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# Universe-scale scope Comprehensive physics guide Advanced math integration The Road to Reality: A Complete Guide to the Laws of the Universe (Vintage)

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## Summary

> 🌟 Unlock the cosmos — one equation at a time!

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- **What is this?** The Road to Reality: A Complete Guide to the Laws of the Universe (Vintage)
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## Key Features

- • **Master the Universe’s Blueprint:** Dive deep into the mathematical laws shaping reality, from quantum particles to cosmic expanses.
- • **Math Meets Physics Like Never Before:** Explore calculus, differential geometry, Lie groups, and tensor analysis seamlessly woven into physics.
- • **Unlock a Gateway to Scientific Mastery:** A unique blend of popular science and rigorous math that opens doors to advanced physics literature.
- • **Challenge Your Intellect, Join the Elite:** Not for the faint-hearted—this book demands persistence and rewards with profound understanding.
- • **From Ancient Wisdom to Cutting-Edge Science:** Trace the evolution of physics from Greek foundations to modern theories including relativity and quantum mechanics.

## Overview

Roger Penrose’s 'The Road to Reality' is a definitive, richly detailed guide to the laws governing the universe, blending advanced mathematics with physics. It covers foundational concepts from classical geometry to quantum mechanics and general relativity, designed for readers ready to engage deeply with the science behind reality. Highly acclaimed and ranked top in Mathematical Physics, this book is a must-have for ambitious minds seeking to transcend popular science and truly understand the universe’s fabric.

## Description

Nobel Prize-winner Roger Penrose, one of the most accomplished scientists of our time, presents the only comprehensive—and comprehensible—account of the physics of the universe. A "guide to physics’ big picture, and to the thoughts of one of the world’s most original thinkers.”— The New York Times From the very first attempts by the Greeks to grapple with the complexities of our known world to the latest application of infinity in physics, The Road to Reality carefully explores the movement of the smallest atomic particles and reaches into the vastness of intergalactic space. Here, Penrose examines the mathematical foundations of the physical universe, exposing the underlying beauty of physics and giving us one the most important works in modern science writing.

