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Frank Wilczek's book on beauty

Spoiler: this blog post is full of spoilers'

Ann has sent me Frank Wilczek's new popular book, "A Beautiful Question" (thank you again, Ann!) and it is very nice, entertaining, multi-dimensional, and mostly correct. The question reads "Is the Universe built upon beautiful ideas?" and Wilczek's highly secret answer is a combination of the letters Y, E, S.

If you don't know the name, Frank Wilczek is a Twitter follower of mine. (You probably despise honors but he is also a 2004 Nobel prize winner.)

The book describes the history and recent developments in "generalized physics" and is nicely built around the condensation nucleus called "beauty". It's possible but sometimes, one can see that the presentation of the topics is a bit stretched. Is the history of physics a history of beauty? The answer is as ambiguous as in the case of history of physics as the history of light. Yes, no, a little bit, perhaps. Well, the beauty is great if you see it in the laws of Nature. But if you don't see it, it's likely that it's your fault, not Nature's. A person's sense of beauty is only a good guide to understand Nature if the person has a good sense of beauty, one that is correlated with Nature's. ;-)

Instead of the beauty, we could perhaps choose a different, if not the opposite, principle, for example the Larry Summers postulate:

The world is a šitty place.
In some situations, the explanations of the physical insights would have to be a bit different than the attitude chosen by most books, including Wilczek's, but this pedagogic strategy could very well succeed, too. Wilczek, to be able to claim that every important advance in physics made the beauty central, sometimes needs to redefine the beauty, fall in love with something that would be considered ugly just a minute ago, or talk about complicated aspects of beauty that most people find ugly. The beauty is only a good guide if it is defined to be close to the truth – and if the definition is being constantly adapted for the beauty to remain close to the truth. That's why the presentation may be seen as a loaded one. But it's one that I would ultimately pick, too.

Quite generally, I love lots of things about Wilczek's book and Wilczek's attitude and I would describe many things very similarly. His sentiments and thinking are almost certainly closer to mine than those of an average author of popular physics books. You should have this point in mind when you read about the "often disharmonious" remarks below.

The cover of the book is beautiful but if you expect that every part of it will try to be as beautiful as possible as well, you are quickly proven wrong once you read the other people's recommendations at the very beginning. Much more than beauty, the choice of the pundits conforms to the Summers-Shleifer postulate: the world is a šitty place. The first three pundits are Krauss, Muller, and... Not Even W*it is missing. I kid you not. I have never vomitted while reading the first page of a book and even in this case, I managed to keep my record (and clothes) clean. But I must tell you: it's an achievement that I am proud about. Not to vomit when the third name that jumps at you in a "book about beauty" is Peter W*it, you need a very resilient stomach, indeed.

Let me say in advance that the book is equipped with some fifty pretty colorful images – called "Plates" A-Z, AA-ZZ, not to mention AAA (they're concentrated in two segments of the book and printed on a fancier paper), as well as some monochromatic illustrations. OK, let's finally talk about the content.

The book is focused on modern physics, like many books of this sort, but it is "generalized physics". The concept of beauty in physics is clearly a bit "metaphysical", in the sense of the previous blog post (and others), so you simply can't quite separate it from the broader topics that refined humans have been interested in for thousands of years. Wilczek's story doesn't start with Galileo. It starts in:

Ancient Greece

His choice is clearly a wise one. The ancient Greeks were doing some kind of physics. I would quote Einstein who has said that before mathematics was divorced from physics a century ago, they were intertwined and (the Euclidean) geometry could have been understood as the oldest branch of physics – the science of perfectly solid static bodies and their relationships.

The first big hero of Wilczek's was Pythagoras. We are told about two different Pythagorases: the historical one and the legend that his followers created in order to help themselves. Funnily enough, Wilczek uses the adjective "real" for the legendary one. Pythagoras found the theorem – which seems to be a link between mind and matter. When I saw this sentence, it made me breathless. What? ;-) Well, it quickly became clear what he meant. The "mind" is the numbers, like the lengths (and algebra), and "matter" is the geometry (the shapes of the real-world triangles).

