Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace
320
Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace
320Paperback(Reprint)
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Overview
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.
Product Details
| ISBN-13: | 9780684865249 |
|---|---|
| Publisher: | Free Press |
| Publication date: | 04/09/2002 |
| Edition description: | Reprint |
| Pages: | 320 |
| Sales rank: | 626,697 |
| Product dimensions: | 5.30(w) x 9.18(h) x 0.68(d) |
About the Author
Read an Excerpt
Chapter One: The First Revolution
Euclid was a man who possibly did not discover even one significant law of geometry. Yet he is the most famous geometer ever known and for good reason: for millennia it has been his window that people first look through when they view geometry. Here and now, he is our poster boy for the first great revolution in the concept of space — the birth of abstraction, and the idea of proof.
The concept of space began, naturally enough, as a concept of place, our place, earth. It began with a development the Egyptians and Babylonians called "earth measurement." The Greek word for that is geometry, but the subjects are not at all alike. The Greeks were the first to realize that nature could be understood employing mathematics — that geometry could be applied to reveal, not merely to describe. Evolving geometry from simple descriptions of stone and sand, the Greeks extracted the ideals of point, line, and plane. Stripping away the window-dressing of matter, they uncovered a structure possessing a beauty civilization had never before seen. At the climax of this struggle to invent mathematics stands Euclid. The story of Euclid is a story of revolution. It is the story of the axiom, the theorem, the proof, the story of the birth of reason itself.
Copyright © 2001 by Leonard Mlodinow
Table of Contents
ContentsIntroduction
I THE STORY OF EUCLID
1. The First Revolution
2. The Geometry of Taxation
3. Among the Seven Sages
4. The Secret Society
5. Euclid's Manifesto
6. A Beautiful Woman, a Library, and the End of Civilization
II THE STORY OF DESCARTES
7. The Revolution in Place
8. The Origin of Latitude and Longitude
9. The Legacy of the Rotten Romans
10. The Discreet Charm of the Graph
11. A Soldier's Story
12. Iced by the Snow Queen
III THE STORY OF GAUSS
13. The Curved Space Revolution
14. The Trouble with Ptolemy
15. A Napoleonic Hero
16. The Fall of the Fifth Postulate
17. Lost in Hyperbolic Space
18. Some Insects Called the Human Race
19. A Tale of Two Aliens
20. After 2,000 Years, a Face-lift
IV THE STORY OF EINSTEIN
21. Revolution at the Speed of Light
22. Relativity's Other Albert
23. The Stuff of Space
24. Probationary Technical Expert, Third Class
25. A Relatively Euclidean Approach
26. Einstein's Apple
27. From Inspiration to Perspiration
28. Blue Hair Triumphs
V THE STORY OF WITTEN
29. The Weird Revolution
30. Ten Things I Hate About Your Theory
31. The Necessary Uncertainty of Being
32. Clash of the Titans
33. A Message in a Kaluza-Klein Bottle
34. The Birth of Strings
35. Particles, Schmarticles!
36. The Trouble with Strings
37. The Theory Formerly Known As Strings
Epilogue
Notes
Acknowledgments
Index
What People are Saying About This
Edward Witten, California Institute of Technology
Mlodinow leads the reader on a fascinating tour through the history of geometry, from ancient times to our modern-day fumblings in trying to understand string theory. The book is written with grace and charm.
Euclid's Window is a very good introduction to geometry, from Euclid to Einstein. Readable and entertaining.
—(Amir Aczel, author of Fermat's Last Theorem)
Amy Brunvand, University of Utah Lib, Salt Lake City
This surprisingly exciting history of how mathematicians and physicists discovered geometric space beyond Euclid's three dimensions ... does an excellent job of explaining the importance of the study of geometry without making the reader learn any geometry. For all math and science collections.
How often can you say that a book on math-on math!-is a real page-turner? Well, this one is. As engaging as a soap opera, as fascinating as a whodunnit, as funny as the Sunday comics, Mlodinow's book is story-telling at its best.