Review: Likely will become one of the Great Books: Buy it! - The short version: If you are reading reviews in consideration of purchasing this book then just BUY IT. It has long been my wish for someone to a write a popular treatment of modern physics, one which includes the math, starts at the beginning, and then covers whatever is needed so that the reader can understand the theories described. For me, reading physics is fun; it is not a path to "becoming a physicist", but I want something beyond the popular science level. My goal is to become a READER of real physics, and I am willing to work to reach this goal. This wish describes "The Road to Reality" almost perfectly. Penrose literally intends to take the reader from basic math through calculus, and on to field theory, Lie Groups/Algebras, calculus of variations (Lagrangians & Hamiltonians), differential geometry with fiber bundles, and tensor analysis. He plans and prepares to explain both quantum physics and general relativity (gravity.) This book is both a popular science guide and introductory mathematics text (including introductions to advance subjects) at the same time. The book is a wondrous delight, while simultaneously being maddening for its flaws. If there were there a thousand similar books, it would be easy to criticise the flaws. The writing is at times simply awful (the worst and most common offense is 'pre-shadowing' for no useful purpose -- and without clearly warning the perhaps already struggling reader). Much of the math details are simple skipped or hand-waved, but the outline and structure provided for mathematical physics is both useful and significant With great persistence by the reader it is understandable. One reads this book both for what it contains, and also for the gateways it will open to other books. The book likely deserves only four stars, but due to it's unique nature I awarded the fifth as a reward for attempting and coming very close to what most would consider impossible. On the other hand, any criticism that it is "incomplete" (the subtitle says the "complete guide to the laws of the universe") is unrealistic and similar to criticizing a "complete guide to Europe" or "a complete guide to fishing" for not listing every hotel or restaurant, or for not including a picture of every fish and a map of every body of water. 'Complete' here means comprehensive and full in coverage and scope, not that every detail is specified. As to criticisms concerning Penrose's idiosyncratic views on physics, he is ABSOLUTELY clear when stating a personal opinion, or covering topics from his own point of view. His own less popular ideas for final theories in physics are a very small portion of the entire book. Pensore clearly delineates his own ideas whenever he mentions them in other sections. [A little about me, but only as a point of reference, might help you evaluate this review since those with significant college math and physics or those with no background in these subjects will approach this book differently: My prior background only includes high school calculus and physics, though I've read many popular physics titles. At the start, I was mathematically naive at the university level, but I was also completely undaunted by the prospect of learning the math and physics.] If you buy this book [highly recommend you do] just read it. Promise yourself that you will keep reading; determine to force your way through no matter what obstacles you encounter. If you have an interest in physics the rewards are immense. Using the book as a tour guide, outline, overview, and foundation you can find resources freely available on the Internet, or available for sale here on desertcart, to actually LEARN to READ physics. You should not expect to "become a physicist" without much more study, but you can develop a reading knowledge of the subject beyond the popular treatments, including the mathematics of tensor analysis, differential geometry, and group theory. An encouragement and warning to young people interested in Physics and Math (as well as those who might buy this book for them) is warranted: If you really want to read this book and work very hard it is possible, but forcing yourself (or being forced) to read it before you have either significant mathematicsal knownlodge and/or the ability to study and develope such know on your own is not a good idea. This book could convince the beginning student of physics or math that these subjects are more difficult than they actually are. Instead they are rather more like any significant skills: they takes some ability, some time to develop, and above all they require careful and persistent work on your part. Currently (three months after starting), I have finished the book (took two months for this) and also reached a rough, reading competence with advanced calculus, differential equations, lie groups/algebras, complex analysis, Lagrangians & Hamiltonians, and can now read introductory quantum mechanics texts and papers. Since reading this book, I have made a good start on Tensor Analysis and Differential Geometry. My estimate is about one year for me to fully understand the book and its topics, but the effort is well worth the results already. Even though, I have sought and used many other sources to improve my understanding, my successes are directly due to the incredible foundation provided by Penrose. In addition, I highly recommend "Deep Down Things" by Schumm, who is much more clear, but less mathematical, on Lie Groups and Gauge Theory. Schumm relates Lie theory directly to Gauge Symmetries, going beyond mere hand-waving while still remaining mathematically simple and clear. I further recommend "Understanding Quantum Physics" by Morrison which offers a much better guided, and step-by-step, introduction to the mathematics and