Rating: 
Summary: serious scientific contemplation and reader-friendly
Review: A good easy to read book which maps the history of the development of geometry right from euclid to Descartes to Gauss ... and finally to string theory. There are some very good points about this book, which include the presentation of the original developments of geometry very clearly and in a very highly understandable manner. The book is also full of little trivia and delves cursorily into some of the aspects of the personalities behind the mathematics. But it flounders as it reaches string theory and becomes very abstract. Perhaps the author could have spent lesser time in the initial parts by cutting out some of the analogies he would draw with the charactors in the example bearing his childrens names. Having said that, this is a good book and is worth a buy for the Maths lover. In terms of analogies, if you liked Eli Maors "e The story of a number", Mlodinov will fit in well in his ease of development of a complex history.
Rating: 
Summary: A Great Read, and New Insights, Too
Review: Although the title of this book made it seem like just another math book, I noticed, skimming the intorduction and back cover, that it is really also about both math and physics, the story of the development of geometry and how geometry is used in modern physics. That intrigued me, so I bought the book. I wasn't disappointed! I have to say that I start to read many popular math and physics books but often do not finish them. They end up getting bogged down in detail, or are just too dry, or seem as if they are written by some stuffy or priggish academic, and I lose interest. But Mlodinow manages to explain some rather difficult mathematical concepts clearly, and in an entertaining way employing humor and a very accessible writing style, making this one of the few books on this subject that, after you put it down, you actually look forward to getting back to! Mlodinow has a real knack for story-telling - the book pulls you along from concept to concept in a way that few popular accounts of science or math ever do. If only I had read this years ago, before taking high school geometry, I might have actually developed some interest for the subject! As for the physics, I now feel I understand much more of what they are talking about when I read about modern physics in magazines or the newspaper - or in other books. I highly recommend this great book!
Rating:
Summary: Amazing Book
Review: Anyone who thought geometry was boring or dry should prepare to be amazed. Despite its worthy cover this book is exactly what its title says - a story - and the plot of this story involves life, death and revolutions of understanding and belief, and stars the some of the most famous names in history. The book opens with Aristotle watching ships at sea disappearing hull first over the horizon. "On a flat earth, ships should dwindle evenly until they disappear", and so he came to the realisation that the earth must be curved. This sets the scene for Mlodinow's tale of how geometry has shaped human history - "to observe the large scale structure of our planet, Aristotle had looked through the window of geometry." The book recounts how we have continued to look through this window to understand the reality we live in, and how the window has changed along the way. The book is arranged as a series of five tales of the "five geometric revolutions of world history". These are told as the story of their main figures - Euclid, Descartes, Gauss, Einstein and Witten - in the context of their time, place and culture. This is one of the things that makes this book stand apart from others on the history of mathematics and science. It is told as a series of personal stories, of discoveries and leaps of understanding made by human beings. And this perhaps unexpectedly human side of geometry is enhanced by Mlodinow's accessible style. He is able to bring historical situations and mathematical concepts to life with the language of the present day. For example he explains the importance of applied geometry to Egyptians: "In building a pyramid, just a degree off from true, and thousands of tons of rocks, thousands of person-years later, hundreds of feet in the air, the triangular faces of your pyramid miss, forming not an apex by a sloppy four pointed spike. The Pharaohs, worshipped as gods, with armies who cut the phalluses off enemy dead just to help them keep count, were not the kind of all-powerful deities you would want to present with a crooked pyramid." This book also contains some of the clearest explanations of relativity and string theory that I have ever read. Placed in the context of the evolution of geometry, and told as human triumphs of discovery by Einstein and Witten and their peers, these theories offer answers to obvious questions arising from our struggle to understand our reality. They also contain some very amusing examples such as Mlodinow explaining the entropy of black holes in terms of the messiness of his son, Alexei's bedroom. "Before Hawking, black holes, thought to have no internal structure, were thought to be something like an empty room. But now it seems they are like Alexei's actual room. Had Hawking asked, I could have confirmed this: I have always told Alexei that his room was like a black hole." This is an excellent book not just for those select few fascinated by geometry, but for anyone interested in history of science, philosophy and humanity. In fact I would recommend it to anyone who enjoys a good story. Who would have thought that the story of geometry would include tales of life, death, sex and taxes?
