Wow, this book is truly remarkable as it encompasses an extensive range of materials. I find myself at a loss for words, not knowing precisely where to commence. Moreover, I'm not entirely confident in my ability to review some of the more intricate and advanced topics. It's important to note that this isn't a traditional textbook. Instead, it reads with greater fluidity and purpose compared to the dull theorem-proof structure.

Just for your information, approximately the first third of the book is essentially a comprehensive survey or seminar on modern mathematics. It commences innocuously with philosophy, Euclidean geometry, an introduction to hyperbolic geometry, and some rather elementary number theory. However, quite abruptly, Penrose immerses you in a series of advanced topics across various fields. These include real and complex analysis, advanced number theory, differential geometry, Fourier analysis, Riemann surfaces, group/symmetry theory, advanced linear algebra, representation theory, Lie algebras, tensors, n-manifolds, gauge theory, and the calculus of variations. I believe you can envision the breadth of the content. All of this is covered within less than 400 pages, and then the physics section begins.

As I mentioned earlier, some of this mathematics is extremely advanced, reaching the PhD level. The physics section appears to cover the standard model of particle theory and quantum mechanics/QFT, general relativity, and then grand unification. It concludes with some topics in supersymmetry and string theory. However, you have the option to skip these if you so desire, as you would only need to know them if you aspire to obtain a PhD in theoretical physics.

In summary, this is an enormous undertaking, and Penrose doesn't make it easy for you. Therefore, it's crucial to ensure that you have a genuine interest in studying these topics in-depth before delving into it. Because if you do, you'll discover that it's truly a brilliant work.

By his own admission in the preface, this is Penrose' attempt to popularize the current thinking in theoretical physics, including quantum mechanics, relativity, and unification theories such as string theories and quantum gravity.

In the introduction, he says (paraphrased) that he has intentionally chosen the more mathematical route, despite contrary advice. However, he hopes that those without a mathematical inclination can simply skip the equations and still grasp the essence of the concepts.

With due respect to the author, his attempt at “popularization” is a failure. I wonder who his intended audience is when he expends great effort to explain basic concepts like imaginary numbers and why they aren't so mysterious, yet presents higher-level abstractions that even a college graduate may struggle to understand. His target audience seems to range from the average person on the street to professional mathematicians and physicists.

In my opinion, this book of over 1000 pages, filled with equations and complex diagrams, would be excellent material for several graduate courses in mathematical physics. He introduces abstractions such as complex analysis, abstract algebra, manifolds, bundles on manifolds, tensor fields, Lie groups, Hilbert spaces, etc. with only a few words of explanation and a plethora of complex equations. After that, he proceeds with advanced theoretical concepts and their applications to physics. I studied most of these topics to some extent in graduate school, and I was particularly comfortable with Hilbert spaces and complex analysis. Nevertheless, it was all I could do to keep up with him. Perhaps it's because I was in my early thirties then and had more functioning neurons, but my 63-year-old brain found it difficult to follow his descriptions, let alone the equations. I confess that (despite considering myself a former mathematician) I often tried to take the route he had suggested for those without a mathematical bent: just skim over the equations and get the general concept. But his descriptions are so detailed that I found it challenging to extract the general concepts from the text alone.

I also decided that I don't particularly like his writing style. It ranges from overly wordy and redundant to extremely cryptic and abstract. Moreover, the edition I have was poorly edited: there were numerous mistakes such as unclosed parentheses, repeated words, missing words, and grammar errors. Given how dense his prose is, I don't envy the editors' job, but it's still their responsibility to catch such obvious mistakes.

The one positive thing to come out of this is that I think I finally (sort of) understand tensors and tensor fields. For some reason, I didn't quite get into tensors when I studied differential geometry in the early 80s, and I've felt this was a gap in my education over the years (for example, understanding the field equations of general relativity). His description prompted me to study tensors further, and I found a couple of excellent websites with detailed explanations and examples.

In any case, I do recommend this book if you have a talent for or interest in advanced mathematics and theoretical physics. However, I do not recommend it if you're looking for an easy read in popular science to make you feel intelligent.

Penrose delves into the direction that modern theoretical physics has taken as it endeavors to develop a unified model of the sub-atomic realm through the pursuit of multi-dimensional mathematical models. His stance is not solely based on mathematics. Instead, from the perspective of the straightforward progress achieved by theoretical physics in uncovering the mathematical elegance of the relationships among various observed constants, he presents cogent arguments against unnecessary complexity.

His most profound criticism of String Theory is its inability to make any predictions that can be verified through observation. He passionately advocates for the exploration of alternative models rather than allowing mathematical physics to become trapped in what might ultimately turn out to be a dead end. He suggests several possible paths forward, with his own twistor theory being one of them.

Penrose reminds me of Feynman due to his innovative thinking. Feynman's formulation of quantum-electrodynamics necessitated a novel way of conceptualizing observed phenomena. For those who are willing to explore the surface of modern physics, Penrose offers a clear and understandable primer. In fact, he dedicates nearly a third of the book to presenting the mathematics, enabling the reader to begin to comprehend the subsequent content. Spanning over a thousand pages, this is indeed a comprehensive primer.

I am grateful to the friend who lent me this book a few years ago and gave me sufficient time to assimilate what I could. Yes, I did return it, but I might need to obtain my own copy.

This is an extraordinary book. There is indeed nothing else on the market quite like it.