—(Michael Guillen, Ph.D., author of Five Equations That Changed the World)
This is an exhilarating book, one that celebrates geometry as one of mathematics' shining suns. And it is an important book, if only because that sun has for too long been covered by a numver of scudding clouds. And it is, finally, a lovely book, one that reflects the radiance of its subject and so warms even as it instructs.
—(David Berlinski, author of A Tour of the Calculus)
If there is one thing that progress in physics confirms again and again, it is that geometry is a powerful conceptual framework for describing and understanding the universe. In Euclid's Window, Leonard Mlodinow tells the intriguing story of geometry, from antiquity through the exciting and mind-bending developments of superstring theory. There is perhaps no better way to prepare for the scientific breakthroughts of tomorrow than to learn the language of geometry, and Euclid's Window makes this task lively and enjoyable.
—(Brian Greene, author of The Elegant Universe)
Interviews
Exclusive Author Essay
No matter who is credited with a scientific discovery, in the end, each of us involved in science has to rediscover it anew. Playing baseball when I was 8 years old, I began to think that, while there was obviously a geometry to the diamond, there must also be a geometry to the arc of the ball in the air. Maybe understanding it would make me a better hitter... I learned about spin and air resistance. Soon, geometry had opened the door to a love of physics. I discovered non-Euclidean geometry a couple of years later while picking through stacks of battered old books in a rummage sale. I knew what Euclidean geometry was by then, and something about trajectories, but somehow the term "non-Euclidean" seemed enchanting. It was as if I had been eating hamburgers all my life and suddenly stumbled upon a bacon cheeseburger. I paid a quarter for the book and flipped through it. I marveled at the part about how "parallel" lines could intersect. An 11-year-old doesn't forget things like that. What was "curved space"? How would a baseball fly in that?
I went on to get an advanced degree and conduct research in mathematical physics. I pretty much gave up baseball and started writing stories when I wasn't doing mathematical physics (or doing my laundry). To me, telling stories and doing science never seemed that different. One is phrased in language, the other in mathematics; but the thrill of each resides in creating or exploring new worlds. Eventually, I got to merge writing and science when I was offered a job writing for Star Trek: the Next Generation. I ended up writing for numerous shows, even sitcoms such as Night Court, in which I was prone to building plots around mad scientists and baseball.
Then kids came, and a responsible job as a vice president with an office in downtown New York. A couple of years ago I decided to write Euclid's Window for the child I hoped still lurked somewhere inside me. Could I recapture that excitement about the way geometry underlies everything? From standing on that baseball diamond to arguing physics with Richard Feynman at Cal Tech to dreaming up a Star Trek story to discussing math with my two boisterous boys, it has always seemed to me that geometry -- just understanding the space around us near and far -- is at the heart of much of human civilization. The best way to convey my vision of this wonderful art was to tell the stories of the five people I see as the poster boys of the great revolutions that occurred over the last 3,000 years or so: Euclid, Descartes, Gauss, Einstein, and Witten -- the last of whom is still very much alive, wasn't happy about being set up alongside these hall of famers, and will probably never really forgive me for doing it anyway.
My plan was ambitious: to take the reader on a voyage of 3,000 years, through all the revolutions in thought that brought us from Euclid to today's twisted 11-dimensional world of string theory, and to do it without letting the mathematics interfere with the story, which really is a page-turner. It was a far bigger project than I imagined. But I'm still alive and look forward to the time when, in a few years, my eldest will be able to understand my book. While I hope that it will inspire him as I was inspired, I know one thing is certain: To find it he won't have to go searching through any bins at the rummage sale. (Leonard Mlodinow)
Introduction
Twenty-four centuries ago, a Greek man stood at the sea's edge watching ships disappear in the distance. Aristotle must have passed much time there, quietly observing many vessels, for eventually he was struck by a peculiar thought. All ships seemed to vanish hull first, then masts and sails. He wondered, how could that be? On a flat earth, ships should dwindle evenly until they disappear as a tiny featureless dot. That the masts and sails vanish first, Aristotle saw in a flash of genius, is a sign that the earth is curved. To observe the large-scale structure of our planet, Aristotle had looked through the window of geometry.