This polarized representation of concepts gets even more extreme later in the book when Wilczek presents many conventional topics in terms of the Chinese yin-yang. Yin-yang may be helpful in some cases but it's stupid, too. There are many questions that have "two possible answers". But the labeling of one answer as "yin" and the other as "ying" is often arbitrary. If we reversed the labeling of yin-yang in a whole discipline, it would still work equally well. But back to Pythagoras.

Wilczek offers us two simplest proofs of the Pythagorean theorem. One of them is this one and the other one is all about the division of the big triangle to two smaller ones, both of which are similar to the original one, and \(c^2=a^2+b^2\) arises because the areas of the similar triangles scale like or are equal to \(KR^2\) where \(R\) is the shortest side.

Pythagoras has led to a big irony. He found the theorem and one of the simplest applications tells us that the side of a simple triangle is \(\sqrt{2}\). Wilczek proves that this is an irrational number – contradicting the philosophy of the Pythagorean school which was basically the "discrete physics", the idea of some crackpot physicists (who are still around, in the 21st century) that everything has to be reduced to integers.

Pythagoras has also been playing with music. Wilczek dedicates quite some attention to music, even in the end notes. Several tones sound nicely together if their frequency ratio is rational. Wilczek says that those chords "feel good" because we're more capable of predicting the pattern. It's a nice idea. My alternative idea is that some of their higher harmonics (frequencies) coincide and what is pleasant to our ear is that on the frequency spectrum, there are pretty nice and big enough gaps between the frequencies that are being detected. On the contrary, when you play H1 and C2 together, with the frequency \(15/16\), these frequencies are so close that they press on two nearby places of the ear (on the frequency axis), and this is painful. ;-)

The second big guy is Plato. He loved some ideals – everything is built from Platonic polyhedra (some atoms). The classification of the Platonic polyhedra is presented pretty much completely (including the infinite "grids", non-compact, planar cousins of the polyhedra.) Plato didn't care much that this version of the atomic theory didn't work accurately. Accuracy was only good for cattle, Plato stressed. Top minds primarily care about the idealized ideals. Wilczek also discusses Plato's cave metaphor – perhaps, we only see some shadows or reflections of the deeper truth.

Throughout the book, Plato's musings are used as the benchmark to discuss the relationship between "real" and "ideal" things. Almost every section of the book tells you something about the relationship between the "real" and "ideal" things!

Aristotle is briefly described just as some low-brow superficial data-collecting worker and he has no impact on the rest of the book. I tend to agree with this viewpoint. ;-)


Nothing happens in the following 1500 years or so and then we have renaissance. People start to understand the projective geometry, produce paintings respecting the perspective, and all this stuff. They're excited and so is Wilczek. Everyone starts to realize that we don't see "the world directly" but only "how the world looks to us", through all the transformations that are in between the world and ourselves.

It doesn't take too much time before Wilczek gets to Newton. Newton has three parts: Newton I, Newton II, and Newton III. The most beautiful painting ever is an ugly irregular Newton's paining of a dirty potato – which should be the Earth – and someone standing on a hill on the potato, tossing a stone ever more rigorously. When his strength is really impressive, he throws the stone to the orbit. That, and not the apple, is the probably actual imagery that Newton had in mind when he unified the terrestrial and celestial gravity.

Before Newton, they looked at the resulting orbits of planets etc. – and the circular ones were the prettiest ones. But those were not accurate which is why epicycles etc. were needed. (Kepler is also briefly mentioned.) From Newton's era on, the beauty was searched for elsewhere. What should be beautiful are not the solutions – orbits – but the laws that we have to solve.

The second part of Newton is about the addition of colors and the perception color as seen by the human eye. It's a funny piece of pre-physics that should be taught at basic schools or kindergartens, I think. Red plus green gives yellow – when added positively, like in light (i.e. displays). (No one gave me the right answer during the latest popular talk I was giving. The degree of ignorance about these basic matters is very high.)

