Euclid's Window The Story of Geometry from Parallel Lines to Hyperspace
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Summary
Author Biography
Table of Contents
Introduction  ix  
I The Story of Euclid  

3  (1)  

4  (7)  

11  (6)  

17  (12)  

29  (10)  

39  (14)  
II The Story of Descartes  

53  (2)  

55  (5)  

60  (10)  

70  (9)  

79  (11)  

90  (5)  
III The Story of Gauss  

95  (3)  

98  (9)  

107  (8)  

115  (6)  

121  (6)  

127  (9)  

136  (7)  

143  (10)  
IV The Story of Einstein  

153  (4)  

157  (6)  

163  (13)  

176  (6)  

182  (11)  

193  (12)  

205  (5)  

210  (7)  
V The Story of Witten  

217  (2)  

219  (4)  

223  (5)  

228  (3)  

231  (4)  

235  (4)  

239  (10)  

249  (6)  

255  (8)  
Epilogue  263  (4)  
Notes  267  (26)  
Acknowledgments  293  (2)  
Index  295 
Excerpts
Twentyfour 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 largescale 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 probablysomedayattainable speed of onetenth 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 largescale structure of spacetime, 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 spacetime to matter and energy, represented a paradigm change of a magnitude not seen in physics since Newton. It sureseemedradical. 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 largescale structure of spacetime 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
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 isgeometry,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 windowdressing 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
Chapter Seven: The Revolution in Place
How do you know where you are? After the realization that space itself exists, this is perhaps the next natural question. It may seem that the answer is provided by cartography, the study of maps. But cartography is only the beginning. A proper theory of place leads to ideas far deeper than simple statements like "To find Kalamazoo, look in F3."
There is more to location than naming a spot. Imagine an alien emissary landing on earth, a stringy bubbleheaded creature living on oxygen, or perhaps a hairy, apelike individual partial to nitrous oxide. If we wished to communicate, it would be nice if the alien had brought a dictionary. But would that be enough? If your idea of good communication is "Me Tarzan you Jane," it might be, but for an exchange of intergalactic ideas we'd also have to learn each other's grammar. In mathematics, too, the "dictionary"  a system of naming the points in the plane, in space, or on the globe  is just a beginning. The real power of a theory of location resides in the ability to relate different locations, paths, and shapes to each other, and to manipulate them employing equations  in the unification of geometry and algebra.
Today, as one old textbook on the subject states, "With relatively little effort the student may now reach out and grasp these tools." It is hard to imagine what yet greater theories the great astronomer/physicists Kepler and Galileo could have created had the tools of coordinate geometry been familiar to them, but they had to do without. With this knowledge, their successors Newton and Leibniz created calculus and the modern age of physics. Had geometry and algebra remained unrelated, few of the advances of modern physics and engineering would have been possible.
Like the revolution of proof, the first signpost along the way to the revolution of place came in preGreek times, with the invention of maps. Though the Greeks added their particular genius, the end of their civilization left the subject unfinished, and the power unleashed. The next step along the way was the invention of the graph, but this awaited the revival of the intellectual tradition following the Dark Ages. In the end, this revolution trailed by a dozen centuries the last great Greek mathematicians and cartographers.
Copyright © 2001 by Leonard Mlodinow
Chapter Thirteen: The Curved Space Revolution
Euclid aimed to create a consistent mathematical structure based on the geometry of space. The properties of space derived from his geometry are therefore the properties of space as the Greeks understood it. But does space really have the structure described by Euclid and quantified by Descartes? Or are there other possibilities?
We don't know if Euclid would have raised an eyebrow had he been told that hisElementswould remain sacrosanct for 2,000 years, but as they say in the software business, 2,000 years is a long time to wait for version 2. A lot changed in that time: we discovered the structure of the solar system; we gained the ability to sail around, and map, the globe; we stopped drinking diluted wine for breakfast. And, in that time, the mathematicians of the Western world had developed a universal aversion to Euclid's fifth postulate, the parallel postulate. Alas, it was not the content they found distasteful, it was its place as an assumption rather than a theorem.
