Principal
Shape of Inner Space
Shape of Inner Space
Shingtung Yau, Steve Nadi
String theory says we live in a tendimensional universe, but that only four are accessible to our everyday senses. According to theorists, the missing six are curled up in bizarre structures known as CalabiYau manifolds. In The Shape of Inner Space, ShingTung Yau, the man who mathematically proved that these manifolds exist, argues that not only is geometry fundamental to string theory, it is also fundamental to the very nature of our universe. Time and again, where Yau has gone, physics has followed. Now for the first time, readers will follow Yau’s penetrating thinking on where we’ve been, and where mathematics will take us next. A fascinating exploration of a world we are only just beginning to grasp, The Shape of Inner Space will change the way we consider the universe on both its grandest and smallest scales.
Año:
2010
Editorial:
Basic Books
Idioma:
english
ISBN 13:
9780465023141
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MOBI , 2.43 MB
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Table of Contents Title Page SPACE/TIME PREFACE PRELUDE One  A UNIVERSE IN THE MARGINS Two  GEOMETRY IN THE NATURAL ORDER Three  A NEW KIND OF HAMMER Four  TOO GOOD TO BE TRUE Five  PROVING CALABI Six  THE DNA OF STRING THEORY Seven  THROUGH THE LOOKING GLASS Eight  KINKS IN SPACETIME Nine  BACK TO THE REAL WORLD Ten  BEYOND CALABIYAU Eleven  THE UNIVERSE UNRAVELS Twelve  THE SEARCH FOR EXTRA DIMENSIONS Thirteen  TRUTH, BEAUTY, AND MATHEMATICS Fourteen  THE END OF GEOMETRY? Epilogue Postlude A FLASH IN THE MIDDLE OF A LONG NIGHT NOTES GLOSSARY INDEX Copyright Page SPACE/TIME Time, time why does it vanish? All manner of things what infinite variety. Three thousand rivers all from one source. Time, space mind, matter, reciprocal. Time, time it never returns. Space, space how much can it hold? In constant motion always in flux. Black holes lurking mysteries afoot. Space and time one without bounds. Infinite, infinite the secrets of the universe. Inexhaustible, lovely in every detail. Measure time, measure space no one can do it. Watched through a straw what’s to be learned has no end. SHINGTUNG YAU BEIJING, 2002 PREFACE Mathematics is often called the language of science, or at least the language of the physical sciences, and that is certainly true: Our physical laws can only be stated precisely in terms of mathematical equations rather than through the written or spoken word. Yet regarding mathematics as merely a language doesn’t do justice to the subject at all, as the word leaves the erroneous impression that, save for some minor tweaks here and there, the whole business has been pretty well sorted out. In fact, nothing could be further from the truth. Although scholars have built a strong foundation over the course of hundreds—and indeed thousands—of years, mathematics is still very much a thriving and dynamic enterprise. Rather than being a static body of knowledge (not to suggest that languages themselves are; set in stone), mathematics is actually a dynamic, evolving science, with new insights and discoveries made every day rivaling those made in other branches of science, though mathematical discoveries don’t capture the headlines in the same way that the discovery of a new elementary particle, a new planet, or a new cure for cancer does. In fact, save for the proof of a centuriesold problem from time to time, they rarely capture headlines at all. Yet for those who appreciate the sheer force of mathematics, it can be viewed as not just a language but as the surest path to the truth—the bedrock upon which the whole edifice of physical science rests. The strength of this discipline, again, lies not simply in its ability to explain physical reality or to reveal it, because to a mathematician, mathematics is reality. The geometric figures and spaces, whose existence we prove, are just as real to us as are the elementary particles of physics that make up all matter. But we consider mathematical structures even more fundamental than the particles of nature because mathematical structures can be used not only to understand such particles but also to understand the phenomena of everyday life, such as the contours of a human face or the symmetry of flowers. What excites geometers perhaps most of all is the power and beauty of the abstract principles that underlie the familiar forms and shapes of our contemporary world. For me, the study of mathematics