Wilczek presents many details about the projection to the 3D color space relevant for the human perception of colors. He mentions the mantis shrimps that see many more independent colors – and even polarization – but also dogs and color-blind people who only see two independent colors. In fact, he even playfully discusses the possibility of extending our color perception higher-dimensional by representing the "fourth basic color" as blinking colors, and so on. ;-) Wilczek compares hearing and vision – hearing is more time-oriented (better resolution for the frequencies etc.) while the vision is better when it comes to the spatial resolution (place from where the light arrives).

James Clerk Maxwell is a favorite physicist of Wilczek's – he had some Christian background and wrote poetic things to his relatives, among other virtues. Maxwell and especially Newton are quoted as arguments against eugenicists. They had either a poor background or imperfect health etc.

But back to physics: there is some fast discussion of classical electromagnetism. Maxwell has added his correction term to Maxwell's equations. It's pretty much the only "really new" thing he added – the rest was synthesis. But this correction term was added thanks to considerations about consistency or beauty, not experiments. Another nice example of the importance of "non-empirical" thinking.

Quantum era

Ancient Greece and classical physics are kind of clumped together – which is sensible – and quantum mechanics first appears on page 166, before the middle of the "main part" of the book. Wilczek presents the energy eigenstates of atoms as analogies of the eigenfrequencies of a classical guitar string. There are standing waves, and those are the same as the energy eigenstate wave functions etc.

He doesn't seem to give a damn about the completely different interpretation of the "waving" mathematical objects in these two situations. It may be troublesome but in some sense, this sloppiness about the interpretation may be justifiable in a book about the beauty. The equations are beautiful and if you primarily care about the beauty, you may ignore what the quantities in the equations actually mean.

But yes, I do think that the intrinsic coherence and cleverness of the quantum mechanical framework of physics is extremely beautiful as well and this beauty is completely omitted in the book.

At one point, Wilczek quotes Einstein's enthusiastic praise for Bohr's old theory of the atom. It's great music, Einstein says. However, Wilczek correctly corrects Einstein: the new quantum mechanics is much better music still! There is a rather original explanation of the relationship between Bohr's old theory and new quantum mechanics. Quantum mechanics isn't described as a replacement of the old model; instead, it is a/the solution to the inverse problem "what are the equations that produce Bohr's old rules as their solution" or "what is the internal mechanism behind Bohr's old theory".

Just to be sure, Wilczek's explicit statements about quantum mechanics (QM) are almost universally correct. He isn't an anti-quantum zealot. But even in this book, one may see seeds of the widespread anti-quantum sentiments. At several places, it's being suggested that you "should" try to measure the wave function (even though he admits that it can't be). Even though it cannot be, it "should" be interpreted as a real thing, Wilczek implicitly says, and that's also why he mentions the "many worlds" at one moment.

He also says that QM isn't a completed theory that you learned once but a "process to which you are constantly adapting". I disagree with that. QM is as "complete" a framework for physics as classical physics – it just takes some extra effort for people to learn it because before people start to learn classical physics, they already have some informal "pre-courses" on classical physics – marketed as common sense and practical basic thinking in the kindergarten – while they have no pre-training in QM.

Atoms are equal to each other, Wilczek observes, and sells this insight as a vindication of Plato (atoms as several Platonic polyhedra). It may be a controversial interpretation but I tend to agree with it, too.

Relativity achronologically begins after QM. Einstein's "God had no chance" is described as an unscientific quote; Einstein has apparently returned to ancient Greece. Symmetry begins to play a central role: that's a broader, conceptual, less technical part of the Einsteinian revolution. Physics revolving around symmetry takes life of its own.

While describing special relativity, he picks the Doppler effect (without naming it) because it shows that all Newton's "atoms of light" – different frequencies – are really the same photons seen from different frames. General relativity is all about the "metric fluid" – a fun explanation that there's some fluid ("metric fluid", also in the glossary), instead of the bare tensor, that deforms the effective metric for all phenomena around. (Real fluid wouldn't be good).