postulates of Quantum Mechanics. (Only real criticism of Morrison is that there are NO solutions for exercise, but he does work many other problems in detail.) Although I bought Morrison's book several years ago and was unable/unwilling to read it, I can now read this one comfortably -- it's not a novel, but it is no longer a fight to read. Neither of these excellent books offers the scope of Penrose however, so read "The Road to Reality" first. (I might have missed the beauty of Schumm's treatment of Lie groups and Gauge theory had I encounted it first.) I am also working through "Quantum Mechanics Demystified" and "Relativity Demystified" both by David McMahon, and "A First Course in General Relativity" by Bernard F. Schutz. [Five months after starting Penrose's book, I now feel comfortable in reading the very imposing "Gravitation" by Misner, Thorne, Wheeler (MTW).] When I have finished these, my plan is to read "Quantum Field Theory in a Nutshell" by A. Zee and move on to Zwiebach's String Theory book. Notice that if you don't have the background in math or physics then this book is going to lead you to reading many others and learning many new topics. This truly great book doesn't end the journey but rather opens new worlds and capabilities for the interested reader. If you are asking "Should I buy it?", then: Yes, JUST BUY IT. If you do buy it, then JUST READ IT. No matter how long it takes you or how difficult it seems at time just keep reading.... You will be delighted to finish this book, and disappointed that it ends -- expect both emotions at the same time. Thank you Roger Penrose!
Review: A stupendous piece of work for understanding both the physical world and the stock market!!! - I spent about 4 months, on and off, and finally finished reading this great book. I have a dual purpose: (a) I wanted to quickly recover my knowledge in math and physics I acquired during my prior physicist career, and (b) I wanted to see if I could apply anything I learnt from here to the machine trading models as introduced in the book "Forecasting and Timing Markets: A Quantitative Approach." I really enjoyed this book and indeed found a lot of similarities and dissimilarities between building mathematical models for interpreting the real physical world and building models for forecasting market. Here is a summary of what I have found out to be very applicable and useful: (1) p.7: What laws govern our universe? How shall we know them? How may this knowledge help us to comprehend the world any hence guide its actions to our advantage? ... Eventually, even the much more complicated apparent motions of the planets began to yield up their secrets, revealing an immerse underlying precision and regularity. (2) p.18: Fig. 1.3 Three 'worlds' - the Platonic mathematical, the physical, and the mental - and the three profound mysteries in the connections between them. ... everything in the physical universe is indeed governed in completely precise detail by mathematical principles. ... all actions in the universe could be entirely subject to mathematical laws. (3) p.28: Euclid's first postulate effectively asserts that there is a (unique) straight line segment connecting any two points. His second postulate asserts the unlimited (continuous) extendibility of any straight line segment. His third postulate asserts the existence of a circle with any centre and with any value for its radius. Finally, his fourth postulate asserts the equality of all right angles. (4) p.45: We are to think of a light, straight, stiff rod, at one end P of which is attached a heavy point-like weight, and the other end R moves along the asymptote. (5) p.67: The system of complex numbers is an even more striking instance of the convergence between mathematical ideas and the deeper workings of the physical universe. (6) p. 109: What about the places where the second derivative f''(x) meets the x-axis? These occur where the curvature of f(x) vanishes. In general, these points are where the direction in which the curve y = f(x) 'bends' changes from one side to the other, at a place called a point of inflection. (7) p. 115: Armed with these few rules (and loads and loads of practice), one can become an 'expert' at differentiation without needing to have much in the way of actual understanding of why the rules work! This is the power of a good calculus. (8) p.151: Air, of course, consists of enormous numbers of individual fundamental particles (in fact, about 10^20 of them in a cubic centimeter), so airflow is something whose macroscopic description involves a considerable amount of averaging and approximation. There is no reason to expect that the mathematical equations of aerodynamics should reflect a great deal of the mathematics that is deeply involved in the physical laws that govern those individual particles. (9) p. 211: It seems that Nature assigns a different role to each of these two reduced spin-spaces, and it is through this fact physical processes that are reflection non-invariant can emerge. It was, indeed, one of the most striking unprecedented discoveries of 20th-century physics (theoretically predicted by Chen Ning Yang and Tsung Dao Lee, and experimentally confirmed by Chien-Shiung Wu and her group, in 1957) that there are actually fundamental processes in Nature which do not occur in their mirror-reflected form. (10) p. 217: For example, the configuration space of an ordinary rigid body in Euclidean 3-space is a non-Euclidean 6-manifold. (11) p.223: As in Sec 10.2, we have the notion of a smooth function (Phi), defined on manifold M. (12) p.388: He (Newton) had originally proposed five (or six) laws, law 4 of which was indeed the Galilean principle, but later he simplified them, in his published Principia, to the three 'Newton's laws that we are now familiar with. (13) p. 390: It