Rating:
Summary: Mlodinow follows the "straight" line of geometry development
Review: Born in the mudflats of the Nile and Tigris rivers, organized and codified by Euclid and "warped" by Einstein to describe the universe, geometry is the second oldest area of mathematics. Only basic arithmetic was used before geometry was first used to (re)mark flooded territories. While it is commonly claimed that religious books are the most widely published, it is often stated that "Euclid's Elements" is the second most widely published book in history. Think of the consequences to society and learning if a copy of the Elements was placed in a drawer in every hotel room in the United States!
Geometry is also a pure science in the sense that in all but a few cases, you are not actually working with the objects, only an idealized abstraction of the figure is available. This forces the user to apply an intellectual rigor that is unnecessary in most other areas of human endeavor.
Mlodinow starts you out with the annual rising of the Nile river, which is the lifeblood of Egypt. He then moves on to the story of Euclid, where surprisingly little is known about him, given that he did so much to advance civilization. Mlodinow also points out that Euclid also gave birth to a revolution in the power of thought. Euclid, obviously being a perfectionistic cynic, insisted on starting with the simplest possible initial set of assumptions and then proving every specific detail after that. Although it was proven later that Euclid did make some unwarranted assumptions, these were very minor in comparison to his demonstration of the power of analytical thought.
In the history of mathematics, there is no discovery more powerful than that of analytical geometry by Descartes. Ranking with the use of decimal numbers, it allows people to combine numbers and geometry in ways that opened up an enormous number of different avenues of research and proof. It is hard to see how one could do calculus without it. Mlodinow describes the life of Descartes, and his story of Descartes' relationship with Swedish Queen Christina is very funny.
The remainder of the book describes the development of Non-Euclidean geometry and how it was used by Einstein in his development of relativity. This is one more instance of mathematicians developing new mathematics that appears at first to be only an intellectual curiosity, but ultimately proves to be the model used to describe aspects of physical reality. I continue to find it astounding that the extremely non-intuitive features of Non-Euclidean geometry were developed over fifty years before Einstein found a practical use for them. This section of the book should be mandatory reading for any fool who thinks that they will never find a use for mathematics.
Written in a style that is very amusing and historically accurate, Mlodinow takes you through the history of geometry and the ride is gentle and informative.
Rating:
Summary: A MUST BUY FOR ANYONE INTERESTED IN MATHEMATICS OR PHYSICS!!
Review: Euclid's Window by Leonard Mlodinow is an outstanding book. From the discoveries of Pythagoras and Isaac Newton to John Schwarz's String Theory, you can learn so much about the history of mathematics and physics through Euclid's Window. Mlodinow basically provides the reader with a summary of the evolution of mathematics and science, yet he does it in such a way that it is like reading a novel. The genius of Mlodinow is seen through his ability to take a topic that would take most authors thousands of pages to cover and convert it into a concise, easy to read story. Most people turn and run when they see a math history book, but this is no ordinary math book. Mlodinow's use of real life examples, graphic images, and stories from his own experiences with his children turn complex, abstract math concepts into concrete ideas that the ordinary person walking down the street can understand. Also, this is the first math/science book that I have ever read that actually provides some background information about the men who invented the formulas and theories. Most books either do one or the other. They either discuss the theories and formulas or they talk about the life of the person who invented them. Mlodinow does both. For example, Mlodinow not only discusses the mathematical discoveries made by Carl Gauss, but he gives an overview of his childhood, schooling, and life.