Penrose is a pre-eminent mathematician whose work has had a profound impact on my own thinking. I've often been surprised to discover that a viewpoint I learned from another source actually originated from Penrose himself, as the author of that other source turned out to be his student. For example, consider "Visual Complex Analysis" by Tristan Needham. I came across this book in high school, and its powerful exposition stuck with me throughout college and to the present day. While reading "The Road to Reality," I couldn't help but notice that Penrose engaged in a similar exploration of complex numbers. It was a pleasant surprise to find out that Needham had studied directly under Penrose! Although I've had this book in my collection for a few years, I only recently discovered this connection.

With this tangential introduction out of the way, let me be clear from the start that I can't recommend this book to just anyone. It demands a great deal of mathematical acumen or, more precisely, a great deal of explicit mathematical interest. You don't have to be a mathematician, but you must truly delight in the opportunity to learn the mathematics of the universe. Even if you have a degree in mathematics, you're likely to encounter something new here. For instance, I had never seen the concept of a hyperfunction before, which Penrose enthusiastically presents in relation to the Fourier transform.

The book's mathematical explanations are quite idiosyncratic, reflecting the depth of Penrose's intuition on the topics. This elevates the book from a mere encyclopedic reference to a personal outpouring of the author's vision of the world. It not only features unique topics that aren't commonly taught, like hyperfunctions, but also explicates commonly taught ones in a unique manner. Penrose's style is unabashedly geometric, and he almost always thinks of partial differential operators on manifolds as vector fields. The book is filled with many beautiful hand-drawn diagrams, showing everything from a fanciful vision of the creation of the cosmos to a field of one-forms on a surface to the conservation of electric charge in spacetime. It also prominently features his diagrammatic notation for tensor algebra, something I had never seen elsewhere.

I should mention that this book approaches its physical topics from an exceptionally high level. You probably won't be able to learn electrodynamics from Penrose here. I don't even recall the common form of Maxwell's equations appearing in this book. Instead, he writes them down in the simplest, most beautiful, and most abstract form, using differential forms and exterior calculus. To go deeper, rather than expanding these equations, he focuses on how they can arise as the curvature of a bundle connection via gauge theory. This is both stunningly beautiful and extremely detached from practice.

Despite the overwhelming emphasis on theory and mathematics, the book is insistent that physics must still be physics. Penrose is not optimistic about string theory, though he does give it some treatment towards the end for completeness. He's perhaps like Feynman in that regard – he has a deep appreciation for the mathematics and the theory, but at the same time is very uncomfortable with the lack of testability of much of modern theoretical physics. His own theory, twistor theory, is just as untestable, but he's honest about that.

This book was initially extremely foreboding to me. Having no mathematical training beyond high school and self-diagnosed numerical/mathematical dyslexia, I felt a great deal of trepidation. I truly thought my friend had recommended it as a cruel joke. However, I was determined to venture into these interesting yet hopelessly deep waters.

Opening the book and getting through the alarming preamble, I was introduced to what was clearly the first part of a frame narrative. (Readers can be forgiven if they forget it by the epilogue, some 1043 pages later.) Then came some Euclidean notions of geometry, followed by the Pythagorean theorem, which is much-hated by those who think school should only teach practical skills like filing taxes. Within about 60 pages, I realized I had almost completely left behind the concepts I considered comfortable or at least familiar.

Completing this book required a great deal of fortitude, and it remains to be seen how much actually stuck. There were numerous times when I didn't want to continue. I didn't want to struggle through another chapter. But then I'd catch a tantalizing glimpse of something that made sense or a mention of a physical process, and my focus would return.

I won't deceive you: it's not a good idea to read this book in the summer. It's not light summer reading. But then again, I probably would never have bothered with it unless challenged. Overall, I'm glad I did. It might have been even worse to read in a less lively season. Fortunately, my passing interest in astronomy and various physics made the ideas behind the math just about understandable, so I managed to get by as a somewhat conscious reader, if not a learned one. And Penrose does his best to make the ideas accessible, with well-explained illustrations that sometimes disrupt your concentration or make you think about things you probably shouldn't.

I'm glad I read it. It was a huge undertaking, and I'm glad I persevered. Penrose's love of mathematics is evident on every page. You get a sense of the scope and interplay between all these massive ideas that are often carelessly referenced and used in popular culture. When he talks about the magic of it all, I don't precisely understand why, but the excitement is palpable. With some mental effort, it's possible to navigate through this book and feel like you've enriched your mind a little. But if you're like me, you'll mostly feel extremely challenged, and some equation-heavy pages will almost break your determination. So, I guess I can't recommend this book to many people without a mathematical or physics background.

However, if you're as perverse as I am and want to know how little you know, the specifics of what math and science have been up to in recent centuries, and what a minuscule part you play in the grand scheme of things, then this is a wonderful book that I highly recommend.

This book is not intended for those who lack a solid background in mathematics and physics. It is most definitely not suitable for lay readers.

The content of the book, except for the first 40 pages or so, was extremely complex and difficult for me to understand. Today, I made the decision to stop reading as for the past 30 pages or so, I could only comprehend a small portion of what I was reading.

I am now putting this book on hold. I may come back to it at a later time after I have established a stronger foundation in mathematics and physics. This way, I hope to be better equipped to handle the sophisticated material and gain a deeper understanding of the book's contents.

Amazing.

While I can not exactly call Road to Reality a popularization of general relativity and quantum theory, it is a peerless introduction to and review of those topics. I have a PhD in mathematics, and studied physics and math as an undergraduate, and there was plenty for me to learn from this book. There are very few people in the world who would not learn much from reading it.