is remarkable that, from just these simple ingredients (Newton's formula GmM/r^2), a theory of extraordinary power and versatility arises, which can be used with great accuracy to describe the behavior of macroscopic bodies (and, for most basic considerations, submicroscopic particles also), so long as their speeds are significantly less than that of light., (14) p. 392: Galileo's insight does not apply to electric forces; it is a particular feature of gravity alone. (15) p. 410: We shall also begin to witness the extraordinary power, beauty, and accuracy of Einstein's revolutionary theory. (16) p. 412: The geometries of Euclidean 2-space and 3-space are very familiar to us. Moreover, the generalization to a 4-dimensional Euclidean geometry E^4 is not difficult to make in principle, although it is not something for which 'visual intuition' can be appealed to. (17) p. 455: Einstein's famous equation E = mc^2 tells us that mass and energy are basically the same thing and, as Newton had already informed us, it is mass that is the source of gravitation. (18) p. 462: Einstein originally introduced this extra term, in order to have the possibility of a static spatially closed universe on the cosmological scale. But when it became clear, from Edwin Hubble's observations in 1929, that the universe is expanding, and therefore not static, Einstein withdrew his support for this cosmological constant, asserting that it had been 'his greatest mistake' (perhaps because he might otherwise have predicted the expansion of the universe!). Nevertheless, ideas once put forward do not necessarily go away easily. The cosmological constant has hovered in the background of cosmological theory ever since Einstein first put it forward, causing worry to some and solace to others. Very recently, observations of distant supernovae have had most theorists to re-introduce /\ (greek lambda), or something similar, referred to as 'dark energy', as a way of making these observations consistent with other perceived requirements. (19) p. 466: The timing of these signals is so precise, and the system itself so 'clean', that comparison between observation and theoretical expectation provides a confirmation of Einstein's general relativity to about one part in 10^14, an accuracy unprecedented in the scientific comparison between the observation of a particular system and theory. (20) p.490: He (Hilbert) appears to have believed that his total Lagrangian gives us what we would now refer to as a 'theory of everything'. (21) p. 503: ... it took many years for Einstein's original lonely insights to become accepted. (22) p. 523: Heisenberg's uncertainty relation tells us that the product of these two spreads cannot be smaller than the order of Planck's constant, and we have Delta-p Delat-x >= h_bar / 2. (23) p. 528: I denote Schrodinger evolution by U and state reduction by R. This alternation between these two completely different-looking procedures would appear to be a distinctly odd type of way for a universe to behave! (24) p. 541: As the state of the arts stands, one can either be decidedly sloppy about such mathematical niceties and even pretend that position states and momentum states are actually states, or else spend the whole time insisting on getting the mathematics right, in which case there is a contrasting danger of getting trapped in a 'rigour mortis.' (25) p.686 (Chapter 27 The Big Bang and its Thermodynamic Legacy): What sorts of laws shape the universe with all its contents? The answer provided by practically all successful physical theories, from the time of Galileo onwards, would be given in the form of a dynamics - that is, a specification of how a physical system will develop with time, given the physical state of of the system at one particular time. These theories do not tell us what the world is like; they say, instead: 'if the world was like such-and-such at one time, then it will be like so-and-so at some later time'. (26) p.687: The usual way of thinking about how these dynamical laws act is that it is the choice of initial conditions that determines which particular realization of the dynamics happens to occur. Normally, one thinks in terms of systems evolving into the future, from data specified in the last, where the particular evolution that takes place is determined by differential equations. (27) p.689: What about evolution into the past, rather than the future? It would be a fair comment that such 'chaotic unpredictability" is normally much worse for the 'retrodiction' that is involved in past-directed evolution than for the 'prediction' of the normal future-directed evolution. This has to do with the Second Law of thermodynamics, which in its simplest form basically asserts: Heat flows from a hotter to a cooler body. ... This procedure of dynamic retrodiction is clearly a hopeless prospect in physics. ... For this kind of reason, physics is normally concerned with prediction, rather than retrodiction. (28) p. 760: Of course, it might indeed ultimately turn out that there is simply no mathematical way of fixing certain parameters in the 'true theory', and that the choice of these parameters is indeed such that the universe in which we find ourselves must be so as to allow sentient life. But I have to confess that I do not much like that idea! (29) p. 850: But to take this position is to part company with one of the basic principles of Einstein's theory, namely the principle of general covariance. (30) p. 935: ... A lot of these stem from the fact Einstein's theory is 'generally covariant' (Sec 19.6). Finally, I have to say that I really like so many drawings in the book, which are simplistic yet stupendously expressive. Thanks Professor Penrose for sharing your knowledge and achievements of many decades, which will benefit many on this planet called Earth!