However, if there is one draw back to this book it is the physic's side of the story. I come from a mathematical background and even I found it difficult to understand the physic's theories like String Theory and M-theory. The author continually throws out new theories and new terms like quarks and positrons without much explanation. On the other hand, I think you have to give Mlodinow some credit for trying to discuss these extremely complex ideas in layman terms. Most authors would just leave out the part about String Theory altogether. According to Mlodinow, we all live in a giant puzzle known as the Earth, and since the Egyptian and Greek civilizations we have been trying to use reason, observation, and experimentation to piece the puzzle together. Through Euclid's Window, Mlodinow shows just how far humanity has come in its search to complete the puzzle.
Rating:
Summary: Brilliantly written
Review: Euclid's Window is an astounding book. It takes you on a ride through History, where you explore the origins of Geometry & Mathematics. From the early Babylonians to the Egyptians who used Geometry & Mathematics, but didn't ask the deeper questions which the Greeks did. With the Greeks came Thales, Pythagoras, and Euclid (Not necessarily in that order) who changed our view of the world by developing Geometry as we know it today in High School books. Than came the Dark Ages, and Europe plummeted into more than 1000 years of intellectual silence. The book than talks about the revolutions led by Galileo, Descartes, Gauss, and Riemann. Finally the Author describes the later developments in physics. The revolution that Einstein made with the Special Theory or Relativity and the General Theory of Relativity. From then on we knew that mass curves space creating gravity, that nothing can travel at the speed of light, and that time is a privet matter rather than universal. Than the Revolution of Quantum Physics which, was developed in particular by Heisenberg and Schrodinger. Leonard Mlodinow explains the conflicts that arise when quantum physics and the General Theory of Relativity are combined, they fail. Quantum Mechanics works perfectly on the small scale, General Relativity works fine on the large scale, yet there is no way physicist and mathematicians could combine the two. And then came the birth of the String Theory, rather five different String theories that turn out to be approximations to the much larger M-Theory, the theory that would be able to describe everything in the Universe, from subatomic particles to distant galaxies in the Universe. There is only one problem nobody knows what it looks like, and mathematicians and physicist can only calculate approximations of the theory. Leonard Mlodinow takes the reader on a fascinating ride through the history of Mathematics and Physics, the book is enlightening, even to me who constantly tries to keep up with new developments in Physics. A must read for anybody interested in Mathematics & Science, or just plain old History, this book is essential.
Rating: 
Summary: gross historical error makes one question entire book
Review: I was enjoying all of Mr. M's anecdotes of Ancient Greek mathematicians and then I got to the part on Charlemagne. I am no religious scholar, but when M refers to Dominicans and Franciscans as providing teachers to Charlemagne's church schools (page 61) I started to wonder if most of Mr. M's book is fiction, albeit a nice readable fiction. Mr. M (and his editors) failed to grasp that Dominicans and Franciscans were orders founded in the 13th century and of course Charlemagne lived in 8th and 9th centuries. That is only a small tiny error of about 4-500 years.
Well this is another example of what happens when one tries too hard to popularize material best left to nerds. I don't object to all the "made up stuff". It makes for a good story. I only wish authors like Mr. M would be clear that their work belongs in the fiction not the non-fiction section. Mr. M should refer back to his comments on recidivism on page 46. Like Topographia Christiana maybe Mr. M is shooting for the 500 year best seller list.
Rating:
Summary: Amazing Book
Review: In the past few years I have read several books about Einstein's theories, the beginning of the universe, and string theory. Though they were understandable at times, they were also often obtuse, and in a couple cases too dry. But most important, the one thing they didn't explain well was "what is curved space anyway?" Then, recently I read Mlodinow's book, Feynman's Rainbow, and I thought - he really writes well. So next I purchased Euclid's Window. I wasn't disappointed! In Euclid's Window Mlodinow finally gives all us non-scientists a good feeling for what physicists are all talking about in modern physics and astronomy as he tells an entertaining tale of the development of human ideas about space from the early Greeks puzzling over parallel lines to Einstein's theory and even the extra dimensions of string theory. His accounts of the mathematical developments are punctuated with wit and humor, and with tales of the times, the people, and the cultural history surrounding the advances. But, best of all, I understood it, and kept wanting to read more!