Many years ago, I read Penrose's Emporer's New Mind which was good as far as it went, but earned my derision with doubtful, hand-waving arguments for quantum origins of consciousness. Knowing Penrose is no dummy, I permitted a friend to convince me his work deserved another chance.

I am very glad to have read RtR, even though the process took me most of 2010. I now have a far better understanding of mathematical physics than I could have achieved with any other reading list. I did not complete most of the exercises; I suspect that if I had, then (A) I would not have finished until 2012, and (B) the exercises would have been sufficient to bring me to a fairly professional level of competence. Kudos to Mr. Penrose for including them.

The book begins with quite a few chapters of mathematics, quickly progressing to advanced undergraduate topics such as calculus on manifolds. In some ways I liked Penrose's clear treatment and drawings better than, say, Michael Spivak's beautiful but sparse texts. In order to provide a foundation for his chapters on physics, more mathematics is interspersed where necessary.

Penrose introduces complex manifolds, continuous groups (Lie groups), and principal bundles. This machinery is all truly essential to the physics, and it was enlightening to see it collected in a single place, however briefly explained. It is especially useful because most graduate students in physics or math end up missing a formal introduction to one or more of these topics.

The grand themes in RtR are the two major 20th century discoveries of general relativity and quantum theory. Penrose is particularly interested in probing how the two may be made consistent. He covers some of the cosmological work that treats both (e.g. Hawking radiation from black holes), and then discusses theories that seem to join them, including various level of detail on spin foams, string theory and M-theory, loop quantum gravity and his own invention, twistors.

A great strength of this magnum opus is Penrose's ability and willingness to discuss philosophical and aesthetic issues of the physics. Four of these stand out. First, I quite like his perspective on the futility of obtaining unified theories by (more-or-less) trying to guess a tractable Lagrangian. Second, his detailed treatment of entropy, especially the universe's original low-entropy state with respect to gravity and the cosmological implications, was really fascinating.

Third, Penrose seriously considers the various interpretations available for (apparent) collapse of the quantum wavefunction. His bias is toward objective collapse (environmental collapse), rather than being spookily dependent on \\"observation\\" by a conscious observer, and I agree with him that far. He suggests that general relativity, being the only other physical theory we have of similar stature to quantum theory) may somehow provide the mechanism. This is not an assertion but merely a suspicion on his part, and personally I lend it little credence. I do agree with him that general relativity is likely to remain a permanent Newtonian-style large-scale limit of the physics, while quantum theory seems ripe for some kind of fundamental reinterpretation.

Finally, Penrose revels in the aesthetics of what he labels complex number magic. That is, he considers the interesting ways in which physical reality is so well described not just by mathematics but specifically by complex analytic structures, a simple example being the phases of quantum wavefunctions. His fascinating twistors are the coolest example of this, where he changes the physical perspective utterly. No longer are points in spacetime the essential quantities; rather the physics is on the manifold of (potential) light rays. A spacetime point is a confluence of rays, and the interesting part is how fundamentally a point can be represented by, and treated as, a Riemann sphere (a compactified 1-dimensional complex line). As an ex-complex-manifolds guy, this was wonderful stuff to me.

I will conclude by noting that Penrose even redeemed, somewhat, his handwaving arguments from New Mind. I now understand that he is essentially pointing out that our conscious observations of physical phenomena are (or appear to be) collapsing the wavefunction. Since there must be a mechanism for that collapse, Penrose is arguing that something fundamental about conscious minds (as opposed to highly sophisticated computers) is triggering it. I still don't agree, because I believe consciousness is computational and emergent from complex systems, but his point no longer seems so silly.

What is reality?

How is it that mathematics is so conveniently used to describe it?

And often more acutely than our eyes can perceive?

What are the laws that govern everything?

Are we understanding it all correctly?

What if we are focusing on the wrong questions and seeking the wrong answers?

These and many other questions are what the 2020 Nobel Prize winner Roger Penrose is giving the reader a glimpse of working on: string theory, physics, the mind-boggling quantum effects...

All of this is explored through the prism of the mathematical apparatus.

Roger Penrose delves deep into these complex and fascinating areas, using mathematics as a powerful tool to try and unlock the mysteries of reality.

His work not only challenges our current understanding but also opens up new avenues of exploration and discovery.

By presenting these ideas in an accessible way, he allows readers to engage with the cutting-edge research in physics and mathematics.

Whether you are a scientist or simply someone with a curious mind, Penrose's work offers a unique perspective on the nature of reality and the role of mathematics in understanding it.

It is without a doubt the greatest science book ever penned in the entire world, dating back to the dawn of time. This remarkable work is by no means ordinary popular science. Instead, it delves deep into the realms of hardcore science and mathematics, yet is presented in a way that can be understood by the general audience. It offers a comprehensive and in-depth exploration of complex scientific and mathematical concepts, challenging the reader's intellect and expanding their knowledge. The author's masterful writing style makes even the most difficult topics accessible, engaging, and thought-provoking. Whether you are a science enthusiast or simply curious about the mysteries of the universe, this book is a must-read. It has the power to inspire, educate, and change the way you think about the world around you.

Dave Langford, an SF&F critic and reviewer, once made an interesting observation in his long-defunct column for White Dwarf magazine. He said that there is a tendency to over-praise big books simply because one has managed to get through them. I concur that this tendency exists, but I note that Langford didn't provide a reason for it. I believe the reason is more or less related to a sense of macho intellectual pride. People want to show off that they have read this huge and perhaps difficult saga. They think that if they have put in the effort to read it, it must be great, otherwise they would have to admit to wasting their time. And they also want to flaunt their intellectual credentials. Now, imagine if the book is not only huge but also really difficult, perhaps because it is dense with obscure references or full of complex mathematics that is not for the faint of heart. The temptation to over-praise such a book must be even stronger.