## Features

- The Road to Reality By Penrose Roger

## Technical Specifications

| Specification | Value |
|---------------|-------|
| Best Sellers Rank | #42,108 in Books ( See Top 100 in Books ) #14 in Mathematical Physics (Books) #59 in Cosmology (Books) #64 in Astrophysics & Space Science (Books) |
| Customer Reviews | 4.6 out of 5 stars 1,267 Reviews |

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## Customer Reviews

### ⭐⭐⭐⭐⭐ Likely will become one of the Great Books: Buy it!
*by H***N on May 19, 2006*

The short version: If you are reading reviews in consideration of purchasing this book then just BUY IT. It has long been my wish for someone to a write a popular treatment of modern physics, one which includes the math, starts at the beginning, and then covers whatever is needed so that the reader can understand the theories described. For me, reading physics is fun; it is not a path to "becoming a physicist", but I want something beyond the popular science level. My goal is to become a READER of real physics, and I am willing to work to reach this goal. This wish describes "The Road to Reality" almost perfectly. Penrose literally intends to take the reader from basic math through calculus, and on to field theory, Lie Groups/Algebras, calculus of variations (Lagrangians & Hamiltonians), differential geometry with fiber bundles, and tensor analysis. He plans and prepares to explain both quantum physics and general relativity (gravity.) This book is both a popular science guide and introductory mathematics text (including introductions to advance subjects) at the same time. The book is a wondrous delight, while simultaneously being maddening for its flaws. If there were there a thousand similar books, it would be easy to criticise the flaws. The writing is at times simply awful (the worst and most common offense is 'pre-shadowing' for no useful purpose -- and without clearly warning the perhaps already struggling reader). Much of the math details are simple skipped or hand-waved, but the outline and structure provided for mathematical physics is both useful and significant With great persistence by the reader it is understandable. One reads this book both for what it contains, and also for the gateways it will open to other books. The book likely deserves only four stars, but due to it's unique nature I awarded the fifth as a reward for attempting and coming very close to what most would consider impossible. On the other hand, any criticism that it is "incomplete" (the subtitle says the "complete guide to the laws of the universe") is unrealistic and similar to criticizing a "complete guide to Europe" or "a complete guide to fishing" for not listing every hotel or restaurant, or for not including a picture of every fish and a map of every body of water. 'Complete' here means comprehensive and full in coverage and scope, not that every detail is specified. As to criticisms concerning Penrose's idiosyncratic views on physics, he is ABSOLUTELY clear when stating a personal opinion, or covering topics from his own point of view. His own less popular ideas for final theories in physics are a very small portion of the entire book. Pensore clearly delineates his own ideas whenever he mentions them in other sections. [A little about me, but only as a point of reference, might help you evaluate this review since those with significant college math and physics or those with no background in these subjects will approach this book differently: My prior background only includes high school calculus and physics, though I've read many popular physics titles. At the start, I was mathematically naive at the university level, but I was also completely undaunted by the prospect of learning the math and physics.] If you buy this book [highly recommend you do] just read it. Promise yourself that you will keep reading; determine to force your way through no matter what obstacles you encounter. If you have an interest in physics the rewards are immense. Using the book as a tour guide, outline, overview, and foundation you can find resources freely available on the Internet, or available for sale here on Amazon, to actually LEARN to READ physics. You should not expect to "become a physicist" without much more study, but you can develop a reading knowledge of the subject beyond the popular treatments, including the mathematics of tensor analysis, differential geometry, and group theory. An encouragement and warning to young people interested in Physics and Math (as well as those who might buy this book for them) is warranted: If you really want to read this book and work very hard it is possible, but forcing yourself (or being forced) to read it before you have either significant mathematicsal knownlodge and/or the ability to study and develope such know on your own is not a good idea. This book could convince the beginning student of physics or math that these subjects are more difficult than they actually are. Instead they are rather more like any significant skills: they takes some ability, some time to develop, and above all they require careful and persistent work on your part. Currently (three months after starting), I have finished the book (took two months for this) and also reached a rough, reading competence with advanced calculus, differential equations, lie groups/algebras, complex analysis, Lagrangians & Hamiltonians, and can now read introductory quantum mechanics texts and papers. Since reading this book, I have made a good start on Tensor Analysis and Differential Geometry. My estimate is about one year for me to fully understand the book and its topics, but the effort is well worth the results already. Even though, I have sought and used many other sources to improve my understanding, my successes are directly due to the incredible foundation provided by Penrose. In addition, I highly recommend "Deep Down Things" by Schumm, who is much more clear, but less mathematical, on Lie Groups and Gauge Theory. Schumm relates Lie theory directly to Gauge Symmetries, going beyond mere hand-waving while still remaining mathematically simple and clear. I further recommend "Understanding Quantum Physics" by Morrison which offers a much better guided, and step-by-step, introduction to the mathematics and postulates of Quantum Mechanics. (Only real criticism of Morrison is that there are NO solutions for exercise, but he does work many other problems in detail.) Although I bought Morrison's book several years ago and was unable/unwilling to read it, I can now read this one comfortably -- it's not a novel, but it is no longer a fight to read. Neither of these excellent books offers the scope of Penrose however, so read "The Road to Reality" first. (I might have missed the beauty of Schumm's treatment of Lie groups and Gauge theory had I encounted it first.) I am also working through "Quantum Mechanics Demystified" and "Relativity Demystified" both by David McMahon, and "A First Course in General Relativity" by Bernard F. Schutz. [Five months after starting Penrose's book, I now feel comfortable in reading the very imposing "Gravitation" by Misner, Thorne, Wheeler (MTW).] When I have finished these, my plan is to read "Quantum Field Theory in a Nutshell" by A. Zee and move on to Zwiebach's String Theory book. Notice that if you don't have the background in math or physics then this book is going to lead you to reading many others and learning many new topics. This truly great book doesn't end the journey but rather opens new worlds and capabilities for the interested reader. If you are asking "Should I buy it?", then: Yes, JUST BUY IT. If you do buy it, then JUST READ IT. No matter how long it takes you or how difficult it seems at time just keep reading.... You will be delighted to finish this book, and disappointed that it ends -- expect both emotions at the same time. Thank you Roger Penrose!