I don't "instinctively" tend to think about fluids when I need to imagine the metric tensor, Wilczek does. I think that if he lived in the 19th century, he would clearly be a fanatical champion of the luminiferous aether – which is a completely analogous thing. In this sense, Wilczek's happy attitude to the "fluids" makes me understand (a bit more empathically than before) why the great 19th century physicists were so enthusiastic about the stupid idea of the aether. I think that people like me have a more Platonic view - the mathematics and numbers describing the fields *may* be "separated" from any material carrier, at least from a carrier that resembles the usual objects we often see around us.

In a subsequent short quantum chapter, he says "three features of electrons" (fields, electric forces, exclusion) and this somewhat vague picture is used to discuss the different arrangements of carbon atoms – balls, graphite, graphene, diamond, nanotubes etc. These forms of carbon seem to be his real "goal" – in this sense, his discussion of the related phenomena is dominated by "applied physics" goals.

One more chapter talks about the local \(SU(3)\) transformations: a picture with locally transformed RGB colors.

In another quantum chapter, he is getting the particles from gauge theories. He presents all the four interactions as basically the same, generalizations of the electromagnetism and gravity. Wilczek sketches fiber bundles, without the words, and pictures them as images with various local changes and projections of colors etc. In all four forces, space tells how the charges/masses move (what is "most straight" trajectory), and the charges/masses curve the space around. It is somewhat oversimplified, of course, especially because at the level of equations, the Yang-Mills equations of motion are sort of "qualitatively" different from their gravitational counterparts.

The matter-space couple is also presented as yin-yang, at some point things get too cheesy and non-quantitative. I generally feel that he underestimates the mathematical background of his average reader. Those people almost always can understand things like "a number for each point in space" or a "function" but he carefully avoids these things to make the text even more accessible and non-mathematical. I have doubts about the value of explanations whose degree of oversimplification exceeds a certain point.

At those places, he mentions the Core Theory. He criticizes the term Standard Model because it sounds like "conventional wisdom" or "rules of thumb". I agree – but I think that these are exactly the right connotations we should have associated with the Standard Model. We just measured certain things in some experiments that didn't go much beyond the framework of QFT, and found there were 3 colors of quarks, 6 flavors, 2 components in electroweak multiplets, electric charges of all these particles, and the continuous parameters of the SM. We put these things and interactions together, that's the accumulated wisdom, and these rules of thumb allow us to calculate the results of particle experiments etc. These "practical" labels are adequate because whenever we use the Standard Model as the "most current theory", we don't know (or admit of knowing) any deeper explanation of these rules of thumb, or the origin of almost any relationship between two of them etc. They're independent, they're purely measured, not coming from deeper ideas, which is why the "pragmatically sounding" terminology "the Standard Model" is absolutely adequate!

There are several arbitrary and provoking idiosyncrasies in these interpretations that Wilczek chooses. Not only Weinberg's term "Standard Model" is said to be worse than Wilczek's "Core Theory". But the electroweak theory (Weinberg) is also uglier than the QCD (Wilczek), we read somewhere. Cute. Well, I think that both of them are non-Abelian gauge theories with some matter content, which is about equally pretty and equally arbitrary, and the fates of the symmetry are just different, but these are different consequences of the laws which can't determine the instrinsic beauty of the laws!

Wilczek discusses the discovery of the nucleus and the construction of the hadrons from quarks etc.