I want to start my review with a couple of gripes. This book, which is filled with maths that would make the average undergrad scientist struggle, and in some places even has physics at a post-graduate level, has no glossary of technical terms. There is ample cross-referencing and an index, but these are no substitute. When you are trying to figure out what Clifford Algebra is, for example, having to go back and read through an entire section again can be quite annoying. A list of definitions at the back of the book would have been enormously helpful. Admittedly, this would have made an already big book even bigger, but it would have made it much more user-friendly.

My second gripe is similar. Penrose fails to supply a list of the upper and lower case Greek alphabet symbols and their names, or a similar list for obscure mathematical symbols like del and scri. Given that nobody without training in Greek or science is going to know these, and such a list would only take up one page, its omission is quite egregious.

This leads neatly into a topic that has been discussed quite a bit on Goodreads - namely, who is this book aimed at and what is its purpose? Firstly, I would point out that the subtitle "A Complete Guide to the Laws of the Universe" is not really accurate. Classical Thermodynamics is barely touched upon as we rush straight into the statistical view of the Second Law. Of the other three Laws of Thermodynamics, Zero is never mentioned and the others are only barely name-checked. I doubt many physicists would consider that all the basic theories have been covered in such a situation. But I think this is more of a marketing problem. I don't believe Penrose ever intended to write such a "Complete Guide."

In the preface, Penrose talks about wanting to make cutting-edge physics available to people who struggle to understand fractions. Now, this can only be taken as a joke, considering what one has to face even in chapter 2. But I would guess that Penrose genuinely wants to have the widest possible audience for his book while not compromising his aims.

So, what are those aims? In my view, he wants to give his personal views on the state of cosmology and fundamental physics, but he wants to do it at an advanced technical and mathematical level. Additionally, he wants to share his own philosophies regarding the nature of thought, science, maths, and nature itself. This means that he wanted to deliver chapters 27 - 33 on the physics/cosmology, bracketed by chapters 1 and 34 of philosophizing. The entirety of the rest of the book is simply there to equip readers to understand what he says in those six technical chapters. This requires 15 chapters of maths and ten chapters of physics/cosmology.

When looked at in this way, the book begins to reflect the genius and madness of the author. Many of the explanations in the earlier stages of the book left me wondering why he did it that way. It didn't seem to be the easiest way to understand the concepts if you had never come across them before. He also goes straight to very general mathematical principles, skipping over intermediate levels of abstraction that might have made what comes later easier. He chooses to emphasize the geometrical/topological view of everything, which, surprisingly, is not always the easiest way to understand things. Many of the choices of what to emphasize and what to ignore seem odd - that is, until you get to the later stages of the book.

Once you reach chapters 27 - 33 (i.e., what I think Penrose really wants to talk about), you can see that everything that has come before has been put together in order to provide the most efficient route to understanding. Hardly a page has been wasted. All those strange choices of what to emphasize, all the peculiar, non-standard explanations when easier ones exist, all the leaps to the most general mathematical ideas, all the things that were missed out - all these things were done so that the points he wants to discuss can be followed without wasting time or space in what is already a 1000+ page book. The necessary skill, thought, and effort required to do this impress me enormously.

Inevitably, this means that most of what is covered in the book of "standard" physics has been explained better elsewhere, even at a mathematical level - but not in one volume. The consequence of this is that Penrose's widest possible audience may not be as wide as he hoped. Although he suggests that one could read the book and ignore every equation in it (something I often do when reading technical literature), I suspect that one would rapidly become bored and disenchanted. The unavoidable fact is that the greater your mathematical capabilities, the more you will get from this book. Additionally, the more maths and physics you know before starting, the more you will gain from it.

Furthermore, the more you are willing to study the book, the more you will gain. Manny approached it by reading 3 hours per night until he was done. I would suggest that the closer you can get to that approach, the better off you will be, even though I failed miserably to do so. There are numerous exercises scattered throughout the book, which I did not attempt. But I would suggest that if you are determined to try them, you should read the remainder of each chapter as soon as you hit a hard problem, and then go back and look at the problems again. (And don't forget to note the solutions web address given in the preface!)

So, what did I gain from the unavoidable slog of this book? The general philosophizing in chapters 1 and 34 struck me as a waste of time. I either thought what was being espoused was obviously nonsense or obviously true - and for me, the questions he raises mostly aren't interesting anymore. (They were back when I hadn't reached my own conclusions yet.) Others, however, may feel very differently - and many would not agree about which parts are nonsense! The remainder of the book, however, offered me quite a bit.

For instance, a frankly embarrassing misunderstanding of the EPR paradox that I was laboring under was corrected! (This was a bit of a body-blow to me as it is undergrad physics.) On the other hand, Penrose makes an astounding mistake at one point, where he gets himself horribly messed up with basic (high school) probability theory and time-reversal. (This gave me quite a good laugh!) This is a good reminder that there is no argument from authority in science: just because Penrose says it, doesn't make it right! This wrong argument is then used to go on to explain a completely freaky (and I suspect wrong) prediction about basic quantum theory. I'm not clear that the example, which is definitely wrong, invalidates his whole line of reasoning, though. It may be that other examples show the general argument to be correct.