Through the centuries, the mathematicians who attempted to prove the parallel postulate as a theorem came close to discovering strange and exciting new kinds of space, but each of them was hampered by a simple belief: that the postulate was a true and necessary property of space.
All but one, that is, a teenaged boy of fifteen named Carl Friedrich Gauss, who, as it happened, would become one of Napoleon's heroes. With this young genius's realization in 1792, the seeds of a new revolution were planted. Unlike the previous ones, this would not be a revolutionary improvement on Euclid, it would be an entirely new operating system. Soon the strange and exciting spaces overlooked for so many centuries were discovered and described.
With the discovery of curved spaces came the natural question, is our space Euclid's, or one of those others? That question eventually revolutionized physics. Mathematics, too, was thrown into a quandary. If Euclid's structure isn't simply an abstraction of the true structure of space, then what is it? And if the parallel postulate can be questioned, what about the rest of Euclid's edifice? Soon after the discovery of curved space all of Euclidean geometry came tumbling down, and then  surprise! The rest of mathematics tumbled as well. By the time the dust cleared, not only the theory of space, but physics and mathematics, too, were in a new era.
To understand how difficult a leap it was to contradict Euclid, one has to appreciate how deeply entrenched was his description of space. Already in his own, ancient time, Euclid'sElementswas a classic. Euclid not only defined the nature of mathematics, but his book played a central role as a model of logical thought in education and natural philosophy. It was a key work in the intellectual revival of the Middle Ages. It was one of the first books printed after the invention of the printing press in 1454, and from 1533 until the eighteenth century it was the only one of all the Greek works to exist as a printed text in the original language. Until the nineteenth century, every work of architecture, the composition in every drawing and every painting, every theory and every equation employed in science were all inherently Euclidean.Elementswas not unworthy of its great stature. Euclid transformed our intuition of space into an abstract logical theory from which we could make deductions. Perhaps most of all, we must credit Euclid with attempting to shamelessly bare his assumptions, and never pretending that the theorems he proved were anything more than logical deductions from his few unproven postulates. As we saw in Part I, though, one of these postulates, the parallel postulate, caused consternation in almost every scholar who studied Euclid because it was not as simple and intuitive as Euclid's other assumptions. Recall its wording:
Given a line segment that crosses two lines in a way that the sum of inner angles on the same side is less than a right angle, then the two lines will eventually meet (on that side of the line segment).
Euclid didn't use the parallel postulate at all in proving his first twentyeight theorems. By then he had already proven the converse of the postulate, as well as other statements that seemed far better candidates for "axiomhood"  like the fundamental fact that the lengths of any two sides of a triangle have to add up to more than the length of the third. Why, then, so far down the road, did he need to introduce such an arcane, technical postulate? Did he write that chapter on deadline?
For over 2,000 years, as 100 generations lived and died, as borders changed and political systems rose and fell, as the earth hurtled 1,000 billion miles around and around our sun, thinkers everywhere remained dedicated to Euclid, questioning their god not on any issue of content, but only on this one teeny point: couldn't the ugly parallel postulate be proved?
Copyright © 2001 by Leonard Mlodinow
Chapter TwentyOne: Revolution at the Speed of Light
Gauss and Riemann showed that space could be curved, and gave the mathematics needed to describe it. The next question is, what kind of space do we live in? And, probing deeper: what determines the shape of space?
The answer, given so elegantly and precisely by Einstein in 1915, was actually first proposed in 1854, in broad strokes, by Riemann himself:
The question of the validity of geometry...is related to the question of the internal basis of metric [distance] relationships of the space...we must seek the ground of its metric relations outside it, in the binding forces which act on it....
What makes things far apart or close together? Riemann was too far ahead of his time to be able to develop a concrete theory based upon his insight, too far ahead even for his words to be appreciated. Sixteen years later, though, one mathematician did take notice.
On February 21, 1870, William Kingdon Clifford presented a paper to the Cambridge Philosophical Society entitled "On the Space Theory of Matter." Clifford was twentyfive that year, the same age as Einstein when he published his first articles on special relativity. In his paper, Clifford boldly proclaimed,
