while saving a girl in a swimsuit.

We want predictive principles that have consequences and because they're equated with beauty in this class of situations, we obviously want things to be beautiful, too. To question the desirability of beauty would be stupid because it is the predictive power here. The aesthetic terminology is being used because internally, good physicists excite similar places of their brain as if they see something beautiful in the conventional sense!

I/27: Resolving power of optical systems
To discover the rule that determines how far apart two points have to be so that at the image they appear as separate points can be stated in a very beautiful way associated with the time it takes for different rays... $$t_2-t_1 \gt 1/\nu$$
With a given frequency $$\nu$$, you need the separation between the moments to be greater than $$1/\nu$$ or so if you want to make it possible to distinguish the two events. The inequality – a time-frequency uncertainty principle of a sort – is called "beautiful" because it's very simple and naturally presents the time delay and the inverted frequency as players in the same equation or inequality. It makes complete sense, is widely usable, so it's beautiful. But it's not trivial – you need to invert the frequency etc. There is some degree of surprise here which is beautiful, but not too much surprise. The beauty is largely defined as the simplicity and the predictive power and we just want those things – at least, we want to start with cracking and understanding such possible laws even if we find out that some of them are incorrect at the end.

I/28: Beauty of electrodynamics

For the first time, we read about the beauty of Maxwell's equations, with the elegant cross products everywhere:
What is the formula for the electric and magnetic field produced by one individual charge? It turns out that this is very complicated, and it takes a great deal of study and sophistication to appreciate it. But that is not the point. We write down the law now only to impress the reader with the beauty of nature, so to speak, i.e., that it is possible to summarize all the fundamental knowledge on one page, with notations that he is now familiar with...

The magnetic field is given by $$\vec B = -\vec e_{r'} \times \vec E / c$$. We have written these down only for the purpose of showing the beauty of nature or, in a way, the power of mathematics.
Nature is beautiful because its equations, at least for the electromagnetic phenomena, may be written down on a T-shirt and perfectly understood. He explicitly says that the beauty of Nature in this proper sense is basically the same thing as the power of mathematics. Again, it's a mental flower that has emerged from the fog. Feynman clearly considered Maxwell's equations beautiful. So did Maxwell, I think. Maxwell originally presented them in terms of quaternions because he considered those beautiful, too. And they are beautiful. The combination is a bit fishy but you could see that top physicists have been driven by the beauty for quite some time.

The simple cross product rule for the magnetic field created out of some electric ones after a delay (see the chapter for the details) is also beautiful. The cross product is a beautiful terminology. Seemingly difficult rules for the components perfectly cooperate so that the rotational symmetry is respected while the relationships may be nontrivial and left-right-asymmetric.

I/32: Beauty of ordered atoms

Feynman talks about scattering in light and in
Ordinarily, if the atoms are very beautifully located in a nice pattern, it is easy to show that we get nothing in other directions, because we are adding a lot of vectors with their phases always changing, and the result comes to zero.
the beauty is found in a periodic or otherwise pattern-respecting arrangement of the atoms. More generally, when many building blocks nontrivially respect a pattern, regularity, or law, it's beautiful because it's special. Generic defects make the situation more ugly. That's simply how most people feel about it. Regular and law-abiding things are pretty. Even if some things in Nature end up being irregular, aperiodic, or breaking symmetries and laws, it's a good idea to study and understand the pretty and periodic ones first, so beauty is a good guide once again.

In the text above, he is arguing that the periodic arrangements of atoms only emit light in special directions. It's because the Fourier transform of a periodic function is a linear combination of delta-functions which vanishes almost everywhere. Perfect periodicity is "equally beautiful" as the perfect localization somewhere – these two traits are dual to one another by the Fourier transform.

I/35: Beauty of color pictures translated to spectral graphs
There are many situations in which, if the light intensity were stronger, we could see color, and we would find these things quite beautiful... The former shows a beautiful blue inner part, with a bright red outer halo, and the latter shows a general bluish haze permeated by bright red-orange filaments...