Then Penrose delivers the knockout punch: Conservation of energy/momentum/angular momentum in General Relativity is non-local! Not only that, but it has only been proved to be true at all in a subset of cases! Seriously, how could I have never known this before?! (Non-physicists may well have no clue why I am so thunderstruck by this revelation, but it's not far short of learning that there's a whole continent you'd never heard of before.) It's completely gob-smacking. And I can't see how I didn't get told as an undergrad.

Furthermore, Penrose's main purpose was achieved. I have a much better understanding of the main approaches to tackling the outstanding problems in cosmology/fundamental physics than I did before, and along the way, I gained some insights I previously lacked. Two examples are the Higgs boson explanation of the origin of mass and spinors. The Higgs boson theory is barely touched upon and is one of the rare examples of something being included that is not strictly necessary later. I wish there had been more about it, while recognizing precisely why there isn't. What material there is made the theory seem much less arbitrary than it had previously.

Spinors are a mathematical concept that feature heavily in the book, mainly because they feature extremely strongly in Penrose's Twistor Theory of quantum gravitation. Penrose gives an assessment of his own theory that I respect enormously and cannot praise highly enough. He expresses clearly what it can achieve and equally clearly and forthrightly what it cannot. Every weakness and limitation is mentioned and explained. The only time I have previously come across a scientist giving such an honest and complete assessment of the weaknesses of his own theories in a popular account is when I read Charles Darwin's Origin of Species. I cannot express how much respect Penrose earns from me by doing this. Suffice it to say that most popular science books will make out that the author's ideas are obviously and unassailably correct. Further, many technical papers fail to match this level of dispassionate critical assessment.

But back to the spinors; they feature in the now well-established Dirac Equation for a relativistic electron, but the (non-standard) way Penrose shows this and explains their connection with the left-handedness of the Weak nuclear force and how they link to the Higgs boson ideas are fascinating. However, I'm not clear about them in one (crucial) regard: are they real? Penrose says they are. I'm not sure (because my understanding is still muddy), and I find (somewhat to my horror!) that even though I've read all 1050 pages of the main text, all of chapters 2 - 17 twice, and many individual sections several more times, I'm still not done with this book! I have to go back and see if I can make sense out of these spinors. Also, I owe Manny a discussion of Inflationary Cosmology: I'm going to have to read the relevant chapter again in order to provide it.

Wish me luck as I delve back into the very deep waters of this book!

I am finished, finally. All the 1050+ pages of this ambitious behemoth - including many exercises. What a ride!

Finished? Well you are never really done with such a book, titled “The road to reality” but actually offering far more than that. It provides nothing less than a “road-map” to reality, opening up new and beautiful vistas in modern mathematics and physics for the reader. I'm certain that I'll return to this book in the future, both as a source of inspiration and for future reference.

Before I begin, I must admit that it feels a bit absurd for me to critique such a monumental work by a crazy genius like Penrose, who clearly inhabits a higher plane of consciousness than the vast majority of us ordinary mortals. Nevertheless, the best I can do is describe my personal, deeply humbling experience in reading this amazing display of intellectual prowess. Penrose is undoubtedly the Mozart of mathematical physics: a unique intellect combining genius, craziness, wild imagination, and pure technical virtuosity.

Let me start by highlighting some of the peculiarities of this book (which, in my personal opinion, are due in large part to editorial/commercial choices/pressures, and only partially to the author's own personal choices). This is also crucial for the proper positioning of this book within the overcrowded (and of vastly varying quality) world of popular science books:

A) This book is marketed as a science “popularization” (presumably with the aim of attracting as wide an audience as possible). In reality, however, this book requires a significant amount of background knowledge in mathematical physics. In fact, there are some sections where the equivalent of a full maths/physics degree would be needed to fully appreciate all the subtleties of the subjects being covered. Personally, I would recommend the following minimum prerequisites:

- Calculus (including multivariate)

- Ordinary and partial differential equations (at least at a rudimentary level)

- Complex analysis

- Linear algebra

- Basics of topology

- Ideally, differential geometry at a basic level

- In the physics area, some prior exposure to Maxwell's equations, the Lagrangian and Hamiltonian formalisms, the basics of quantum mechanics and relativity (at least special) would also be extremely helpful.

If a reader attempts this book with no prior exposure to any of these topics, I'm afraid it will prove to be a very steep learning curve indeed. I had all of the above prerequisites, and yet it was still, for me, an occasionally demanding, if not challenging, book in terms of the required focus, time, and intellectual stamina. But this is also a strength of the book - one of its most rewarding aspects is that the author doesn't shy away from exposing the reader to real maths (some of the exercises can be quite hardcore), and delving into the heart of modern physical theories such as QFT and GR. If you want the real thing, you simply can't avoid getting into some real maths.

B) This book is not really a textbook, or at least not an ordinary textbook. It lacks the necessary structure and flow, and (too) many important derivations are left as exercises. The depth of mathematical analysis and rigour is uneven, and seems to be highly dependent on the author's sometimes whimsical personal level of interest in the subject. Moreover, the formal treatment of many subjects is highly original and, unless you have had some previous exposure to them, may be confusing and certainly not easily reconcilable with what standard textbooks present. But this is part of the beauty of this masterpiece.

C) This book, contrary to what is stated in the title, is not a “complete” guide to the laws of the Universe. While managing to cram so many sophisticated and fascinating subjects into a single book, and at a serious level of detail (which is no small feat and something quite unique), I think Penrose should have written two books (one on the mathematical underpinnings and a separate one on the physical aspects) rather than trying to cram such a massive amount of information into one single large book.