### ⭐⭐⭐⭐⭐ A stupendous piece of work for understanding both the physical world and the stock market!!!
*by H***Y on February 23, 2021*

I spent about 4 months, on and off, and finally finished reading this great book. I have a dual purpose: (a) I wanted to quickly recover my knowledge in math and physics I acquired during my prior physicist career, and (b) I wanted to see if I could apply anything I learnt from here to the machine trading models as introduced in the book "Forecasting and Timing Markets: A Quantitative Approach." I really enjoyed this book and indeed found a lot of similarities and dissimilarities between building mathematical models for interpreting the real physical world and building models for forecasting market. Here is a summary of what I have found out to be very applicable and useful: (1) p.7: What laws govern our universe? How shall we know them? How may this knowledge help us to comprehend the world any hence guide its actions to our advantage? ... Eventually, even the much more complicated apparent motions of the planets began to yield up their secrets, revealing an immerse underlying precision and regularity. (2) p.18: Fig. 1.3 Three 'worlds' - the Platonic mathematical, the physical, and the mental - and the three profound mysteries in the connections between them. ... everything in the physical universe is indeed governed in completely precise detail by mathematical principles. ... all actions in the universe could be entirely subject to mathematical laws. (3) p.28: Euclid's first postulate effectively asserts that there is a (unique) straight line segment connecting any two points. His second postulate asserts the unlimited (continuous) extendibility of any straight line segment. His third postulate asserts the existence of a circle with any centre and with any value for its radius. Finally, his fourth postulate asserts the equality of all right angles. (4) p.45: We are to think of a light, straight, stiff rod, at one end P of which is attached a heavy point-like weight, and the other end R moves along the asymptote. (5) p.67: The system of complex numbers is an even more striking instance of the convergence between mathematical ideas and the deeper workings of the physical universe. (6) p. 109: What about the places where the second derivative f''(x) meets the x-axis? These occur where the curvature of f(x) vanishes. In general, these points are where the direction in which the curve y = f(x) 'bends' changes from one side to the other, at a place called a point of inflection. (7) p. 115: Armed with these few rules (and loads and loads of practice), one can become an 'expert' at differentiation without needing to have much in the way of actual understanding of why the rules work! This is the power of a good calculus. (8) p.151: Air, of course, consists of enormous numbers of individual fundamental particles (in fact, about 10^20 of them in a cubic centimeter), so airflow is something whose macroscopic description involves a considerable amount of averaging and approximation. There is no reason to expect that the mathematical equations of aerodynamics should reflect a great deal of the mathematics that is deeply involved in the physical laws that govern those individual particles. (9) p. 211: It seems that Nature assigns a different role to each of these two reduced spin-spaces, and it is through this fact physical processes that are reflection non-invariant can emerge. It was, indeed, one of the most striking unprecedented discoveries of 20th-century physics (theoretically predicted by Chen Ning Yang and Tsung Dao Lee, and experimentally confirmed by Chien-Shiung Wu and her group, in 1957) that there are actually fundamental processes in Nature which do not occur in their mirror-reflected form. (10) p. 217: For example, the configuration space of an ordinary rigid body in Euclidean 3-space is a non-Euclidean 6-manifold. (11) p.223: As in Sec 10.2, we have the notion of a smooth function (Phi), defined on manifold M. (12) p.388: He (Newton) had originally proposed five (or six) laws, law 4 of which was indeed the Galilean principle, but later he simplified them, in his published Principia, to the three 'Newton's laws that we are now familiar with. (13) p. 390: It is remarkable that, from just these simple ingredients (Newton's formula GmM/r^2), a theory of extraordinary power and versatility arises, which can be used with great accuracy to describe the behavior of macroscopic bodies (and, for most basic considerations, submicroscopic particles also), so long as their speeds are significantly less than that of light., (14) p. 392: Galileo's insight does not apply to electric forces; it is a particular feature of gravity alone. (15) p. 410: We shall also begin to witness the extraordinary power, beauty, and accuracy of Einstein's revolutionary theory. (16) p. 412: The geometries of Euclidean 2-space and 3-space are very familiar to us. Moreover, the generalization to a 4-dimensional Euclidean geometry E^4 is not difficult to make in