I hold in fact: (1) That small portions of space are of a nature analogous to little hills on a surface which is on the average flat. (2) That the property of being curved or distorted is continually passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is really what happens in that phenomenon which we call the motion of matter....
Clifford's conclusions went far beyond Riemann's in their specificity. Which would hardly be notable except for one thing: he got it right. The reaction of a physicist reading this today has got to be, "How did he know?" Einstein came to similar conclusions only after years of careful reasoning. Clifford didn't even have a theory. However, Clifford managed to intuit such detailed conclusions, he, Riemann, and Einstein were all guided by the same simple mathematical idea: if objects in free motion move in the straight lines characteristic of Euclidean space, then might not other kinds of motion be accounted for by the curvature of nonEuclidean space? And in the end, it was precisely Einstein's careful reasoning, based onphysics,not mathematics, that enabled him to develop the theory that Clifford could not.
Clifford worked feverishly on his theory, usually all through the night, for the day was too burdened with teaching and administrative duties at University College London. But without the deep understanding of physics that led Einstein
to the intermediate step of special relativity, and the proper role of time, Clifford had little chance of developing his ideas into a workable theory. The mathematics had preceded the physics  a difficult situation, reminiscent, as we'll see, of the state of string theory today. Clifford got nowhere. He died in 1876, some say of exhaustion, at age thirtythree.
One problem Clifford had was that he found himself leading a parade of one. In the world of physics, the sky was sunny and bright, and few saw reason to spend their time attacking laws in which they saw no sign of corruption. For over 200 years it had seemed that every event in the universe was explicable by Newtonian mechanics, the theory based on the ideas of Isaac Newton. In Newton's view, space is "absolute," a fixed, Godgiven framework upon which to lay the coordinates of Descartes. The path of an object is a line or other curve described by a set of numbers, the coordinates that label the points the path covers in space. The role of time is "to paramatrize" the path, mathematician's lingo for "to tell you where you are along it." For instance, if Alexei is walking up Fifth Avenue at a steady speed of one block per minute, starting at 42nd Street, then his position is simply Fifth Avenue and (42 plus the number of minutes)nd Street. By specifying the number of minutes he has walked, you are determining where along the path he is.
With this understanding of time and space, Newton's laws predict how and why an object like Alexei moves  they give his position as a function of the parameter called time. (This of course assumes he is an inanimate object, which is only true some of the time; picture him with Discman earphones on.) According to Newton, Alexei will continue in uniform motion  in a straight lineandat a constant speed  unless acted upon by an external force, such as the attraction of a video game arcade around the corner. Or, given such an attraction, Newton's laws predict how Alexei's path will differ from uniform motion. They will tell you, quantitatively, exactly how he will move, given his personal inertia and the strength and direction of the force. According to these equations, a body's acceleration (which is change in speedordirection) is proportional to the force applied to it and inversely proportional to its mass. But the description of the motion of a body reacting to a force is only half the picture, known as the "kinetics." To form a complete theory, we also need to know the "dynamics," that is, how to determine the strength and direction of the force, given the source (the arcade), the target (Alexei), and their separation. Newton gave such a force equation for only one type of force, the gravitational force.
Putting the two sets of equations together, the force equations (dynamics) and the motion equations (kinetics), one could (in principle) solve for an object's path as a function of time. One could predict, say, Alexei's orbit around the arcade, or (sadly) the path of a ballistic missile flying between two continents. Newton had fulfilled the ambition, which had begun with Pythagoras, to create a system of mathematics that permits the description of motion. And, by explaining how the same law governs motion on earth and in space, Newton did something else that was equally important: he united two old and separate disciplines  physics, which had been thought of as primarily concerned with everyday human experience, and astronomy, which had been concerned with the motion of heavenly bodies.
* * *