Today we explore space as millennia ago we explored the earth. A few people have traveled to the moon. Unmanned ships have ventured to the outer reaches of the solar system. It is feasible that within this millennium we will reach the nearest star — a journey of about fifty years at the probably-some-day-attainable speed of one-tenth the speed of light. But measured even in multiples of the distance to Alpha Centauri, the outer reaches of the universe are several billion measuring sticks away. It is unlikely that we will ever be able to watch a vessel approach the horizon of space as Aristotle did on earth. Yet we have discerned much about the nature and structure of the universe as Aristotle did, by observing, employing logic, and staring blankly into space an awful lot. Over the centuries, genius and geometry have helped us gaze beyond our horizons. What can you prove about space? How do you know where you are? Can space be curved? How many dimensions are there? How does geometry explain the natural order and unity of the cosmos? These are the questions behind the five geometric revolutions of world history.
It started with a little scheme hatched by Pythagoras: to employ mathematics as the abstract system of rules that can model the physical universe. Then came a concept of space removed from the ground we trod upon, or the water we swam through. It was the birth of abstraction and proof. Soon the Greeks seemed to be able to find geometric answers to every scientific question, from the theory of the lever to the orbits of the heavenly bodies. But Greek civilization declined and the Romans conquered the Western world. One day just before Easter in A.D. 415, a woman was pulled from a chariot and killed by an ignorant mob. This scholar, devoted to geometry, to Pythagoras, and to rational thought, was the last famous scholar to work in the library at Alexandria before the descent of civilization into the thousand years of the Dark Ages.
Soon after civilization reemerged, so did geometry, but it was a new kind of geometry. It came from a man most civilized — he liked to gamble, sleep until the afternoon, and criticize the Greeks because he considered their method of geometric proof too taxing. To save mental labor, René Descartes married geometry and number. With his idea of coordinates, place and shape could be manipulated as never before, and number could be visualized geometrically. These techniques enabled calculus and the development of modern technology. Thanks to Descartes, geometric concepts such as coordinates and graphs, sines and cosines, vectors and tensors, angles and curvature, appear in every context of physics from solid state electronics to the large-scale structure of space-time, from the technology of transistors and computers to lasers and space travel. But Descartes's work also enabled a more abstract — and revolutionary — idea, the idea of curved space. Do all triangles really have angle sums of 180 degrees, or is that only true if the triangle is on a flat piece of paper? It is not just a question of origami. The mathematics of curved space caused a revolution in the logical foundations, not only of geometry but of all of mathematics. And it made possible Einstein's theory of relativity. Einstein's geometric theory of space and that extra dimension, time, and of the relation of space-time to matter and energy, represented a paradigm change of a magnitude not seen in physics since Newton. It sure seemed radical. But that was nothing, compared to the latest revolution.
One day in June 1984, a scientist announced that he had made a breakthrough in the theory that would explain everything from why subatomic particles exist, and how they interact, to the large-scale structure of space-time and the nature of black holes. This man believed that the key to understanding the unity and order of the universe lies in geometry — geometry of a new and rather bizarre nature. He was carried off the stage by a group of men in white uniforms.
It turned out the event was staged. But the sentiment and genius were real. John Schwarz had been working for a decade and a half on a theory, called string theory, that most physicists reacted to in much the same way they would react to a stranger with a crazed expression asking for money on the street. Today, most physicists believe that string theory is correct: the geometry of space is responsible for the physical laws governing that which exists within it.
The manifesto of the seminal revolution in geometry was written by a mystery man named Euclid. If you don't recall much of that deadly subject called Euclidean Geometry, it is probably because you slept through it. To gaze upon geometry the way it is usually presented is a good way to turn a young mind to stone. But Euclidean geometry is actually an exciting subject, and Euclid's work 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 through which the nature of our universe has been revealed. And as his geometry has passed through four more revolutions, scientists and mathematicians have shattered theologians' beliefs, destroyed philosophers' treasured worldviews, and forced us to reexamine and reimagine our place in the cosmos. These revolutions, and the prophets and stories behind them, are the subject of this book.
Copyright © 2001 by Leonard Mlodinow