principle, although it is not something for which 'visual intuition' can be appealed to. (17) p. 455: Einstein's famous equation E = mc^2 tells us that mass and energy are basically the same thing and, as Newton had already informed us, it is mass that is the source of gravitation. (18) p. 462: Einstein originally introduced this extra term, in order to have the possibility of a static spatially closed universe on the cosmological scale. But when it became clear, from Edwin Hubble's observations in 1929, that the universe is expanding, and therefore not static, Einstein withdrew his support for this cosmological constant, asserting that it had been 'his greatest mistake' (perhaps because he might otherwise have predicted the expansion of the universe!). Nevertheless, ideas once put forward do not necessarily go away easily. The cosmological constant has hovered in the background of cosmological theory ever since Einstein first put it forward, causing worry to some and solace to others. Very recently, observations of distant supernovae have had most theorists to re-introduce /\ (greek lambda), or something similar, referred to as 'dark energy', as a way of making these observations consistent with other perceived requirements. (19) p. 466: The timing of these signals is so precise, and the system itself so 'clean', that comparison between observation and theoretical expectation provides a confirmation of Einstein's general relativity to about one part in 10^14, an accuracy unprecedented in the scientific comparison between the observation of a particular system and theory. (20) p.490: He (Hilbert) appears to have believed that his total Lagrangian gives us what we would now refer to as a 'theory of everything'. (21) p. 503: ... it took many years for Einstein's original lonely insights to become accepted. (22) p. 523: Heisenberg's uncertainty relation tells us that the product of these two spreads cannot be smaller than the order of Planck's constant, and we have Delta-p Delat-x >= h_bar / 2. (23) p. 528: I denote Schrodinger evolution by U and state reduction by R. This alternation between these two completely different-looking procedures would appear to be a distinctly odd type of way for a universe to behave! (24) p. 541: As the state of the arts stands, one can either be decidedly sloppy about such mathematical niceties and even pretend that position states and momentum states are actually states, or else spend the whole time insisting on getting the mathematics right, in which case there is a contrasting danger of getting trapped in a 'rigour mortis.' (25) p.686 (Chapter 27 The Big Bang and its Thermodynamic Legacy): What sorts of laws shape the universe with all its contents? The answer provided by practically all successful physical theories, from the time of Galileo onwards, would be given in the form of a dynamics - that is, a specification of how a physical system will develop with time, given the physical state of of the system at one particular time. These theories do not tell us what the world is like; they say, instead: 'if the world was like such-and-such at one time, then it will be like so-and-so at some later time'. (26) p.687: The usual way of thinking about how these dynamical laws act is that it is the choice of initial conditions that determines which particular realization of the dynamics happens to occur. Normally, one thinks in terms of systems evolving into the future, from data specified in the last, where the particular evolution that takes place is determined by differential equations. (27) p.689: What about evolution into the past, rather than the future? It would be a fair comment that such 'chaotic unpredictability" is normally much worse for the 'retrodiction' that is involved in past-directed evolution than for the 'prediction' of the normal future-directed evolution. This has to do with the Second Law of thermodynamics, which in its simplest form basically asserts: Heat flows from a hotter to a cooler body. ... This procedure of dynamic retrodiction is clearly a hopeless prospect in physics. ... For this kind of reason, physics is normally concerned with prediction, rather than retrodiction. (28) p. 760: Of course, it might indeed ultimately turn out that there is simply no mathematical way of fixing certain parameters in the 'true theory', and that the choice of these parameters is indeed such that the universe in which we find ourselves must be so as to allow sentient life. But I have to confess that I do not much like that idea! (29) p. 850: But to take this position is to part company with one of the basic principles of Einstein's theory, namely the principle of general covariance. (30) p. 935: ... A lot of these stem from the fact Einstein's theory is 'generally covariant' (Sec 19.6). Finally, I have to say that I really like so many drawings in the book, which are simplistic yet stupendously expressive. Thanks Professor Penrose for sharing your knowledge and achievements of many decades, which will benefit many on this planet called Earth!