This approach (admittedly less appealing from a commercial perspective) would have allowed him to expand on some important areas that he unfortunately neglected in his otherwise magnificent book (such as finiteness and re-normalization issues in QFT), and would also have enabled him to include other subject matters as well - for example, the coverage of thermodynamics focuses mainly on the statistical view of the Second Law, neglecting all the other elements.

But make no mistake - this book feels like a \\"War and Peace\\" of mathematical physics – a colossal enterprise that just keeps giving and giving, a treasure trove of original insights, beautiful hidden connections, amazing stories, and revelations. A fantastic and detailed exploration of quantum physics, relativity, the current trends in the attempts to unify the two, and cosmology. An exhilarating intellectual adventure in the company of a crazy genius.

Let me now make a few comments about the author's peculiar personal style and approach:

- Penrose's prose is beautiful, even when considered from a purely literary perspective. It is often very clear, although sometimes a bit too concise. It usually flows very smoothly, and occasionally even takes on some poetical undertones in sections where aspects of the philosophy of science and the philosophy of mathematics are treated with insightful passion.

- Penrose relies heavily on a diagrammatical/pictorial/visual/topological representation of the concepts being covered. In doing so, many beautiful, eye-catching, and highly informative hand-drawn diagrams/pictures are presented to the user. This approach generally works very well, but there are instances where (in my personal opinion) the most intuitive and simplest way to learn/teach a new concept is to present it analytically (in mathematical format), and where the purely visual approach may be quite limiting if not outright confusing. For example, the visual explanation of the concept of “covariant derivative”: while important and helpful, is in my opinion not sufficient on its own to gain a real appreciation of the mathematical features of this entity. Its analytical derivation, and its expression using the Christoffel symbols, would actually have better clarified its tensorial nature.

- In order to condense as many subjects as possible within the constraints of the space allowed by one single book (even one of 1050+ pages), Penrose has left the derivation of many important results as “exercises”. The exercises are therefore very important; they are essential for a full appreciation of the underlying theory, and some of them are beautiful and rewarding (like exercise 22.32, which involves a beautifully elegant derivation of the Laplacian in spherical coordinates, using the curved metric and covariant derivatives). But some others should not have been left as exercises, and they seem more like an editorial cop-out. There is one exercise that even asks the reader to complete a significant derivation step of the GR field equations! (Thankfully, there is a website where many exercises have been solved by other readers (the majority of them clearly with a professional background in mathematical physics) – see https://sites.google.com/site/vascopr...).

- I love Penrose's great intellectual honesty in not only debunking much of the hype behind String Theory and Multiverse Hypotheses in general, and in destroying the so-called “Strong Anthropic Principle”, but also in making very clear the speculative character of some of his own positions and theories. I also love his great originality and independence of mind, and his nuanced, multi-disciplinary approach that includes aspects of the philosophy of science and considerations of a pure mathematical nature.

- Penrose has reached such a high, rarefied level of proficiency in mathematical physics that he must have completely forgotten how common humans think in relation to mathematical formalism. Just as an example: after delving into pretty advanced stuff such as hyperfunctions and treating it as if it were the simplest thing on Earth, the author then refuses to get into a detailed definition of the differentiability of a many-variable function (which is actually quite simple) because \\"it is too technical\\"!! So his ideas of what is complex can sometimes be significantly at odds with what the common mortal might think.

Finally, let me add here some miscellaneous notes about selected individual chapters of this majestic book (it is a very incomplete list, only a severely reduced sample, as there is simply too much material in this book for me to be able to analyze in just one review):

- In the first few chapters, after a fascinating introduction of an overall philosophical character, Penrose briefly addresses some of the basic fundamentals of mathematics, including a short but intensely interesting discussion about the Axiom of Choice and its relationship to the Zermelo–Fraenkel set theory. A totally fascinating subject that unfortunately Penrose only touches on, without developing it into more detail.

There is also a fascinating treatment of hyperbolic geometry and the number system/s. However, when dealing with the relationship between real numbers and reality, the author does not make any reference to important aspects such as computability and irreducible complexity, as for example addressed by the excellent work done by G.Chaitin.

- The author then delves into one of the most beautiful and fascinating realms of mathematics - complex analysis. The treatment is concise but well done, and I completely share Penrose's enthusiasm and love for the aesthetically as well as functionally beautiful world of complex numbers, which he calls “magical” with good reason.

- Another subject of interest covered is the fascinating quaternions, which extend the complex numbers and provide the uncommon and interesting features of a non-commutative division algebra. There are a couple of minor omissions in the book regarding the algebraic structure of quaternions: an algebra is a vector space that must also be equipped with a bilinear product, and a ring (being also an abelian group under addition) must be provided with an additive inverse.

Moreover, Penrose is dismissive of the utility of quaternions in the development of physical theories – this is correct, but it must also be said that quaternions find important uses in information technology applications, particularly for calculations involving three-dimensional rotations, computer graphics, and computer vision.

- Penrose could not have forgotten the extremely important Clifford/Grassmann algebras, which are fundamental to the entire architectural structure of mathematics, and he didn't. They are treated well, but a bit too succinctly in my opinion, considering their importance.

- Symmetry groups are treated really well, clearly and concisely. Very nice.

- Manifolds and calculus on manifolds: all is treated quite well, but I would have found it very helpful to have some more analytical detail rather than an almost exclusive focus on the visual/topological approach. I confess that here, in order to gain a proper detailed understanding of tensor calculus and exterior calculus, I had to consult other more traditional, “textbook-type” sources.