Bizarrely enough, after he has posted about 30 often pretty yet contrived pictures and metaphors that represent the QCD \(SU(3)\) color (and, sometimes, other Standard Model charges) in terms of red-green-blue perception colors, i.e. after he uses the visual-color metaphor/description of the SM charges more intensely than any other book in the history of popular physics, he says that the name color for the \(SU(3)\) charge was chosen for "no good reason"! ;-)

Well, there is a very good reason why this terminology was picked. Because the group is \(SU(3)\), we need to distinguish 3 different related charges – and it's almost mathematically isomorphic to the case of the 3 independent perception colors. Moreover, the color-neutral "egalitarian" mixtures of the colors/charges have a special status in both situations. They look "grey" (colorless: my grandpa, an academic painter, has always emphasized that neither black nor white are colors!) to the eye and they may be isolated in QCD.

Wilczek describes the Yang-Mills description of the strong and weak force in quite some detail: charges of quarks and leptons, asymptotic freedom, confinement. The colorful "tables" showing the charges of the particles arguably look much uglier than necessary. If one explains what the representation \(({\bf 2},{\bf 3})\) of \(SU(2) \times SU(3)\) is, she may summarize all these things more concisely – and more beautifully. So the messy list of all the charges of all the components seems unexpected for a book about the beauty.

He says that there were many more discoverers of the theory of the weak force than for QCD – so his and Gross' etc. discoveries were more "unique", in a sense. It seems to be the case but I find it strange to use it against the importance of the "electroweak" physicists. There simply seems to be a greater number of independent insights that have to be made in the electroweak case. Yang-Mills theory is shared with QCD but also P-breaking and C-breaking, CP-breaking, virtual W-bosons (he calls them "weakons", not too beautiful words), and the Higgs mechanism. None of those things really make physics more ugly. The P-breaking fermion spectrum is as pretty as the P-conserving one, I think. A spontaneously broken symmetry is as beautiful as the unbroken one (Wilczek actually agrees). So by giving the same "weight" to the whole theory of the strong force as to the electroweak theory is probably inadequate as the description of the amount of insights that have to be made. From that perspective, the electroweak theory should have a "greater weight", I think. QCD also has many aspects (asymptotic freedom, confinement, possibility to do lattice QCD calculations, a discipline Wilczek also mentions) but those directly follow from the same simple theory.

There was one day in 1976 he called the most productive day of his career. His daughter had an infection in the ear and other things sucked, too. However, he made a poetic walk and found the gluon fusion as the dominant production (and similarly decay) channel for the Higgs boson – a top-quark loop; an instanton-generated mass; one or two discoveries that haven't made their impact yet; and the axions (BTW did you realize that "axions" and "axial" are etymologically related?). Betsy Devine told him to wash the dishes before he may continue to play with this childish things, i.e. physics, so he named the axions after the detergent in order to show his fellow physicists how much hard work is being given to him by the head of his family. OK, his official explanation of the "axion" terminology isn't as beautiful as mine but it's still OK.

After a summary of the "Core Theory", including a few words on the three families, he dedicates one chapter to Emmy Noether – who is also presented as a noble human being (recall that Maxwell is Wilczek's favorite physicist as a human personality; Weyl seems personally close to Wilczek, too). It's a good choice, she's often underestimated in popular physics. He lists a few conservation laws and corresponding symmetries and spends more time with the history of the concept of energy (especially with the energy loss via heat and how it was initially neglected). If I were writing this stuff, I simply couldn't omit an explanation – in quantum mechanics – why this bizarre relationship actually works. The reason is beautifully simple. (In the Hamiltonian formalism, \([H,L]=0\) may be interpreted in two ways, either as a conservation law for \(L\) or as a \(L\)-generated symmetry of the laws of physics, the Hamiltonian \(H\).) So this chapter looks incomplete to me for that reason.

The following chapter begins with a few pages of dodecahedrons built from a sheet of paper and you don't know where it's going but at some point, it's clear that it is about grand unification. He tries to make you map the GUT matter multiplets to those in the Standard Model, a component-by-component. I don't believe that a layman reader who hasn't learned any group theory will actually do so right. If you know a counterexample including yourself, let me know. GUT is flavored by Wilczek's traditional idiosyncrasies. First, all the fields are "fluids". And just he coined the "weakons" (W-bosons), he invents a new term for "mutatrons", new GUT gauge bosons. I actually think this new word is both ugly and unhelpful. Lay readers typically have at least the same linguistic skills as particle physicists. They wouldn't have a problem with "gauge bosons" and "gauge fields". Replacing them with a new word that isn't used by physicists doesn't seem to make things prettier or simpler – neither for a physicist reader nor for the layman.