Rating:
Summary: More Math History for the Math-Challenged
Review: This is another great book for us non-mathematicians that in this case allows us to share the excitement of the history of discoveries in the development of geometry. Leonard Mlodinow has opened Euclid's window and demonstrated why we should be in awe of Euclid's achievement and the achievements of the mathematicians and physicists who both preceded and followed him. Thales, Pythagoras, Euclid himself, Descartes, Gauss, Einstein and Witten, along with Heisenberg, Schrödinger, and others are made to live on these pages. Mlodinow adds many background points that help the reader understand the period in which the various geometric concepts were developed.
Mathematics, including geometry, is by no means self evident. Concepts that we take for granted were often hard won. People were even murdered over the square root of 2 because irrational numbers were considered a secret! The struggle over the shape and size of the universe and earlier fights over the geometry of the solar system were difficult and often bitter.
Geometry is one of the crowning achievements of our species and indeed laid the foundation for much art, philosophy and of course science. Anyone who wants to really understand the latter should read this book.
I highly recommend this volume as among the best of the attempts to explain the basic concepts of mathematics upon which much of our civilization is based!
Rating: 
Summary: A prototype of bad popular science
Review: This is by and large the worst popular science book I've ever come across (and being an astronomer and translator of popular science, I've read quite a number of them). Mlodinow violates almost all rules of scientific method and conduct, and even some of rules governing the plain civility in writing. There are literally hundreds of examples of such violations, so let me mention just a few of them. Mlodinow engages in hero-worship to an unprecedented degree: he does not shy away from pronouncing Witten "the most influential physicist and mathematician in the world" (p.253), as if such a grandeloquent statement can ever be proven or even properly supported. He enjoys judging long-dead people without proper history knowledge: he finds Cantor genius and Kronecker "a crab", he outrageously states that medieval Arabs didn't contribute anything original to mathematics (perhaps he should have taken some *history* courses from Witten, who knows better for certain!) and then, a dozen or so pages later, contradicts himself by citing some important results of two Arab mathematicians; he finds geometry and calculus more cognitively important than algebra (a dubious statement and quite improper for a popular work); he censores ancient Romans for their decadent ways (while simultaneously celebrating Athenian often quite promiscuous "symposia"); he does not know that Sirius is the brightest star which can be seen from Greece; he calls Aristotle just a meteorologist (p. 56). In addition, Mlodinow heavily indulges in what serious historians call "Whiggish interpretation of history", i.e. judging of the past by its present utility. Thus, he repeats at least 10 times that middle ages were "dark ages" or "barbaric", or that everything similar to the obnoxious US system of tenuring academic scientists belongs to the best of all possible worlds.
Quite amusing are Mlodinow's factual mistakes: in calculating Kaluza's salary, he errs only for a factor of 5 (p. 233). He expresses Planck's constant in wrong units (p. 226). He confuses arithmetics and number theory (p. 149). Funnily enough, for all this slopiness, he excels in a strange field: telepathy and spiritism. Namely, we are almost at each page treated with something which "Einstein thought" or which "seemed to Euclid", or such. This is the history of science at its nadir.
Finally, a word should be said about underlying ideology of this disappointing book. It serves as a sort of rousing manifesto for a particular branch of physics (theoretical particle physics) and one single theory (string/brane theory). Since it is written and marketed as intended for general public, it is does dangerously misleading in suggesting that this one narrow field, and this one particular narrow theory within this field is everything that exists, or at least everything that is important, beautiful or profound in modern physics. This is megaparsecs distant from the truth. There is no meaningful sense in which string theorists are "closer to God" and eternal truths than are solid-state physicists, or astrophysicists, or geophysicists, or quantum opticians, etc. They are certainly not funnier or more literate than the rest of us. Mlodinow, unwittingly, proved this exactly with this pitiful book.
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