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5.0 out of 5 stars Amazing Book
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...
Published on Oct. 3 2003

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2 of 3 people found the following review helpful
2.0 out of 5 stars gross historical error makes one question entire book
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...
Published on May 31 2003


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2 of 3 people found the following review helpful
2.0 out of 5 stars gross historical error makes one question entire book, May 31 2003
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
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.
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5.0 out of 5 stars Amazing Book, Oct. 3 2003
By A Customer
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
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?
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4.0 out of 5 stars serious scientific contemplation and reader-friendly, May 31 2003
By A Customer
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
Subtitled The Story of Geometry from Parallel Lines to Hyperspace, this luminous book offers the rare combination of serious scientific contemplation and reader-friendly accessibility.
Starting with the mathematicians and geometers of antiquity, Mlodinow traces the progress of rational thought - and irrational numbers - from before Euclid's elucidation of the Elements of geometry to the possibilities which still wait for us to reveal them - from "A point is that which has no part" straight up to the equally puzzling notion that space and time may only be shadowy hints of some more fully flowering, if abstract, function of mathematics on another plane of reality. Sound like science fiction? Rest assured that Mlodinow has both feet planted square on terra firma. The paradoxes and upsets of his discipline are not lost on the author - nor, indeed, are the ironies and jokes of history (say what you like about death, but it was the decidedly un-mystical necessity of taxation which launched geometry as a scholarly pursuit in ancient Egypt) - but the author reminds his reader at various points of the dangers of assuming too readily that any given idea is worthless, too far-out, or obviously and intuitively wrong. Intuition, as it turns out, resists and rebels against much of what has become higher learning in the fields of mathematics and physics.
Mlodinow's dedication to the subject matter at hand matches in beautiful, if heartbreaking, counterpoint to the obscurity in which many of the scholars he discusses labored. Drawing not only on the work of famous theoreticians like Einstein and Hawking, but also on essays and ideas buried in forgotten papers and musty appendices, the author gives full credit wherever it may be due. In the process, whether by design or accident, Mlodinow imparts an even more valuable lesson: the ease with which scientific knowledge can be lost, sometimes for millennia. If Artistotle knew, nearly 2,500 years ago, that the planet must be round, why do we still hear that Columbus' sailors were terrified of sailing off the edge of a flat Earth? (This story in itself is almost certainly apocryphal.) If primitive versions of the Theory of Evolution were kicking around in ancient Greece, how is it we still face voids of serious scientific credibility in modern-day Kansas? Regrettably, superstition, fear, politics, and the manipulation of knowledge - who gets it and who pays the price for seeking too much of it - is also part of the history of geometry, as it is part of the history of science in general.
Your reviewer himself studied a fair amount of the history of mathematics and physics in the Western World (starting, in fact, with Euclid, and progressing then through Ptolemy, Apollonius, Descartes, Newton, et al, right up through Einstein and Minkowski) and found certain parts of the curriculum cheerless, if not downright appalling. What a relief and a joy, then, to find Euclid's Window not only concise and readily understandable, but effervescent as well. Author Mlodinow clearly enjoys the subject matter and - more importantly - enjoys imparting it to others. As a writer and a teacher, Mlodinow demonstrates that he is gifted and enthusiastic.
It's hard to want to be critical of a book like this, which is both charming and brave in the face of apathy, even hostility, toward mathematics and scientific inquiry (a situation far from unique to our times). Indeed, there is little to be critical about here, for the book is nearly perfect in its balance of detail and simplification. If anything, though, the simplification of the material may be a little too rigorously carried out, and the focus on geometry a bit too narrow. Though quantum physics is not the focus of the book, it has become relevant to contemporary geometric notions, and while Mlodinow does incorporate plenty on the field and its proponents, the origins of quantum physics are glossed over a bit too much. A broad sketch of the theories underlying quantum physics would not have been out of place, but there is not so much as a mention of black-body radiation, the effect which first put physicists onto the notion of quantum energy states. As if in compensation, however, Mlodinow's explanation of relativity strikes a perfect balance and his exposition on string theory is wonderfully clear. Don't be scared off if these sound like high-falutin subjects impossibly out of reach: Mlodinow does the invaluable service of grasping a higher limb of the Tree of Knowledge and bending it down until the layman can get a hold on, and enjoy, its fruits.
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4.0 out of 5 stars Euclid's Window, May 19 2003
By 
Ashwin (Bangalore, India) - See all my reviews
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
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.
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5.0 out of 5 stars Mlodinow follows the "straight" line of geometry development, March 15 2003
By 
Charles Ashbacher (Marion, Iowa United States) - See all my reviews
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
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.