and my specialty, geometry, has truly been an adventure. I still recall the thrill I felt during my first year of graduate school, when—as a twentyyearold fresh off the boat, so to speak—I first learned about Einstein’s theory of gravity. I was struck by the notion that gravity and curvature could be regarded as one and the same, as I’d already become fascinated with curved surfaces during my undergraduate years in Hong Kong. Something about these shapes appealed to me on a visceral level. I don’t know why, but I couldn’t stop thinking about them. Hearing that curvature lay at the heart of Einstein’s theory of general relativity gave me hope that someday, and in some way, I might be able to contribute to our understanding of the universe. The Shape of Inner Space describes my explorations in the field of mathematics, focusing on one discovery in particular that has helped some scientists build models of the universe. No one can say for sure whether these models will ultimately prove correct. But the theory underlying these models, nevertheless, possesses a beauty that I find undeniable. Taking on a book of this nature has been challenging, to say the least, for someone like me who’s more comfortable with geometry and nonlinear differential equations than writing in the English language, which is not my native tongue. I find it frustrating because there’s a great clarity, as well as a kind of elegance, in mathematical equations that is difficult, if not impossible, to express in words. It’s a bit like trying to convey the majesty of Mount Everest or Niagara Falls without any pictures. Fortunately, I’ve gotten some wellneeded help on this front. Although this narrative is told through my eyes and in my voice, my coauthor has been responsible for translating the abstract and abstruse mathematics into (hopefully) lucid prose. When I proved the Calabi conjecture—an effort that lies at the heart of this book—I dedicated the paper containing that proof to my late father, Chen Ying Chiu, an educator and philosopher who instilled in me a respect for the power of abstract thought. I dedicate this book to him and to my late mother, Leung Yeuk Lam, both of whom had a profound influence on my intellectual growth. In addition, I want to pay tribute to my wife, YuYun, who has been so tolerant of my rather excessive (and perhaps obsessive) research and travel schedule, and to my sons, Isaac and Michael, of whom I’m very proud. I also dedicate this book to Eugenio Calabi, the author of the aforementioned conjecture, whom I’ve known for nearly forty years. Calabi were an enormously original mathematician with whom I’ve been linked for more than a quarter century through a class of geometric objects, CalabiYau manifolds, which serve as the principal subject of this book. The term CalabiYau has been so widely used since it was coined in 1984 that I almost feel as if Calabi is my first name. And if it is to be my first name—at least in the public’s mind—it’s one I’m proud to have. The work that I do, much of which takes place along the borders between mathematics and theoretical physics, is rarely done in isolation, and I have benefited greatly from interactions with friends and colleagues. I’ll mention a few people, among many, who have collaborated with me directly or inspired me in various ways. First, I’d like to pay tribute to my teachers and mentors, a long line of illustrious people that includes S. S. Chern, Charles Morrey, Blaine Lawson, Isadore Singer, Louis Nirenberg, and the aforementioned Calabi. I’m pleased that Singer invited Robert Geroch to speak at a 1973 Stanford conference, where Geroch inspired my work with Richard Schoen on the positive mass conjecture. My subsequent interest in physicsrelated mathematics has always been encouraged by Singer. I’m grateful for the conversations I had on general relativity while visiting Stephen Hawking and Gary Gibbons at Cambridge University. I learned about quantum field theory from one of the masters of the subject, David Gross. I remember in 1981, when I was a professor at the Institute for Advanced Study, the time Freeman Dyson brought a fellow physicist, who had just arrived in Princeton, into my office. The newcomer, Edward Witten, told me about his soontobepublished proof of the positive mass conjecture—a conjecture I had previously proved with a colleague using a very different technique. I was struck, for the first of many times to come, by the sheer force of Witten’s mathematics. Over