What laws govern our universe? Modern physics doesn't give a unified answer for this question, but only give partial answers. That is, we have two fundamental theories explaining our universe, relativity and quantum mechanics that conflict in some situations. There are theories like string theory which claims that it might be able to give a unified theory (as physicists say, the final theory). Among them is twistor theory formulated by the author, Roger Penrose in 1950s. As the inventor of twistor theory, he shows to readers how mankind has answered to the question from the ancient times until the present.<br><br>Several distinguished features of this book include:<br><br>1. more than 1000 pages with neither typos nor grammatical errors.<br><br>2. almost all major roads to final theory: string theory, loop variables, twistor theory, non-commutative geometry. For descriptions, the book deals with classical mechanics, relativity, quantum mechanics, and chaos theory from the basics. And also it deals with classical and modern mathematics: irrational numbers, Euclidean geometry, hyperbolic geometry, projective geometry, real number calculus, complex number calculus, Riemann surfaces, Fourier decomposition, generalized functions, Clifford algebra, Grassmann algebra, vector fields on manifolds, Riemannian geometry, exterior derivative, Lie groups, Lie algebras, connections, fibre bundle theory, Cantor's set theory, Minkowskian geometry, Lorentz geometry, sheaf theory, and tensors. But in spirit, this book is for general audience.<br><br>What is the most important for a reader? I think it is how much he learned from the reading. In this aspect, I could not give five stars. To finish the book, I spent almost two months. Of course, I learned a quite amount and it was a valuable time. But I think what I gained is just 35 percent of what the book contains. Comparing with Brian Green's popular books, the book is the next or the next-next level of a book. The difficulty was in concepts relating to relativity, in particular, tensors. In the book, the tensor notation is universal. Even the Maxwell equation is expressed in the tensor notation. I graduated in physics department and have a doctoral degree in mathematics. But I've never studied general relativity and its related Lorenz and Minkowskian geometry, and tensor notations. And I've never studied quantum field theory and its related tensor formalism as well.<br><br>If you think that the book is difficult for you also, I would like to give some tips. They are all related to skipping.<br><br>1. As the author says, skip equations and difficult parts if you don't want to read it sometimes the whole chapter. If you realize that the part is important to understand the main stream of the book, you can always go back to that part when necessary. At page 74, the author says,<br><br>My advice to such readers is basically just to read the words and not to bother too much about trying to understand the equations.<br><br>2. This book is not a textbook. If you want to learn relativity or tensor calculus or quantum physics, then you are referred to standard texts or Youtube lectures. You should not try to learn such subjects from this book. So when you meet some parts dealing with such a sophisticated level physics and think that it's too difficult, you should skip it without any regret.<br><br>3. As I said, in spirit, the book is for general audience. But if one can understand more than a half of the book, then I think that he would be at least a graduate student studying quantum field theory and relativity. If you are interested in the question, what laws govern our universe, you are entitled to read the book. But actually, if you are not already familiar with quantum mechanics at least at the level of popular science books, then it would be extremely hard for you to read. You have to make clever choices about what to read if you don't want to spend time frustrated.<br><br>4. Its style is informal and narrative, but in some parts, it is very dense. For example, the Newton mechanics is summarized only in three pages. After the section, the author assumes that you have mastered the Newton mechanics!<br><br>Now I want to share my detailed appreciation.<br><br>1. If your major is related to science, then among many curriculum subjects, the linear algebra would be the most helpful to read this book, such as, basis, eigenvalue, linear transformations and matrices, basis change, dimension of a vector space. And if you know what a phase space is for a dynamical system, then it would be very helpful (Search the Wiki). And if your major is mathematics, I have something more to say. I've read the differential geometry book by O'Neil, Calculus on Manifolds by Spivak and studied one-semester courses of differential topology and Riemannian geometry. So I am familiar with concepts like curvature, 1-form, integration on forms, exterior derivative, Poincare lemma. But that was not so helpful to understand relativity and tensors in the book. Everybody who is interested in the subject knows that they are related, but I think that they live somewhat in different area.