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3.0 out of 5 stars You can tell he wrote for television, Feb. 25 2003
By A Customer
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
Leonard Mlodinow has a good story to tell, but his writing, probably influenced by his stint as a television writer, shows a flippancy that constantly annoys. His ongoing references to sons Alexei and Nicolai make his explications of geometry and relativity too cute - a further annoyance. On page 116, he refers to a spiritual seance as consulting with "ghouls". However the participants receivied their information, it would have been from spirits, as ghouls are those who consume human remains. If I consult with my late wife, I am consorting with a spirit; if I sprinkle her ashes on my cereal in the morning, I become a ghoul. I feel this book could have benefitted from more judicious editing than it was obviously (not) given.
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2.0 out of 5 stars Where was the editor?, Jan. 12 2003
By 
Diogenes (long beach, wa United States) - See all my reviews
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
We agree with the Publishers Weekly comments,"...sloppiness...tells jokes and avoids the issue..."
For example, Eratosthenes may have used a gnomon or sundial, or the shadow from an obelisk to determine noon on the summer solstice; and although shadows are integral to their operation, it is inept and misleading to suggest,"The lenght of the shadow at Alexandria..." contributed Eratosthenes' epiphany(page 41, line 28).
Key to comprehending his insight is to understand he saw the sun's rays had a bit of a slant, about 7 degrees, at Alexandria, and, he had heard, none at Syene, since the sun was directly overhead. It is not about any shadow length, but all about the angle. The shadow length can vary, the angle will be constant.
Finding the angle, Eratosthenes was then able to use Euclid's Proposition 29, and his best guess as to the distance between Alexandria and Syene, to calculate the earth's circumference.
As to the distance, A-S, Mlodinow claims a nameless graduate student (page42, line 1) was sent to pace off the 500 miles. Mlodinow must have thought it was a good idea to insert this bit of fantasy, in an effort, unfulfilled, to inject some humor in the text.(Where was his editor?) No T A was employed. Eratosthenes most likely simply estimated the distance, from travel reports. Mlodinow's invention is silly, frivolous fiction, which further undermines his credibility.
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3.0 out of 5 stars Following yet left behind, Dec 16 2002
Euclid's Window by Leonard Mlodinow is a book about geometry and physics which really suprised me with a title "The Story of Geometry from Parallel Lines to Hyperspace." I figured that the book would pertain to the creation of geometry and the figures who invented it instead it started with geometry and ended with a heavy load of physics. I enjoyed the history of all the inventors of both fields yet the heavy jagor used in some sections threw me off course. At certain parts,I simply skipped over sections to get to a part where I understood what they were trying to explain. Do not get me wrong I gained a lot of knowledge from this book about the creation of geometry and physics alike however I found it difficult to follow at times. I recommend this book to anyone involved in mathematics or physics but to the reader without any background in either a recommend a nice fiction book...
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4.0 out of 5 stars A MUST BUY FOR ANYONE INTERESTED IN MATHEMATICS OR PHYSICS!!, Dec 15 2002
By A Customer
This review is from: Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace (Paperback)
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.
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3.0 out of 5 stars A book more enjoyed by scientists than mathematicians, Dec 12 2002
By 
The title of Leonard Mlodinow's book is "Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace." This title is a little misleading. I assumed this book would trace the development of geometry. Although Mlodinow does describe the history of mathematics, half of the book focuses on the latest developments in the field of physics. I understand the two fields are related, but Mlodinow does not state clearly enough exactly how the two fields tie together. Mlodinow begins with the development of geometry and then explores the latest developments in physics, but he does not return to geometry or speculate how the latest physical developments will affect math. He also does not build one final bridge to unite math and physics. In reading this book, the field of physics slowly crept up on me, and Mlodinow never took me back to the mathematical roots which are the foundation of this book. I was a little confused by the writing towards the end of the book. I never quite understood the latest physical theories, regardless of the multiple examples Mlodinow provides.
Just as there are weaknesses in "Euclid's Window," there are also strengths. Mlodinow interestingly recounts the history of mathematics in the first three sections of the book. He often shows practical applications of math in daily life as well as explaining how developments such as Cartesian coordinates and algebra have simplified math. This book is full of interesting facts and trivia, from the first person to use the sign of infinity to glimpses of the lives of several famous mathematicians. Readers might enjoy learning that the people behind the infamous proofs were not perfect; they had flaws as well as great ideas.
I recommend the first three sections of "Euclid's Window" to those interested in learning about the development of math and the last two sections for the physics buffs among us. Those interested in physics would most likely enjoy the entire book, learning the basic mathematical principles behind today's physics theories. For mathematicians, however, the last two sections of the book leave something to be desired.
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