If Newton's view of space and time is true, then it is easy to see two things that cannot be. First, there can be no limit to the speed at which one thing can approach another. To see this, imagine that there is such a limiting speed; call itc.Next, imagine that an object is approaching you at that speed. Now (for the sake of science) spit at the object. If this drama occurs in a tangible medium called absolute space, it is easy to see that the object is now approaching your saliva faster than it is approaching you. The speed limit law is violated. Second, the speed of light cannot be constant. More precisely, light must approach different observers at different speeds. If you race toward light, it will approach you faster than if you run from it.
If an objective structure for space exists, these two truths are selfevident. Yet these two "truths" are false. This is the basis of special relativity, the ingredient missing from earlier speculations on the physics of curved space. It is a fact that was "observed" long before it was "appreciated."
Copyright © 2001 by Leonard Mlodinow
Chapter TwentyNine: The Weird Revolution
Is there a relationship between the nature of space and the laws governing what exists in space? Einstein showed that the presence of matter affects geometry by warping space (and time). It sure seemed radical at the time. But in today's theories, the nature of space and matter are intertwined at a level far more profound than Einstein imagined. Yes, matter may bend space a teeny bit here and, if it truly concentrates, a larger bit there. But, in the new physics, space gets more than ample revenge on matter. According to these theories, the most basic properties of space  such as the number of dimensions  determine the laws of nature and the properties of the matter and energy that make up our universe. Space, the container of the universe, becomes space, the arbiter of what may be.
According to string theory, there exist extra dimensions of space, so small that any wiggle room we have in them isn't observable in presentday experiments (though, indirectly, it may soon be). Though they may be tiny, they, and their topology  i.e., properties related to whether they are shaped, say, like a plane, or a sphere, or a pretzel, or a donut  determine what exists within them (like you and I). Twist those tiny donut dimensions into a pretzel and  poof!  electrons (and thus humans) could be banished from existence. And there's more: string theory, though still poorly understood, has evolved into another theory, Mtheory, of which we know even less, but which seems to be leading us to this conclusion: space and time do not actually exist, but are only approximations of something more complex.
Depending on your personality, you may have a tendency at this point either to laugh or to scream derisive remarks about academics wasting hardearned tax dollars. As we'll see, for many years most physicists themselves had these same reactions. Some still do. But among those working in elementary particle theory today, string theory and Mtheory, though still not rigorous, are de rigueur. And whether or not they, or some later derivative, prove to be some sort of "final theory," they have already changed both mathematics and physics.
With the advent of string theory, physics has veered back toward its partner, mathematics, that abstract discipline concerned, since Hilbert, with rules and not reality. String theory and Mtheory are driven, so far, not by the tradition of new physical insight or experimental data, which are lacking, but by discoveries of their own mathematical structure. It isn't to toast the divining of new particles that the tequila is poured, it is to cheer the discovery that the theory describes the existing ones. Aware that such discoveries are an inversion of the usual course of science, physicists have coined for them the new scientific termpostdiction.In a strange contortion of the scientific method, the theory itself has become the subject of the (mental) experiments; the experimentalists are the theoreticians. It is no accident that Edward Witten, today the theory's leading proponent, has won not a Nobel Prize but a Fields Medal, its mathematical equivalent. For just as geometry and matter reflect on each other, so, now, must the studies of each. Witten goes even farther, saying that string theory should ultimately be a new branch of geometry.
This is not unlike prior revolutions reforming not only the idea of space but also the way in which research on space is approached. The story of this revolution, though, is unlike the stories of prior revolutions in one important aspect: we are still in the midst of it, and no one really knows how it will turn out.
Copyright © 2001 by Leonard Mlodinow
Excerpted from Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace by Leonard Mlodinow
All rights reserved by the original copyright owners. Excerpts are provided for display purposes only and may not be reproduced, reprinted or distributed without the written permission of the publisher.