- Chapter 16 (Cantor's infinities, continuum hypothesis, Godel's incompleteness, Turing computability, and similar) is okay and well written, and it would be utterly fascinating to a novice – but of course it could not have been detailed or exhaustive, considering the complexity and breadth of such subjects.

- The “Physics” part proper starts with Chapter 17. Chapter 17 on spacetime, and chapter 18 on Minkowskian geometry (and special relativity, of course) are succinct but riveting and beautifully written. Just one small note: Penrose has a penchant for non-standard, or uncommon, notational or representational choices, not always justified. For example, by using a non-standard metric tensor on page 434, Penrose makes an error at the end of page 435 (c^4 should have been used rather than c^2), and I think he would not have made this typo had he used the more standard and easy-to-use metric.

- Chapter 19 (Maxwell and Einstein) are beautiful. Maxwell's equations and Einstein's field equations in tensorial/differential form are mind-blowing for their mathematical conciseness and beauty, and this is where the reader can start to see the value of the mathematical apparatus described in the previous chapters: things such as manifolds, tensors, exterior derivatives, and bundle connections. Einstein's field equations of general relativity are beautiful. Very rewarding. It all seems so neat and perfect, but then, the bombshell (at least for me, who was never told of such a thing!): the energy/momentum stress tensor does not account for the energy density of the gravitational field itself, and it seems that conservation of energy/momentum is non-local!

- Chapter 20: Lagrangians and Hamiltonians: I must say that they are discussed too hastily. The derivation of the Euler–Lagrange equation (one of the most classic proofs in mathematics) is not covered, the Legrende transform to derive the Hamiltonian from the Lagrangian is not covered, and the derivation of Hamilton's evolution equations is not covered either. I'm not very happy with this chapter.

- The subsequent chapters on quantum mechanics are beautiful, and the description of the EPR issue is really nice. The derivation of Dirac's equations is equally beautiful. I also love Penrose's discussion and perspectives on the measurement paradox. Beautiful stuff indeed. An exception to the overall masterful treatment of quantum physics is chapter 26 (Quantum Field Theory), which is a bit too qualitative - more like a \\"popularization\\" rather than a treatment at the same good level as in the other chapters of this book. Unfortunately, the important subject of re-normalization is treated only briefly and qualitatively.

- The chapters on cosmology are very interesting and nicely written.

- Chapter 29 is essentially about the measurement paradox and the various interpretations of quantum mechanics – succinctly but beautifully written.

- The next chapters are essentially about the current attempts at unification to reconcile QFT with GR. Penrose's FELIX experiment proposal (essentially testing the hypothesis of automatic quantum state-reduction as an objective gravitational effect) is utterly fascinating, but I feel that I shouldn't bet any money on this bold hypothesis.

- I must say, however, that I was disappointed by the second part of chapter 33 (twistor theory). In section 33.8, Penrose uses several wordy sentences to express mathematical relationships rather than writing down the actual underlying equations - the result is something quite close to incomprehensible. I also fail to grasp the underlying physical intuition. Pity, as the other 33 chapters and a half are mostly beautifully written and generally very rewarding.

- The last chapter (34) is beautifully written and deals primarily with aspects of the philosophy of science and the philosophy of mathematics.

To summarize: this is an immensely rewarding, even exhilarating book - a fantastic reading and learning experience. It has opened my eyes to many aspects and connections that I was not aware of, and new beautiful vistas in modern mathematics (for example: tensor calculus and exterior calculus in general) that I will now pursue in more depth.

After reading this great book, I can tell you that I now feel that the majority of popular science books I have read are dull and superficial by comparison.

It is not a perfect book by any means, but overall it is a great book, unique in its approach and contents. Not exhaustive, but with a huge and ambitious scope, virtually unrivaled in its category. It is much more than a “standard” popular science book. Very highly recommended, to be bought, studied, enjoyed, and kept for future reference.

4.5 stars (rounded up to 5).

UPDATE: with a link to a beautiful series of online lectures about Tensor Calculus and the Calculus of Moving Surfaces: https://www.youtube.com/playlist?list... (highly recommended to anyone interested in this subject).

The story commences with me in the garage, accompanied by my high school sweetheart. He was the one who, through his guileless intuition, made the playground swing problem seem insignificant, in contrast to my cynical calculations that were in a hideous antinomy. An actor with a relaxed demeanor and gentle earnestness, he effortlessly unravelled the kinks in those rusty chains, and in doing so, tied intractable knots in my own heart. It was a union of opposites, with me chin in hand and he with a wrench. There was no problem we couldn't troubleshoot together. Our separate talents even began to cross-pollinate, with me sometimes looking up at the undercarriage of a vehicle and asking for a socket while he perused the theoretical oddities I was obsessed with. I could, with confidence, tell people that their alternator was misbehaving, their radiator was acting up, their starter motor was being difficult, or my favorite, "Your fuel economy is terrible, bruv." And he could clearly understand the value of switching his choice in the Monty Hall Problem, know that some infinities are larger than others, and comprehend the threat a quantum computer poses to our current encryption protocols.