the years, I’ve enjoyed close collaborations with a number of people, including Schoen (mentioned above), S. Y. Cheng, Richard Hamilton, Peter Li, Bill Meeks, Leon Simon, and Karen Uhlenbeck. Other friends and colleagues who have added to this adventure in many ways include Simon Donaldson, Robert Greene, Robert Osserman, Duong Hong Phong, and HungHsi Wu. I consider myself especially lucky to have spent the past twentyplus years at Harvard, which has provided an ideal environment for interactions with both mathematicians and physicists. During my time here, I’ve gained many insights from talking to Harvard math colleagues—such as Joseph Bernstein, Noam Elkies, Dennis Gaitsgory, Dick Gross, Joe Harris, Heisuke Hironaka, Arthur Jaffe (also a physicist), David Kazdhan, Peter Kronheimer, Barry Mazur, Curtis McMullen, David Mumford, Wilfried Schmid, YumTong Siu, Shlomo Sternberg, John Tate, Cliff Taubes, Richard Taylor, H. T. Yau, and the late Raoul Bott and George Mackey—while having memorable exchanges with MIT math colleagues as well. On the physics side, I’ve had countless rewarding conversations with Andy Strominger and Cumrun Vafa. In the past ten years, I was twice an Eilenberg visiting professor at Columbia, where I had many stimulating conversations with faculty members, especially with Dorian Goldfeld, Richard Hamilton, Duong Hong Phong, and S. W. Zhang. I was also a Fairchild visiting professor and Moore visiting professor at Caltech, where I learned a lot from Kip Thorne and John Schwarz. Over the last twentythree years, I have been supported by the U.S. government through the National Science Foundation, the Department of Energy, and DARPA in my research related to physics. Most of my postdoctoral fellows received their Ph.D.s in physics, which is somewhat unusual in our discipline of mathematics. But the arrangement has been mutually beneficial, as they have learned some mathematics from me and I have learned some physics from them. I am glad that many of these postdoctoral fellows with physics backgrounds later became prominent professors in mathematics departments at Brandeis, Columbia, Northwestern, Oxford, Tokyo, and other universities. Some of my postdocs have done important work on CalabiYau manifolds, and many of them have also helped on this book: Mboyo Esole, Brian Greene, Gary Horowitz, Shinobu Hosono, Tristan Hubsch, Albrecht Klemm, Bong Lian, James Sparks, LiSheng Tseng, Satoshi Yamaguchi, and Eric Zaslow. Finally, my former graduate students—including Jun Li, Kefeng Liu, Melissa Liu, Dragon Wang, and MuTao Wang—have made noteworthy contributions in this area as well, some of which will be described in the pages to come. —SHINGTUNG YAU, CAMBRIDGE, MASSACHUSETTS, MARCH 2010 Odds are I never would have known about this project were it not for Henry Tye, a Cornell physicist (and a friend of Yau’s), who suggested that my coauthortobe might steer me to an interesting tale or two. Henry was right about this, as he has been about many other things. I’m grateful to him for helping to launch me on this unexpected journey and for assisting me at many junctures along the way. As Yau has often said, when you venture down a path in mathematics, you never know where it will end up. The same has been true on the writing end of things. The two of us pretty much agreed during our very first meeting to write a book together, though it took a long while for us to know what the book would be about. In some ways, you might say we didn’t really know that until the book was finished. Now a few words about the product of this collaboration in an attempt to keep any confusion to a minimum. My coauthor is, of course, a mathematician whose work is central to much of the story related here. Sections of the book in which he was an active participant are generally written in the first person, with the “I” in this case referring to him and him alone. However, even though the book has its fair share of personal narrative, this work should probably not be characterized as Yau’s autobiography or biography. That’s because part of the discussion relates to people Yau doesn’t know (or who died long before he was born), and some of the subject matter described—such as experimental physics and cosmology—lies outside his areas of expertise. These sections, which are written in a thirdperson voice, are largely based on interviews and other research I conducted. While the book is, admittedly, an unusual blend of our different backgrounds and