### ⭐⭐⭐⭐⭐ A panorama of science.
*by P***N on February 26, 2005*

It's a delicate balance for book: Encyclopedic vs well focused on a unifying theme! Penrose succeeds admirably. It's not boring! Books like this are few and far between. Indeed, there are preciously few authors who manage to successfully guide beginning students into serious scientific topics; and even fewer who can see the big picture, and do it all. And then keeping our attention through more than 1000 pages! Penrose's book is inspiring, informative, exciting; and at the same time it's honest about what math and physics are. It is modest when modesty is called for. You are not cheated. You do get the equations (not just hand waving!), but you are gently prepared in advance, so you will want the mathematical formulae. Penrose's book is likely to help high school students getting started in science; and to inspire and inform us all. There is something for everyone: for the beginning student in math or in physics, for the educated layman/woman (perhaps the students' parents), for graduate students, for teachers, for scientists, for researchers; and the list goes on. It is one of the very few books of this scope that is not intimidating. Not in the least! I can't begin to do justice to this terrific book. Get it, and judge for yourself. I will also not give away the ending, other than saying that the title of the book is a good hint. And you will be able to form your own take, and your own ideas on the conclusion. Like with all good and subtle endings, they can be understood and appreciated at several levels. I came across Penrose's book in my bookstore by accident, and I was at first apprehensive: The more than 1000 pages, and the 3.3 pounds are enough to intimidate anyone. But when I started to read, I found myself unable to put it down. And I didn't: Bought it; and I had several days of enjoyable reading. I am not likely to put it away to collect dust either. It is the kind of book you will want to keep using, and to return to. It will not surprise that one of Penrose's unifying themes is the compelling and pleasing geometric images that underlie both the mathematics (roughly one third of the book: modern geometry, Riemann surfaces, complex functions, Fourier analysis, visions of infinity), and the physics: Cosmology (the big bang, black holes), gravity, thermodynamics, relativity (classical and modern: loop quantum gravity, twisters), and quantum theory (wave-particle duality, atomic spectra, coherence, measurements). The pictures: In fact, this semester, I was just teaching a graduate course, and I had a hard time presenting of Riemann surfaces in an attractive way. It's a subject that typically comes across as intimidating in many of the classical books: Take Herman Weyl's book, for example. I also found it refreshing to see that Roger Penrose gave the many illustrations his own personal and artistic touch; as opposed to having flashy pictures generated by the latest in color-graphics and special effects. I think readers will relate better to Penrose's own illustrations: They isolate and highlight the core ideas and they are not intimidating: We sense that we ourselves would have been able to make similar pencil sketches. Or at least we are encouraged to try! The common theme in the pictures serves to bring to life the underlying and fundamental ideas;--- another attractive feature of the book! It is otherwise easy to get lost in some of the equations, and in the encyclopedic panorama of topics. Review by Palle Jorgensen, February 2005.

## Frequently Bought Together

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