The most remarkable features of this are, first, that it is in the eye of almost every vertebrate animal, and second, that its response curve fits beautifully with the sensitivity of the eye, ... But since that time, two of them have been detected by Rushton by a very simple and beautiful technique...

So one measures the reflection coefficient of light which has gone twice through the pigment (reflected by a back layer in the eyeball, and coming out through the pigment of the cone again). Nature is not always so beautifully designed. The cones are interestingly designed so that the light that comes into the cone bounces around and works its way down into the little sensitive points at the apex...

The shape of one curve fits beautifully with Yustova’s green curve, but the red curve is a little bit displaced... Nevertheless, by casting different kinds of shadows in the light, with various overlaps of colors, one gets quite a series of beautiful colors which are not in the light themselves (that is only orange), but in our sensations.
In the first sentences, Feynman refers to the regular beauty of colorful photographs from the Universe. They're beautiful according to normal, layman's understanding of the beauty. A subtlety is that we only see colors when the intensity of the light is high enough which is why stars often look white although they are actually colorful.

A technique by Rushton is "beautiful" largely because it is clever. Clearly, true physicists have some feelings and emotional attachments to particular technologies and tricks invented by experimenters, too. Analogously, Nature is an engineer as well and we may consider some of Her products of evolution beautiful while others less so. And Feynman said that both cases are possible – because he was in no way blinded by some belief that everything is always beautiful. And physicists know it's not.

In some examples, graphs that just match are "beautiful", too. I have always found colors beautiful – and when I learned how colors work in the human eyes and on TVs, I found it beautiful, too. It's hard to fully convey why to someone who just never has such sentiments but my and Feynman's sentiments were or are damn real.

I/36: Beauty of rainbows converted to graphs

He continues to discuss colors.
The retina is organized in just the way the brain is organized and, as someone has beautifully put it, “The brain has developed a way to look out upon the world.”
You know, it's beautiful because it's true and clever. All of our bodies evolve from the same original cells and even historically, the retinas in the eyes and brains were "not separated". Retina is an extended part of a brain. So the brain decided to see something. It's a beautiful insight because the insight shows some unification in Nature. The splitting of an old brain to the modern brain plus retina – an event during the evolution of life – is analogous to the symmetry breaking of the electroweak force into the weak force and electromagnetism! We're talking about anatomy and physiology, not theoretical physics, but it's beautiful for reasons that are analogous to beauty in theoretical physics.
Furthermore, experiments also show that flowers vary in their reflection of the ultraviolet over different parts of the petals, and so on. So if we could see the flowers as bees see them they would be even more beautiful and varied!
Here, he talks about the regular beauty as understood by the laymen. Clearly, colors add to the sensation of beauty, and because bees also see in the ultraviolet spectrum, they unavoidably see "more beauty" in this colorful sense. There's no way to avoid the conclusion. Well, the sensation of beauty is subjective which is why we shouldn't compare bees and humans but that's why Feynman talks about humans who are just allowed to see the flowers in the bees' way!
The beauty of the compound eye is that it takes up no space, it is just a very thin layer on the surface of the bee.
It's a beauty in the engineering by Mother Nature because it's clever and efficient in saving space. This beauty is analogous to the beauty of state-of-the-art thin smart phones relatively to some ancient bulky ones. Why would someone question it or demonize this talk about beauty?
So next time we look at a peacock and think of what a brilliant display of gorgeous color it is, and how delicate all the colors are, and what a wonderful aesthetic sense it takes to appreciate all that, we should not compliment the peacock, but should compliment the visual acuity and aesthetic sense of the peahen, because that is what has generated the beautiful scene!
Right. Male peacocks are beautiful because female peacocks can see the beauty and were choosing the pretty ones. Feynman has clearly thought about such things – why beauty has evolved in Nature etc. – and he found and presented many answers. Incidentally, it works just in the opposite way for humans. When you see a beautiful woman, you shouldn't compliment her but the men who are refined enough to find her attractive! ;-)