perspectives, it seemed to be the best way for the two of us to recount a story that we both considered worth telling. The task of actually getting this tale down on paper relied heavily on my coauthor’s extraordinary grasp of numbers and hopefully profited as well from his collaborator’s facility with words. One other point on the issue of whether this ought to be regarded as an autobiography: Although the book certainly revolves around Yau’s work, I would suggest that the main character is not Yau himself but rather the class of geometric shapes—socalled CalabiYau manifolds—that he helped invent. Broadly speaking, this book is about understanding the universe through geometry. General relativity, a geometrybased description of gravity that has achieved stunning success in the past century, offers one example. String theory represents an ambitious attempt to go even further, and geometry is vital to this quest, with sixdimensional CalabiYau shapes assuming a special place in this theory. The book tries to present some of the ideas from geometry and physics needed to understand where CalabiYau manifolds came from and why some physicists and mathematicians consider them important. The book focuses on various aspects of these manifolds—their defining features, the mathematics that led to their discovery, the reasons string theorists find them intriguing, and the question of whether these shapes hold the key to our universe (and perhaps to other universes as well). That, at least, is what The Shape of Inner Space is supposed to be about. Whether it lives up to that billing may be open to debate. But there is no doubt in my mind that this book would never have come to fruition without technical, editorial, and emotional support from many people—too many, I’m afraid, to list in full, but I will mention as many as I can. I received a tremendous amount of help from people already singled out by my coauthor. These include Eugenio Calabi, Simon Donaldson, Brian Greene, Tristan Hubsch, Andrew Strominger, LiSheng Tseng, Cumrun Vafa, Edward Witten, and, most of all, Robert Greene, Bong Lian, and LiSheng Tseng. The latter three provided me with math and physics tutorials throughout the writing process, exhibiting expository skills and levels of patience that boggle the mind. Robert Greene, in particular, spoke with me a couple of times a week during busy stretches to guide me through thorny bits of differential geometry. Without him, I would have been sunk—many times over. Lian got me started in thinking about geometric analysis, and Tseng helped out immensely with lastminute changes in our everevolving manuscript. The physicists Allan Adams, Chris Beasley, Shamit Kachru, Liam McAllister, and Burt Ovrut fielded questions from me at various times of day and night, carrying me through many a rough patch. Other individuals who were exceedingly generous with their time include Paul Aspinwall, Melanie Becker, Lydia Bieri, Volker Braun, David Cox, Frederik Denef, Robbert Dijkgraaf, Ron Donagi, Mike Douglas, Steve Giddings, Mark Gross, Arthur Hebecker, Petr Horava, Matt Kleban, Igor Klebanov, Albion Lawrence, Andrei Linde, Juan Maldacena, Dave Morrison, Lubos Motl, Hirosi Ooguri, Tony Pantev, Ronen Plesser, Joe Polchinski, Gary Shui, Aaron Simons, Raman Sundrum, Wati Taylor, Bret Underwood, Deane Yang, and Xi Yin. That is merely the tip of the iceberg, as I’ve also received help from Eric Adelberger, Saleem Ali, Bruce Allen, Nima ArkaniHamed, Michael Atiyah, John Baez, Thomas Banchoff, Katrin Becker, George Bergman, Vincent Bouchard, Philip Candelas, John Coates, Andrea Cross, Lance Dixon, David Durlach, Dirk Ferus, Felix Finster, Dan Freed, Ben Freivogel, Andrew Frey, Andreas Gathmann, Doron Gepner, Robert Geroch, Susan Gilbert, Cameron Gordon, Michael Green, Paul Green, Arthur Greenspoon, Marcus Grisaru, Dick Gross, Monica Guica, Sergei Gukov, Alan Guth, Robert S. Harris, Matt Headrick, Jonathan Heckman, Dan Hooper, Gary Horowitz, Stanislaw Janeczko, Lizhen Ji, Sheldon Katz, Steve Kleiman, Max Kreuzer, Peter Kronheimer, Mary Levin, Avi Loeb, Feng Luo, Erwin Lutwak, Joe Lykken, Barry Mazur, William McCallum, John McGreevy, Stephen Miller, Cliff Moore, Steve Nahn, Gail Oskin, Rahul Pandharipande, Joaquín Pérez, Roger Penrose, Miles Reid, Nicolai Reshetikhin, Kirill Saraikin, Karen Schaffner, Michael Schulz, John Schwarz, Ashoke Sen, Kris Snibbe, Paul Shellard, Eva Silverstein, Joel Smoller, Steve Strogatz, Leonard Susskind, Yan Soibelman, Erik