This exchange of mental dexterity continued on a rainy day. I slid under a car on a rickety creeper (which I had damaged by using it to luge down steep driveways at high speeds, causing one wheel to catch and become misshapen), and Nicolas Kim Coppola (known professionally as Nicolas Cage) flipped through the book I had brought in. He asked me about "The Road to Reality." I paraphrased and said the following: Have you ever, while skidding down a vertiginous concrete pathway on a damaged creeper, felt the fixed vibrational frequency of healthy rotary friction give way to dissonant melodies that often预示着 the rapid, explosive disassembly of machinery under immense kinetic strain? Was all the blood in your superficial capillaries squeezed out from clutching the sides of the sled in mortal terror, causing your knuckles to emerge from the dark like buds of white hibiscus and your weak arms to ripple like corded steel? Did your sweetheart, who you suspect must have reduced activity in his amygdala (due to his performance in "Ghost Rider: Spirit of Vengeance"), leap from their stool and try to halt your (by now considerable) momentum using a bag of blood represented by sticks of calcium (i.e., the vertebral column covered in viscera and thinly veiled by contractile tissue that we call the human body)? During those tense moments, you may have thought, in the affected High English characteristic of all your poignant inner musings: Nicolas Cage, it's a pity I can't communicate with you more effectively. That our bandwidth is limited by verbal exchanges in which I must compress thoughts into packets of information and utter them as faithfully as I can, hoping you will receive and unpack them with minimal noise during the transfer. Putting aside incidents where my limbs have moved like an excited Neil deGrasse Tyson, this is the extent of my ability to let you experience what it's like to be me. I can't understand your genius for seeing the good in every screenplay, and you can't fathom the depths of my genius when it comes to nearly dying in bizarre ways. In those simple displays of affection where nothing is said, perhaps we say the most.

And what about inhabiting the mind of a true genius, and more difficult still, the mind of a brilliant theoretical physicist and math prodigy? With art, something primal touches the soul, so that even if you don't understand the creative flourishes of virtuosos, you are moved and aware of their power. Who could read Nabokov and not be humbled by its beauty and precision? Who could hear "Hangar 18" by Megadeth and not respect how a guitarist's flanges dance across wires like the legs of camel spiders on a heroic dose of methamphetamines? Who could see the "Venus of Urbino" by Titian and not want to bury their face in her gorgeous tummy and blow a gentle raspberry? And finally, who could hear Anton Chigurh request a "screwgie" in "No Country for Old Men" and not be gripped by palpable dread? But here, the scientific genius faces a problem. When elucidating the purely conceptual matters of their field to the general public, they must conceal much of their brilliance. If they were to write like Dave Mustaine and fill pages with the lateral movements of their mutant intellects, the majority of us could only shrug in the face of these strange symbols. But imagine if there were a book that managed the impossible. One that was accessible to the layman audience but omitted none of the technical jargon that made it possible to appreciate the operations of their brilliant minds? Perfectly comprehensible to curious high school freshmen. What you hold in your hands, Nicolas Cage..." I slid out from under the car and stood next to him. "This.." I said, taking the book gently in my hands. "Is NOT that fucking book!" I threw it viciously to the ground and jabbed my finger at it. "THIS is madness, Nicolas Cage! This is fucking popsci for math and physics PHDs! Calculus, Fourier series, hyper functions, Riemann surfaces, Riemann mapping theorem, quaternions, Clifford and Grassmann algebras, transformation groups, Lie algebras, parallel transport, geodesics, curvature, exterior derivatives, calculus of manifolds, connections, fibre bundles.. In the first third of a book which, in its preface, indicates he wrote it for his innumerate aunt. Show me the numerically deficient aunt who made it through this with even the vaguest idea of what the hell was going on, and I will crawl inside myself and subsist on my own incredulity until the stars grow cold."

Nicolas Cage, picking up the book and carefully inspecting it for damage, handed it back to me, saying, "It's fine. You'll understand it eventually. I know you will." I begrudgingly put the book under my arm and nodded. I felt a rebellious grin break the severity of my expression. I silently cursed his ability to calm my tantrums so easily. I seized this newfound lightness to suggest we take the creeper out for some limit testing. "Wait!" I can now appreciate this book a lot. If you have a strong background in math and physics and are interested, I can't recommend it highly enough. It may be the most wide-ranging book on theoretical physics I've ever seen, not just summarizing material but offering an in-depth treatment of many of the more sophisticated ideas in the subject. The first half of the book takes you through most of the mathematical techniques physicists use today, gradually covering things like Riemann surfaces and complex mapping, hypercomplex numbers, manifolds of n dimensions, symplectic groups, and tensors. Equations are curated magnificently based on explanatory power. And here I would add that Penrose, perhaps due to his genius, has a level of idiosyncrasy in his presentation that I don't think I've seen since Feynman. It's quite obvious that what goes on in his brain is an intensely visual phenomenon. All of his diagrams and drawings are incredibly helpful, especially if you think in this way too. The second half of the book is dedicated to physics, and the ascent is quite dizzying as we approach the various hopeful theoretical bridges we're constructing to reconcile the discontinuities between our two most successful theories (general relativity and quantum field theory). So you have discussions (it may be more appropriate, considering the sheer rhetorical force with which Roger throws these ideas at the wall like wet pieces of bologna, to call it evisceration) of String Theory, M Theory, Loop Quantum Gravity, and Penrose's favorite: Twistor Theory. If you don't have a background in these things and don't pursue high-level mathematics recreationally, I wouldn't recommend it to you as a comprehensive starting point. If you just want to keep it around and flip through it to occasionally enhance your understanding with some clever illustrations, or you simply want to marvel at it as an enduring artifact of the human intellect, by all means. But if you're planning to work through it from front to back, caveat emptor. This is one of those rare gems that has brought me very close to experiencing the intuitive ease with which a mathematical genius operates. A beautiful work of staggering breadth and depth.