Well, this comment of mine isn't quite true because, if you haven't figured it out yet, women are really buying expensive clothes and lipsticks because of other women, to make them jealous, not because of men who really don't care about clothes etc. so much. ;-)
Finally, we shall briefly describe the more elaborate work, the beautiful, advanced work that has been done on the frog. Doing a corresponding experiment on a frog, by putting very fine, beautifully built needlelike probes into the optic nerve of a frog, ...
The engineering beauty of an experiment to study frogs' vision. The beauty in engineering requires good enough shapes and precision. Of course that engineers and experimental physicists still have some sense of beauty and it's positively correlated with the quality of their experiments because messy ones with random shapes are more likely to fail.

I personally have very negative feelings, not beautiful ones, when I see needles in a frog.
Now, by taking an electrode and moving it down in succession through the layers, we can find out which kinds of optic nerves end where, and the beautiful and wonderful result is that the different kinds of fibers end in different layers!
Beauty as an insight about the anatomy, an insight that makes things non-random and that allows the gadget to work.

I/41: Planck's law

Feynman presents Planck's black body formula, for one state:
$\langle E \rangle = \frac{\hbar\omega}{e^{\hbar\omega / kT} -1}$ This, then, was the first quantum-mechanical formula ever known, or ever discussed, and it was the beautiful culmination of decades of puzzlement.
You know, the black body formula is also beautiful. It's arguably the nicest interpolation between a power law and a dropping exponential that you could design. The curve is nontrivial enough but the total energy is convergent i.e. integrable. And Planck's insight replaced fog with a clear theory. That's just beautiful by itself. The knowledge itself is beautiful and so is the simplicity. It's right for physicists to focus on studying beautiful things in both senses.

I/44: Heat engines
It is also possible to obtain the rule by a purely logical argument, using no particular substance at all. This is one of the very beautiful pieces of reasoning in physics and we are reluctant not to show it to you, so for those who would like to see it we shall discuss it in just a moment. But first we shall use the much less abstract and simpler method of direct calculation for a perfect gas.
He just couldn't resist to derive the efficiency of a heat engine. These derivations – and much of thermodynamics – has been considered beautiful because everything follows from some principles and is independent of the particular substances poured to the engines or other details of the engines. This universality of thermodynamics is one aspect of beauty in physics, as I have already mentioned when I talked about "many applications".

This is a really nontrivial partial definition of the beauty. Einstein once classified relativity as a "theory based on principles" that was analogous to thermodynamics – i.e. a theory from the opposite type than the "constructive" theories such as statistical physics. Principled theories and derivations in them are beautiful due to the universality. A physicist may avoid the term "beauty" but every sane physicist will understand why some colleagues could use the term "beauty" for the "universality". It's so normal to equate them here. Every competent theoretical physicist appreciates the value of the universality.

I/46: Irreversibility
There is nothing more beautiful about one of the motions than about the other. So it is impossible to design a machine which, in the long run, is more likely to be going one way than the other, if the machine is sufficiently complicated.
Here, Feynman discusses the time reversal. There is a $$\ZZ_2$$ symmetry between the two processes related by the time reversal so there's no reason to pick one. He describes the hypothetically preferred one as "beautiful". "Preference", perhaps by the laws of Nature, is just called "beautiful", so it's a different and obviously technical notion of "beauty". Later, he reveals that entropy etc. prefer one process over another, of course, the evolution is irreversible.

OK, that was the first volume. I will speed things up and won't write the full quotes unless it's necessary. You're invited to click and study the material in Feynman's original text – which people should do, anyway.

II/2: Liberating nabla from a particular field is pretty

The operator $$\nabla$$ is said to be beautiful because it allows you to generalize something that holds for the gradient of any scalar field. As I previously mentioned, the ability of mathematics to generalize and apply patterns in many situations is beautiful by itself. Theoretical physicists like it and it has good consequences, patterns that may be reused are capable of producing lots of intellectual work while you only learn them once. They're good things.