Swanson, Max Tegmark, Ravi Vakil, Fernando Rodriguez Villegas, Dwight Vincent, Dan Waldram, Devin Walker, Brian Wecht, Toby Wiseman, Jeff Wu, Chen Ning Yang, Donald Zeyl, and others. Many of the concepts in this book are difficult to illustrate, and we were fortunate to be able to draw on the extraordinary graphic talents of Xiaotian (Tim) Yin and Xianfeng (David) Gu of the Stony Brook Computer Science Department, who were assisted in turn by Huayong Li and Wei Zeng. Additional help on the graphics front was provided by Andrew Hanson (the premier renderer of CalabiYau manifolds), John Oprea, and Richard Palais, among others. I thank my many friends and relatives, including Will Blanchard, John De Lancey, Ross Eatman, Evan Hadingham, Harris McCarter, and John Tibbetts, who read drafts of the book proposal and chapters or otherwise offered advice and encouragement along the way. Both my coauthor and I are grateful for the invaluable administrative assistance provided by Maureen Armstrong , Lily Chan, Hao Xu, and Gena Bursan. Several books proved to be valuable references. Among them are The Elegant Universe by Brian Greene, Euclid’s Window by Leonard Mlodinow, Poetry of the Universe by Robert Osserman, and The Cosmic Landscape by Leonard Susskind. The Shape of Inner Space might never have gotten off the ground were it not for the help of John Brockman, Katinka Matson, Michael Healey, Max Brockman, Russell Weinberger, and others at the Brockman, Inc., literary agency. T. J. Kelleher of Basic Books had faith in our manuscript when others did not, and—with the help of his colleague, Whitney Casser—worked hard to get our book into a presentable form. Kay Mariea, the project editor at Basic Books, shepherded our manuscript through its many stages, and Patricia Boyd provided expert copyediting, teaching me that “the same” and “exactly the same” are exactly the same thing. Finally, I’m especially grateful for the support from my family members—Melissa, Juliet, and Pauline, along with my parents Lorraine and Marty, my brother Fred, and my sister Sue—who acted as if sixdimensional CalabiYau manifolds were the most fascinating thing in the world, not realizing that these manifolds are, in fact, out of this world. —STEVE NADIS, CAMBRIDGE, MASSACHUSETTS, MARCH 2010 PRELUDE THE SHAPES OF THINGS TO COME God ever geometrizes. —PLATO In the year 360 B.C. or thereabouts, Plato published Timaeus—a creation story told in the form of a dialogue between his mentor, Socrates, and three others: Timaeus, Hermocrates, and Critias. Timaeus, likely a fictitious character who is said to have come to Athens from the southern Italian city of Locri, is an “expert in astronomy [who] has made it his main business to know the nature of the universe.”1 Through Timaeus, Plato presents his own theory of everything, with geometry playing a central role in those ideas. Plato was particularly fascinated with a group of convex shapes, a special class of polyhedra that have since come to be known as the Platonic solids. The faces of each solid consist of identical polygons. The tetrahedron, for example, has four faces, each a triangle. The hexahedron, or cube, is made up of six squares. The octahedron consists of eight triangles, the dodecahedron of twelve pentagons, and the icosahedron of twenty triangles. Plato did not invent the solids that bear his name, and no one knows who did. It is generally believed, however, that one of his contemporaries, Theaetetus, was the first to prove that five, and only five, such solids—or convex regular polyhedra, as they’re called—exist. Euclid gave a complete mathematical description of these geometric forms in The Elements. 0.1—The five Platonic solids, named for the Greek philosopher Plato: the tetrahedron, hexahedron (or cube), octahedron, dodecahedron, and icosahedron. The prefixes derive from the number of faces: four, six, eight, twelve, and twenty, respectively. One feature of these solids that no other convex polyhedra satisfy is that all their faces, edges, and angles (between two edges) are congruent. The Platonic solids have several intriguing properties, some of which turn out to be equivalent ways of describing them. For each type of solid, the same number of faces meet at each of the corner points, or vertices. One can draw a sphere around the solid that touches every one of those vertices—something that’s not possible for polyhedra in general. Moreover, the angle of each vertex, where two edges meet, is always the