Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace [Hardcover]

Leonard Mlodinow (Author)

Book Description

Publication Date: April 17, 2001

Through Euclid's Window Leonard Mlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space -- in the living room or in some other galaxy -- have been the hidden engine of the highest achievements in science and technology.

Mlodinow reveals how geometry's first revolution began with a "little" scheme hatched by Pythagoras: the invention of a system of abstract rules that could model the universe. That modest idea was the basis of scientific civilization. But further advance was halted when the Western mind nodded off into the Dark Ages. Finally in the fourteenth century an obscure bishop in France invented the graph and heralded the next revolution: the marriage of geometry and number. Then, while intrepid mariners were sailing back and forth across the Atlantic to the New World, a fifteen-year-old genius realized that, like the earth's surface, space could be curved. Could parallel lines really meet? Could the angles of a triangle really add up to more -- or less -- than 180 degrees? The curved-space revolution reinvented both mathematics and physics; it also set the stage for a patent office clerk named Einstein to add time to the dimensions of space. His great geometric revolution ushered in the modern era of physics.

Today we are in the midst of a new revolution. At Caltech, Princeton, and universities around the world, scientists are recognizing that all the varied and wondrous forces of nature can be understood through geometry -- a weird new geometry. It is a thrilling math of extra, twisted dimensions, in which space and time, matter and energy, are all intertwined and revealed as consequences of a deep, underlying structure of the universe.

Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.

Editorial Reviews

Amazon.com Review

"How do you know where you are?" asks Leonard Mlodinow in his charming mathematical history, Euclid's Window. This question and others about space and time grew out of simple observations of the environment by a select group of thinkers whose lives and brains Mlodinow dissects. Starting with Euclid, geometry has flowed out over the centuries, describing the universe, and, Mlodinow argues, making modern civilization possible.

This is not just a history of geometry--it's a timeline of reason and abstraction, with all the major players present: Euclid, Descartes, Gauss, Einstein, and Witten, each represented by a minibiography.

Lots of examples pepper the narrative to help readers achieve their own "eureka!" And it's impossible not to be staggered at the mathematical feats of these geniuses, accomplished as many of them were in the absence of anything but observation and intense thought. Each story builds satisfactorily on the last, until at the end of this delightful book, one has a sense of having climbed a peak of understanding.

A working knowledge of basic geometry is helpful but not essential for enjoying Euclid's Window, and Mlodinow's chatty style lends itself remarkably well to explaining these deep and revolutionary concepts. --Adam Fisher

From Publishers Weekly

Mlodinow's background in physics and educational CD-ROMs fails to gel in this episodic history of five "revolutions in geometry," each presented around a central figure. The first four Euclid, Descartes, Gauss and Einstein are landmarks, while the fifth, Edward Witten, should join their ranks if and when his M-theory produces its promised grand unification of all fundamental forces and particles. Mlodinow conveys a sense of excitement about geometry's importance in human thought, but 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. Occasionally Mlodinow delivers succinct explanations that convey key insights in easily graspable form, but far more often he tells jokes and avoids the issue, giving the false, probably unintentional impression that the subject itself is dull or inaccessible. More substance and less speculation about the Greeks could have laid the foundations for an equally spirited but far more informative book. 11 figures, two not seen by PW. (Apr.)Forecast: The Free Press may be looking for a math popularizer in the mold of Amir Aczel, but Mlodinow falls short. Don't look for big sales here.

Copyright 2001 Cahners Business Information, Inc.

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Product Details

• Hardcover: 320 pages
• Publisher: Free Press (April 17, 2001)
• Language: English
• ISBN-10: 0684865238
• ISBN-13: 978-0684865232
• Product Dimensions: 9.3 x 5.4 x 1.1 inches
• Shipping Weight: 1.1 pounds
• Average Customer Review: 4.0 out of 5 stars  See all reviews (60 customer reviews)
• Amazon Best Sellers Rank: #402,733 in Books (See Top 100 in Books)

More About the Author

Leonard Mlodinow was born in Chicago, Illinois, to immigrant Jewish parents who were holocaust survivors. He received his PhD in theoretical physics from the University of California at Berkeley, and is now at Caltech. His book The Drunkard's Walk: How Randomness Rules our Lives was a New York Times Bestseller, Editor's Choice, and Notable Book of the Year, and was short-listed for the Royal Society book award. His other books include two co-authored with physicist Stephen Hawking -- A Briefer History of Time, and The Grand Design. In addition to his books and research articles, he has written for the Wall Street Journal, the New York Times, and Forbes magazine, among other publications, and for television series such as McGyver and Star Trek: the Next Generation. Visit my web site at: http://www.its.caltech.edu/~len/