II/4: Mirror charges are pretty
One can get into some rather subtle and interesting problems. So although this chapter is to be on electrostatics, it will not cover the more beautiful and subtle parts of the subject. It will treat only the situation where we can assume that the positions of all the charges are known.
We can't be quite sure what portions of electrostatics with incompletely known charges would be considered beautiful – probably some problems with metals where charge density arises on the metallic surfaces, like the electric fields in the presence of a metallic ball etc. Those are beautiful because one solves them by beautiful image charges, spherical inversion etc. Pretty mathematical tricks.

II/9: A beauty test of a theory of lightnings
Negative charge will be brought down to the bottom part of the cloud by the drops, and the positively charged ions which are left behind will be blown to the top of the cloud by the various updraft currents. The theory looks pretty good, and it at least gives the right sign.
The theory of some thunderstorm "looks pretty good" but it's just some aesthetic jargon chosen for the theory's passing some quick, perhaps instinctive, tests by Feynman. It's obvious why physicists prefer theories that "look pretty good" in this sense. The "good appearances" in this sense are just "promising results of quickly done tests". So despite the seemingly artistic terminology involving "beauty", it's a fuzzy version of technical meritocracy.

II/16: Beauty in architecture and engineering

He praises the beauty of the Boulder Dam – now Hoover Dam – and its shapes and shapes of metallic components. Archaeologists would see this beauty as well. The shapes aren't random, they seem to be good for something, obeying patterns, having the optimum shapes for some purpose. Clearly, there is a sense of beauty in architecture and engineering, as I have said, although it's a more technical type of beauty than the beauty according to painters.

II/17: Generalizations of a principle are pretty, sometimes two parent principles are needed
Usually such a beautiful generalization is found to stem from a single deep underlying principle. Nevertheless, in this case there does not appear to be any such profound implication. We have to understand the “rule” as the combined effects of two quite separate phenomena.
Generalizations are beautiful, as I have often discussed – physicists just want laws to be maximally generalized and it's clearly a good thing to prefer general laws. Here, Feynman discusses that there seem to be two independent laws with cross products saying how the electric and magnetic fields get converted to each other. Well, they're not quite independent but I don't want to discuss the subtle issues in this particular discussion.

II/18: Maxwell's equations are pretty, scaffoldings are gone

Feynman really loved the beauty of Maxwell's equations:
If we take away the scaffolding he used to build it, we find that Maxwell’s beautiful edifice stands on its own. He brought together all of the laws of electricity and magnetism and made one complete and beautiful theory.
OK, the equations are clearly beautiful and said to be beautiful. He also mentions that the path to such beautiful equations wasn't – and often isn't – beautiful. It may be ugly and the incomplete steps before the final product appears in its full glory may resemble "scaffolding" which is an ugly thing. Clearly, one part of the dreaming about the future is to get rid of things that look like scaffolding. That's a good thing – the incomplete arguments in Maxwell's equations looked like scaffolding and they were less correct than the final theory without scaffolding!

Incidentally, hidden variables and other wrong things that people are imagining behind quantum mechanics are also an ugly scaffolding (just like the luminiferous aether breaking relativity would be an ugly scaffolding) – and their preference for these ugly things shows that they have no refined physicist's sense of beauty, and it's indeed a reason why they're constantly going in a wrong direction. Physicists prefer clean, unique, and predictive theories without scaffolding etc., not messy theories with scaffolding that agree with an arbitrary unjustified ideology ("realism" in this case).

Later in the chapter, Feynman also says that the box operator is beautiful, and when the box of the electromagnetic potential gives the current, it's beautiful that the components decouple from each other, each has its own equation. Clearly, a box operator is beautiful and indeed reused at many places in relativistic physics. And the clean separation of components of vectors is also beautiful and omnipresent in correct, relativistic equations – we "explain" this repeated appearance of the beauty by saying that the Lorentz symmetry that demands the clean separation of components is a principle of Nature.
Now we are ready to cross over to the other side of the peak. Things will look different—we are ready for some new and beautiful views.
Something else is beautiful and he wanted some students to be excited for the following material. The following chapter 19 was about the principle of least action which Feynman considered beautiful, as shown elsewhere, but the chapter doesn't contain "pretty" or "beauty".

