on October 3, 2003
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?
on December 2, 2014
I must confess to be one of those kinds of students of math that really did not “get” much of the material presented in the various courses that I took over the years until, after repeated courses in statistics, the penny dropped late in my graduate career. After that, I became fairly fluent in statistical procedures as they applied to data analysis and, in fact, went on to make a good deal of money in that area.
Nevertheless, if you are one of those people who do not immediately understand that a knotted rope will axiomatically produce a right angle triangle, this book will be hard sliding. Perhaps, it is too much to expect that a book on geometry would have illustrations to help the uninitiated understand some of the more basic concepts.
It may have been my manner of reading; I tend to read at night before dropping off to sleep. But, I found the transition from topic to chapter and chapter to chapter choppy.
I felt badly about not enjoying this book more. A friend of mine had read it and recommended it highly as easily understood. Of course, he is an electrical engineer and some of the concepts might come more easily to him.
Bottom line? Give it a read if you like hard work.
on May 31, 2003
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.
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.
on January 12, 2003
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.
on December 14, 2002
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.
on December 12, 2002
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.
on December 4, 2002
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.
on November 18, 2002
From the perspective of one not schooled in mathematics, I found the book much less clear then other books intended for the lay audience. The bios are ok (although the anti-Christian comments seem impertinent) but the discussions of the theories and their significance often leapfrog just when this reader would appreciate more information. The books also have errors of fact. Before buying, read the Publisher's Weekly review (link on this site) which says, in part, ". . . Sloppiness and distracting patter combine with slipshod presentation to bestow a feel for, rather than a grasp of, the subject. Certain misses are peripheral but annoying nonetheless confusing Keats with Blake, repeating a discredited account of Georg Cantor's depression, etc. Some of them, however, undermine the heart of the book's argument. Strictly speaking, Descartes, Einstein and Witten didn't produce revolutions in geometry but rather in how it's related to other subjects, while Gauss arguably produced two revolutions, one of which non-Euclidean geometry is featured, while the other differential geometry though equally necessary for Einstein's subsequent breakthrough, is barely developed. Mlodinow completely ignores another revolution in geometry, the development of topology, despite its crucial role in Witten's work."
on September 24, 2002
"Euclid's work is a work [is] a work of beauty whose impact rivaled that of the bible, whose ideas were as radical as those of Marx and Engels. For with his book, Elements Euclid opened a window thgrough which the nature of our universe has been revealed." Strong words, but Mlodinow backs them up with this surprisingly exciting history of how mathematicians and physicists discovered geometric space beyond Euclid's three dimensions. Each advance in mathematical geometry has been followed by unexpected discoveries proving that the strange mathematics actually describe measurable physical properties. Mlodinow, a physicist and former faculty member of the California Institute of Technology, has also written TV screenplays for Star Trek: the Next Generation and other shows. He has a good sense of popular science writing, and he personalizes geometric abstractions by endowing them with personalities of his adolescent sons Alexei and Nicolai. Euclid, Descartes, Gauss, Einstein, and Witten are among the mathematicians profiled, and each of them also emerges with a distinct personality based on the style of their writing and historical anecdotes. This engaging history does an excellent job of explaining the importance of the study of geometry without requiring the reader to be a mathematician.