<br><br>2. The book is so concise that sometimes you can't understand what the author says. For example, I think that the Mach-Zehnder interferometer at page 514 cannot be understood only by the explanations of the book if the reader does not already know it. And for many extremely important experiments including EPR-experiment, the book describes them so briefly that if you are not already familiar with them, you may have difficulty to understand them. And as for quantum entanglement, it is a really amazing phenomenon of quantum mechanics. But if you didn't already know it, then you may not fully understand it only with this book. As one more example of physics part, while I read the book, I come to know that there is a projective postulate in quantum mechanics that seems to be a very important issue in the book. But I couldn't understand it even though I tried to read the related parts several times. There are such things on the mathematical part. First of all, although there are explanations about tensors, if you are not already familiar with it, you would have a rare chance of understanding it. As the second instance, at the extremely interesting section on covariant derivative on a fibre bundle (Section 15.8), the explanation is not sufficient for actually calculating the example of A=ik the conjugate of z. As another instance, in the sections on complex numbers, we see some logically vague explanations. I found that the author didn't explain the fact that if two holomorphic functions on a domain D coincide on a continuous set (in a sense), then they coincide on the domain D.<br><br>3. There seem to be unsatisfactory explanations. At section 14.3, introducing covariant derivative, the author introduces the concept of parallel transport. But there is some ambiguity whether covariant derivative is derived from the parallel transport or the converse is true (this case is true in mathematical literature). And at section 21.4, it explains the Blackbody radiation. Wien's formula was already there giving insufficient interpretation of Blackbody radiation and several years after Wien, Planck succeeded in explaining Blackbody radiation with introducing his Planck constant. If so, it is absurd that the Planck constant appears in the Wien's formula.<br><br>4. If you are a mathematician, I strongly recommend that you read sections on analytic continuation, hyper-functions, Fourier decomposition, fibre bundles. They are worth reading in the aspect that the book explains to readers the geometric meaning. Once you read them, you will not forget it for a long time. For example, we know that a conformal map is an angle preserving map. The author says that a conformal map preserves shape locally. Maybe this is a common sense to many researchers, but to me it was an astonishing insight. There are many things like that in the book. And there are some differences of point of view to mathematicians like me. For example, in Chapter 5, the author explains complex numbers from the basics, and in Chapter 13, symmetry groups. I thought I knew them. But the way he describes them seems to be strange. As for complex numbers, I skipped some parts and made a decision to retain my understanding. As for symmetry groups, I thought I have to learn more.<br><br>5. Until the author introduces the generalized function, he asks us what function is. What definition of a function can be satisfactory in theoretical meaning and in practical applications? In fact, I used to ask the same question also, although I was not so explicit. I think anyone who studied mathematics for several years may have conceived the question. His argument is very interesting and really thought-provoking. More than that, he gives an explicit answer about the question.<br><br>6. Diagrammatic notations and conformal diagrams don't seem to be helpful for non-specialists.<br><br>7. While reading through the book, I hoped that I could understand the following sentence.<br><br>... according to modern physics, all physical interactions are governed by 'gauge connections' which, technically, depend crucially on spaces having exact symmetries. (page 289)<br><br>But even now after finishing the book, I still don't understand what the above sentence means concretely. My future goal would be to understand it.<br><br>8. What are the merits of the book? Does the book have a merit that other books do not have? I think it has. The author has no hesitation in expressing his explicit opinion about major current theories.<br><br>Quantum field theory - mathematically inconsistent<br><br>Inflation cosmology - suspicious<br><br>String theory - Doubtful, especially due to its higher-dimensional spacetime. String theory regards spacetime as continuum but the author seems to believe that ultimately, spacetime also should be quantized.<br><br>Loop variables - At this stage, it is far from being quantum gravity theory.<br><br>Non-commutative geometry - The model does not incorporate special and general relativity.<br><br>Quantum group - There is no very clear relation between a quantum group and quantum theory.<br><br>Topological quantum field theory - It is hard to see them playing direct roles as models of serious physical theories.<br><br>Another merit of the book is that it gave me a motivation to study relativity, tensors, and quantum field theory by showing a big picture of modern attempts to final theory. I am a group theorist, and I want to meet more various groups in physics. And I am interested in the question: what laws govern our universe? This is the motivation that I chose the book. I hope that I learn more about the area. As remarked above, ultimately, the spacetime also seems that it should have a discrete property at extremely small scale. So some scientists suggested a discrete number system other than real numbers. I thought that's the right way! But the author suggests that rather than a discrete number system, complex number system would be the right number system. Now I think that it is possible that to describe a discrete object, we may use a continuum like the complex number system.<br><br>As a whole, the reading was valuable. While I read this book, I received the impression that the author is a very gentle, sincere, honest, kind, careful, and friendly person. Even though I was not totally satisfied with the book, I come to respect his attitude as a scholar.