II/20: Imagination, beauty of rainbows converted to other formats

There is a whole extremely relevant section in this chapter, about the scientific imagination. He addresses how to imagine the electric and magnetic fields, whether they are real, and a fun exercise is to translate a beautiful thing – a rainbow – to the language of graphs or quantitative data. It's very creative. Does the beauty of the rainbow disappear if you express it by the spectral graphs seen at angles 40, 41, and 42 degrees away from the center LOL?

He mostly concludes that he doesn't feel the beauty in the graphs but he sees the beauty in the equations, in adding the fourth dimensions, and more. "So there is plenty of intellectual beauty associated with the equations." The chapter is recommended, like others.

II/21: Maxwell was a beauty builder, in a frame

A rectangle with Maxwell's equations and the solutions integrating retarded sources is accompanied by
Here is the structure built by Maxwell, complete in all its power and beauty.
Feynman said it was beautiful because it was. You may write it on a T-shirt, the equations are balanced, have some symmetry, and everything seems to play some role.

II/25: Unwordliness' simplicity is fake

He again discusses the beauty of Maxwell's equations – here involving the box of the electromagnetic potential – and says that the beauty is understood by the Lorentz symmetry which Einstein realized to be a good starting point.

Feynman also talks about the equation for a theory of everything, $$U=0$$, where the unworldliness $$U$$ is defined as $$(F-ma)^2$$ plus the squares of all the things that must vanish according to the known laws of physics. Here, the simplicity and beauty of $$U=0$$ is fraudulent – the complications are hiding in the definition of $$U$$ – but the simplicity and beauty of Maxwell's equations is real.

II/26: Visualizing pretty electromagnetic fields

Here he just says that a picture of electric field lines is beautiful and the beauty may be damaged by some magnetic perturbation. It's close to a layman's understanding of the beauty because it refers to pictures.

II/27: Differential operators in 3D,4D are pretty

In some Maxwell's context, derivatives and divergences are beautiful.

II/28: Divergences from point charges, my comments on beauty of renormalization
But we want to stop for a moment to show you that this tremendous edifice, which is such a beautiful success in explaining so many phenomena, ultimately falls on its face.
He discusses the beauty of Maxwell's equations which are nevertheless failing due to the pointlike charges' divergences. The beauty starts to disappear when you try to make the electron non-pointlike etc. – especially because the deformation of the electron to a finite-sized object may be done in so many ways (at least classically).

Quantum field theory and renormalization don't really need any of these ugly arbitrary "fixes" of the divergences. While many considered renormalization ugly, I would say it is beautiful exactly because it avoids the arbitrary, non-unique smearing of the point-like charges. Renormalization preserves the point-like charges and tells us how to compute with the apparent divergences to get meaningful finite results – so how to compute in a way that makes the details of the "regularization of the electron's size" inconsequential. It's really pretty because renormalization ultimately boils down to the universality of the long-distance behavior. The particles are not quite point-like at the end, they are e.g. stringy, but their long-distance properties are independent of the short-distance details how the infinities are removed.

II/35: Another beautiful experiment
Such atomic-beam or, as they are usually called, “molecular” beam resonance experiments are a beautiful and delicate way of measuring the magnetic properties of atomic objects.
Again, some beauty of the design in experimental physics. We've discussed it a few times.

II/37: Ferromagnetism and pretty challenges

Ferromagnetism presents "beautiful challenges", like why it exists at all. The question is beautiful because it's so simple to phrase and there are hints that the answer could be analogously crisp. Like theories, questions may be beautiful, too. It also means that they're largely fundamental and reappearing in many contexts.