Customer Reviews

Most Helpful Customer Reviews

60 of 65 people found the following review helpful:

2.0 out of 5 stars Entertaining but flawed, July 30, 2002

By 

Marvin J. Greenberg (Berkeley, CA USA) - See all my reviews
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This review is from: Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace (Hardcover)

Mlodinow ('M' below) writes entertainingly, as most of the other reviews here testify. It's good that the general public get a taste of the excitement of discovery/invention in these fields. He should just correct, in a subsequent edition, the serious distortions that IAS Professor Langlands (Notices of the American Mathematical Society, vol. 49, number 5, p. 554 - referred to as 'L' below) has pointed out; then the book could be a useful, reliable introduction to whet the appetite of people who might want to study the subjects in more depth. Here are a few of Langlands' criticisms:

1. M's portrayals of Proclus, Kant, Kronecker and Gauss' father are unfair caricatures. L provides evidence in their defense.

2. M strives for sensationalism, not fact. E.g., M speculates that Thales traded in leather dildos. Veracity is sacrificed to effect.

3. M missed the main point of Riemann's great 1854 habilitation lecture. L wrote: "I could hardly believe my eyes, but it seems [M] is persuaded that the introduction of elliptic geometry was the principal achievement of the lecture."

Since M acknowledges on p.205 that Einstein's general theory of relativity was based on Riemann's work, M owes the reader much more explanation of Riemann's new ideas, expanding on his p. 207 discussion, not dismissing Riemann by saying his work "wasn't pretty."

4. L criticizes M's account of Einstein's early years, saying: "...to represent Einstein as an academically narrow, misunderstood or mistreated high-school dropout is a cruel disservice to any young reader or to any educator who swallows such falsehoods."

5. L concludes that M's book is "thoroughly dishonest ... simply because the author shrinks from nothing in his desperation to be readable and entertaining."

I didn't dislike the book as much as L did. It would be a significant achievement to make great breakthroughs in mathematics and physics somewhat comprehensible to lay people in an entertaining way and to do so with integrity.

Even L concedes "There would be little point in reviewing the book, were it not that the germ of an excellent monograph is there."

55 of 62 people found the following review helpful:

5.0 out of 5 stars The Shape of Our Universe, May 5, 2001

By 

Timothy Haugh (New York, NY United States) - See all my reviews
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This review is from: Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace (Hardcover)

As a teacher of geometry, I always keep an eye out for books that offer coherent explanations of the importance of this most intriguing and ancient branch of mathematics. This books offers that in spades. In fact, it is one of the best basic overviews of the field I have ever come across.

Mlodinow divides the history of the development of geometry into five major "revolutions." Starting first with Euclid and his Greek contemporaries, Mlodinow traces the field through Descartes and the development of analytic geometry, Gauss and the development of "non-Euclidean" geometries, Einstein and the physical application of these geometries, to Witten and the development of string theory--the attempt to understand the universe as a consequence of geometry. In high school we teach the basics of plane and analytic geometry but few people are aware of how the field has matured since then. This book takes us on that journey.

And it is a wonderful one. Along the way he gives insight not only into the mathematics but also into the personalities that created it. We too often forget that it is people who created this magnificent structure and that it was not just handed down to us perfectly formed. Even more, we need to be reminded that the development continues and people are still contributing to it.

The real achievement of this book, however, is its acessiblity. Despite the fact that most people will only have experience with the material from the first two sections of the book (Euclid and Descartes), Mlodinow's writing is understandable by anyone who has successfully navigated a course in high school geometry. In my view, he offers one of the most lucid explanations of Einstein's work and string theory that I have ever read. His style is engaging and very readable. The ability to bring the importance of mathematics to a general audience is a great gift that Mlodinow seems to have. I hope he continues to exploit it.

If I have a complaint about this book, it is that it could have used a few more diagrams to help the less mathematically minded visualize his explanations. This is a minor point, however. Overall, this book is a great piece of work.

29 of 31 people found the following review helpful:

5.0 out of 5 stars A work both charming and profound, April 16, 2001

By 

Frederick B. Rose (Los Angeles, CA United States) - See all my reviews

This review is from: Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace (Hardcover)

I found Euclid's Window charming and profound. Though Mlodinow's topic is ambitious, his ability to explain with incredible clarity some of the world's most abstruse thinking reminded me of Jim Gleick's work on Chaos. Mlodinow's storytelling seemed something altogether different--a lot more fun than Gleick.

Building on historical anecdote, Mlodinow has stitched together a history of human thought for me in a way no one else has, explaining how millenia of physical observation have translated into conceptual thought. I worried that this might be heavy going, but Mlodinow glories in the fine line of tale and humor;it spills over in his prose with such ease that it appears unconscious. I suspect this may become something of a classic.