There is also a brief introduction to SUSY and anticommuting variables etc. Gauge coupling unification fails (Wilczek nicely says: Popper demands theories to be falsifiable and we passed his test with the non-SUSY gauge coupling unification: it's not just falsifiable, it's even false, mission accomplished). "His friends and Wilczek" himself have found a solution. With SUSY, the unification works more nicely. Wilczek says that Savas Dimopoulos likes complicated models and isn't afraid to add features. I don't have this feeling at all. After all, the SUSY GUT and the MSSM are the minimum models in their "qualitative classes", too. They're at least as beautiful as the Standard Model itself. Wilczek's own contribution to this advance is said to be in neglecting the electroweak symmetry breaking. Yup, it's useful to neglect things sometimes but if this were the only contribution, I wouldn't think its relative importance would be so profound. At any rate, the final realization – gauge coupling unification in minimal SUSY GUT – is a huge insight.

One more brief chapter follows. You read it and it's hard to determine whether it's a summary of "the universe does embody beautiful ideas" or a lesson in yin-yang complementarity or a story about Wilczek's stolen laptop or Niels Bohr's happy marriage. It's probably everything and nothing and this chapter turns out to be unexpectedly short. Even more surprisingly, the main part of the book is over and (not too exciting) acknowledgments begin on page 328. Knowing that the book has over 400 pages in total, I didn't expect the bulk to be over at page 328. It was shorter reading than expected at the end but it was cute.

Afterlife: bonus parts of the book

A short timeline is added after the acknowledgements – there is usually one sentence is dedicated to each event, starting from a few breakthroughs in ancient Greece and European renaissance but mostly in modern physics. The adjective "core" in front of electroweak theory and "anamorphous" are among the trademark Wilczek buzzwords. The last event is in 2020 when Wilczek's (Yes) bet on SUSY will have to be decided. Good for him, my analogous bet will be decided well before 2020.

If you allow me to return to the main chapters as well as some bonus material: There have been about 1-2 short sentences about string theory. Wilczek has said, for example, that its attempt to achieve gauge coupling unification including gravity seems "elusive" to me. (String theory unquestionably unifies gravity with the other forces at the level of consistency; at the level of the quantitative gauge coupling unification, there exists no "canonical" example that it works but because both the volume moduli and the dilaton are ultimately determined in each vacuum, each vacuum clearly says whether all the 4 coupling constants behave well enough to meet at very high energies.) It is not quite clear from the sentence what he exactly wanted to say but regardless of the detailed content, it's misleading. I think that it's obvious that Wilczek has never seriously studied string theory, he plans to live without that forever, and all his comments about "this or that [achieved by string theory] hasn't been achieved" must be ignored as opinions of a layman.

More than 50 useful pages of glossary, "terms of art", are added afterwards. There is 1-5 items per page in a smaller font (and in two columns per page) so the amount of stuff written to explain almost every word is rather extensive. Some of the explanations are so long that they could be considered sections if not chapters (e.g. "energy" or "coordinate"; "parity" takes almost two pages, too). It is a matter of taste whether this approach to a glossary is better than e.g. Brian Greene's concise dictionary-like slogans that may be memorized by a reader who is sufficiently interested. Even though Wilczek's explanations are pretty nice, I would probably vote for the concise glossaries. In many cases, I was forgetting the term that the text was supposed to be explaining.

Much of the ideas in "terms of art" is a neatly done but rather standard mostly popular presentation of the words. Some of them deal with a bit controversial stuff and it is usually explained intelligently. For example, I liked Wilczek's separation of the anthropic principle to its being (obviously) true and to its explanatory power. The laws must allow the existence of "me", the principle wants to say. A limited definition of "me" or "us" has to be adopted for the principle to be non-vacuous (otherwise the sentence only says that everything I could have observed must agree with the laws of physics which is not helpful).