II/40: Pretty proofs in dry hydrodynamics

Hydrodynamics with dry water (no viscosity). Pictures of vortices and smoke rings are beautiful, for the layman's reasons, while the mathematical problems in "hydrodynamics without turbulence" are beautiful mathematically. They make conformal transformations and similar things useful. It's a mathematical beauty even if it cannot be taken as a sign of a physical correctness. We are really talking about mathematical methods to find solutions to some equations that are given in advance and not questioned, not about the validity of the original physical theory. A "beautiful" proof why the efflux coefficient is 50 percent is given. Memorable, clever arguments are "beautiful".

The third volume is very important because it's about quantum mechanics. Feynman insisted that the students should better learn quantum mechanics fast.

III/3: QM gets a lot from a little: pretty
...we can show you one of the most beautiful things about quantum mechanics—how much can be deduced from so little.
Right, very generally, the predictive power and constrained character of theories is a major aspect of the beauty and it's clearly desirable because we do want to predict a lot from assuming a little. It's the case in quantum mechanics as he shows in many examples. Many of the systems are, for example, described by low-dimensional Hilbert spaces where the operators are also restricted and with a few conditions such as symmetries, they may be completely determined. So the evolution and many other things may be determined almost by pure thought and there's no freedom. That's clearly "beautiful" and it's clearly desirable.

Also, "one of the beautiful consequences of quantum mechanics" is that particles of the same type are identical and the amplitudes for intermediate histories are really added or subtracted with or without the exchange of the identical particles. I find this trick of QM really cute because there is a nice symmetry or balance between the two processes and the fact that both intermediate processes contribute shows that the claim that the particles are identical is more than a bureaucratic, purely formal classification in identity politics. Their being identical has beautiful consequences which really settle that they are identical if you verify it. When two electrons come close to each other and then repel, you can't really say whether they exchanged places – both processes, with and without the exchange, have contributed to the probability distributions according to which the outcome of the final observation was chosen.

III/6: Prettily mastering a 3-state system
Showing you such arguments at this early stage has a disadvantage in that you are immersed in another set of abstractions before we get “down to earth.” However, the thing is so beautiful that we are going to do it anyway.
Feynman just couldn't resist showing the transformation of 3-component spin-1 amplitudes which are completely determined by the rotational symmetry. The three complex amplitudes are really just components of a vector expressed in a complex basis. Quantum mechanics relates all the processes by which "three seemingly different objects" transform to themselves and each other.

The Wigner-Eckart theorem (relating many seemingly independent amplitudes to each other through the Clebsch-Gordan coefficients) is beautiful, too. These constraints and relationships prove a cosmic order that may be invisible to a beginner. And that's beautiful.

III/11: Beauty seen in latest particle physics

A footnote talks about a beautiful example of a solution to a two-state system from high-energy physics that was fresh at that time. Clearly, it was still cool to see such things in particle physics. By now, we have seen lots of them, like neutrino oscillations etc.

III/17: A pretty gem of QM: 1-line proof of Noether's theorem
The most beautiful thing of quantum mechanics is that the conservation theorems can, in a sense, be derived from something else, whereas in classical mechanics they are practically the starting points of the laws.
Right, completely independently, I have said the same thing many times. This is one of the prettiest things about quantum mechanics that really makes quantum mechanics simpler: $$[H,L]=0$$ may be interpreted either as $$dL/dt=0$$ by the Heisenberg equation of motion, i.e. $$L$$ is a conserved quantity because it commutes with the Hamiltonian. Or $$L$$ may be interpreted as a generator of a symmetry of the Hamiltonian. So a symmetry is equivalent to the conservation law. This proof of Noether's theorem in quantum mechanics is way prettier and simpler than the original one in advanced classical mechanics.

Well, a classical proof using the Poisson bracket looks almost the same except that the commutators may be defined more easily than the Poisson brackets.