Also, he says that the widespread Popperian "falsifiability" doesn't correspond to the real-world scientific practice that is often about strengthening of good ideas instead of the liquidation of the bad ones. Wilczek seems to understand that his new words are unlikely to become popular. For example, he says that "we [more precisely, I] call this particle mutatron". LOL. He gets a little plus for choosing Prague and New Delhi as two benchmark points while explaining the "metric fluid".

Some seemingly "ordinary" words such as "qualitatively" and "quantitatively" (and the "Universe" or the "vacuum" – a synonym is Woit) are defined as well – which I find very appropriate. Wilczek funnily says that people who claim to have made a "quantum leap" and who know what they're talking about have only made modest achievements. (As a kid, I used to love a home computer, Sinclair QL i.e. quantum leap, and memorized a long 1-page article about it, although I've never touched this particular product in my life. Clearly, it ended up being a rather modest story for the company, too.)

"Quantum mechanics" gets a 2- or 3-page-long definition, too. Rather vaguely, Wilczek still seems to vaguely suggest that there is something non-universal or vague about quantum mechanics (that it means different things in different contexts) and I obviously don't like those vague and completely invalid hints! There are many more things about Wilczek's definition of QM that I like, however, for example the point that QM is often making extremely robust and accurate statements (e.g. about the spectra), too.

Perhaps the shortest definition is one of "reductionism" which, Wilczek thinks (also in the main text of the book), is a pejorative term for "analysis and synthesis". Well, I don't think that they're quite synonymous (at least, "reductionism" is the general belief that "analysis and synthesis" is a good method to understand [almost?] all observations we can make – not the process of "analysis and synthesis" itself). And I don't think that it's currently considered "pejorative" by most users of the word. Well, the big bang used to be pejorative as well – but its proponents have proudly embraced it. The case of reductionism is similar, I think.

The notes at the end of the book occupy about eight pages.

One third is dedicated to perception of music, octaves etc. Can you hear the relative phase of the two sounds. How does it matter whether you have perfect pitch or not? Wilczek reports the results of several experiments done by himself and by his colleague Benjamin Franklin. ;-)

On a following page, he discusses the Platonic grids (infinite "polyhedra") on the hyperbolic spaces of negative curvature (while avoiding all these simple technical terms).

And there are cool things like: genetics of color-blindness, interactive multimedia explaining Maxwell's equations on the Internet, Weyl's poem on eternalism (another favorite guy of the author), Weyl's origin of the term "gauge symmetry" (Weyl wanted to use his Weyl, local scaling, transformations to explain electromagnetism although the group is noncompact and geometric while it should be compact and non-geometric i.e. internal).

A remark on Noether's theorem suggests that Wilczek doesn't really understand the/a simple explanation why the theorem works – the theorem only becomes really simple in QM – although I can't believe this suggestion.

There are also some more URLs of visualizations of various things, a speculative comment on the action and emergence of conciousness. Some technicalities about the history and meaning of forces. The page most frequently referred to by its number is page 404, the second page from the end of the end notes. If the publisher were witty, the page would say "HTTP error 404, page not found, try the following page". ;-) On that page, Wilczek claims that spinors basically cannot be understood without dry algebra.

His recommended reading: Wilczek recommends you to read geniuses (he somewhat obnoxiously counts left-wing economist Keynes among those); Dirac, Feynman, and Boorse on QM (good choices; by the way, I was surprised not to see anything about Dirac's comments on the mathematical beauty of the physical laws); and some other sources, website, one at Stanford, and so on.

The origin of pictures and the index aren't missing.

This is a playful and deep book that will enrich almost all readers. Despite my proximity and knowledge about all these things, it has enriched me, too. Thanks, Frank and Ann.

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