III/18: Bell's state before Bell: beauty of perfect entanglement

"We would like next to take an example which is very pretty," Feynman says when he starts to discuss the decay of the positronium. It's an example of the maximal entanglement between two qubits – polarizations of two photons – i.e. "Bell's state" although he obviously discussed it a year before Bell wrote his paper (again, Bell's original contribution was basically zero). Feynman shows where Einstein's thinking about locality went wrong. The description of the photons coming from positronium is pretty because the maximum entanglement is pretty. It's really funny that quantum mechanics may keep the perfect correlation or anticorrelation, like yin-yang, basically for any property of one photon that you also measure on the other one. Something – the potential for correlations – is maximized here. It makes it balanced and pretty.

III/20: Beautiful expectation value formulae

The formula $$\langle E\rangle_{\rm av} = \bra\psi \hat H \ket\psi$$ was written "prettily". Well, the beauty of this formula is a combination of the simplicity, symmetry, and the purely visual traits of Dirac's chosen bra-ket notation – the beauty comprehensible to the layman even if he doesn't understand the bra-ket notation.

Also, Feynman derives that the expectation values of operators obey $$\langle\dot x\rangle = \langle p\rangle /m$$ just like in classical physics. It's pretty because the reappearance of old laws with new players is pretty. The formula is simple etc.

III/21: Superconductivity and control over elementary building blocks is beautiful

He discusses quantum mechanics of superconductivity and some interference patterns there and concludes:
We are really getting control of nature on a very delicate and beautiful level.
Right. It's great to control the fundamental building blocks of Nature and it's really pretty that the properties of these building blocks are completely determined by the postulates of QM, low dimension or other special traits of some Hilbert spaces, and symmetries. It's pretty and it's important for applications, too.

Summary

I have omitted some material outside chapters, like the homework exercises written by others, sometimes in an ugly ASCII form, and these other people have also used the aesthetic language at some moments.

If you have read my almost complete review of "beautiful, beauty, and pretty" in the Feynman Lectures on Physics and if you are a sensible person, you must agree that there is absolutely nothing illegitimate, dangerous, or criticizable about Feynman's usage of the comments about beauty.

He just combines the understanding of the beauty that the laymen have – the beauty of girls, pictures, rainbows, and architecture of flowers, bees, and their eyes – with the beauty of dams and various experimental gadgets that are carefully designed and have preferred shapes – and with the natural counterpart of these sentiments in theoretical physics.

In theoretical physics, the beauty refers to a certain balance, symmetry, high predictive power, ability to be fully constrained i.e. to predict a lot from assuming a little, reappearance of the same patterns and tricks, multiplicity of ways how a certain special function or structure may be defined, isolated, or observed, the universality, generality, applicability, and more. Some of them are obviously desirable features of the laws of physics because "beauty" is sometimes used as a complete synonym of the predictive power, perfect match between predictions and experiments, and other traits that are unquestionably positive.

In other cases, we're not quite guaranteed that the beauty and simplicity are reliable guides. But even when it's so, it's natural for a theoretical physicist to first try the simple and/or beautiful candidates for the laws of Nature. They often become natural benchmarks and realistic laws may be found in their vicinity etc.

And many contemporary physicists refer to the beauty in contexts that are broadly analogous to Feynman's. It's common sense, it's often helpful, and it's also a reflection of their basic humanity and civic rights. Not only it would be harmful to ban all "beauty" in physics discussions; the partial ban on just one chosen subtype of the references to the beauty would still be harmful and unjustifiable.

The people who want to demonize if not ban any talk about "beauty in physics" – or blame a hypothetical (unreal) decline of theoretical physics on the concept of beauty – are just ignorant ideologues, alarmists, inventors of excuses for their own inadequacy, and would-be postmodern Inquisitors whose ideas have nothing to do with the reasoning of great minds in physics or science in general – or any great minds in any field, for that matter. Their skulls are demonstrably filled with sawdust and no one should pay any attention to these ignorant, lazy, superficial, spoiled, jealous, obnoxious, narrow-minded, and uncreative sourballs and would-be censors whatsoever.

And that's the memo.