same. The number of vertices plus faces equals the number of edges plus two. Plato attached a metaphysical significance to the solids, which is why his name is forever linked with them. In fact, the convex regular polyhedra, as detailed in Timaeus, formed the very essence of his cosmology. In Plato’s grand scheme of things, there are four basic elements: earth, air, fire, and water. If we could examine these elements in fine detail, we’d notice that they are composed of minuscule versions of the Platonic solids: Earth would thus consist of tiny cubes, the air of octahedrons, fire of tetrahedrons, and water of icosahedrons. “One other construction, a fifth, still remained,” Plato wrote in Timaeus, referring to the dodecahedron. “And this one god used for the whole universe, embroidering figures on it.”2 As seen today, with the benefit of 2,000plus years of science, Plato’s conjecture looks rather dubious. While there is, at present, no ironclad agreement as to the basic building blocks of the universe—be they leptons and quarks, or hypothetical subquarks called preons, or equally hypothetical and even tinier strings—we do know that it’s not just earth, air, fire, and water embroidered upon one giant dodecahedron. Nor do we believe that the properties of the elements are governed strictly by the shapes of Platonic solids. On the other hand, Plato never claimed to have arrived at the definitive theory of nature. He considered Timaeus a “likely account,” the best he could come up with at the time, while conceding that others who came after him might very well improve on the picture, perhaps in a dramatic way. As Timaeus states midway into his discourse: “If anyone puts this claim to the test and discovers that it isn’t so, his be the prize, with our congratulations.”3 There’s no question that Plato got many things wrong, but viewing his thesis in the broadest sense, it’s clear that he got some things right as well. The eminent philosopher showed perhaps the greatest wisdom in acknowledging that what he put forth might not be true, but that another theory, perhaps building on some of his ideas, could be true. The solids, for instance, are objects of extraordinary symmetry: The icosahedron and dodecahedron, for instance, can be rotated sixty ways (which, not coincidentally, turns out to be twice the number of edges in each shape) and still look the same. In basing his cosmology on these shapes, Plato correctly surmised that symmetry ought to lie at the heart of any credible description of nature. For if we are ever to produce a real theory of everything—in which all the forces are unified and all the constituents obey a handful (or two) of rules—we’ll need to uncover the underlying symmetry, the simplifying principle from which everything else springs. It hardly bears mentioning that the symmetry of the solids is a direct consequence of their precise shape or geometry. And this is where Plato made his second big contribution: In addition to realizing that mathematics was the key to fathoming our universe, he introduced an approach we now call the geometrization of physics—the same leap that Einstein made. In an act of great prescience, Plato suggested that the elements of nature, their qualities, and the forces that act upon them may all be the result of some hidden geometrical structure that conducts its business behind the scenes. The world we see, in other words, is a mere reflection of the underlying geometry that we might not see. This is a notion dear to my heart, and it relates closely to the mathematical proof for which I am best known—to the extent that I am known at all. Though it may strike some as farfetched, yet another case of geometric grandstanding, there just might be something to this idea, as we’ll see in the pages ahead. One A UNIVERSE IN THE MARGINS The invention of the telescope, and its steady improvement over the years, helped confirm what has become a truism: There’s more to the universe than we can see. Indeed, the best available evidence suggests that nearly threefourths of all the stuff of the cosmos lies in a mysterious, invisible form called dark energy. Most of the rest—excluding only the 4 percent composed of ordinary matter that includes us—is called dark matter. And true to form, it too has proved “dark” in just about every respect: hard to see and equally hard to fathom. The portion of the cosmos we can see forms a sphere with a radius of about 13.7 billion lightyears. This sphere is sometimes referred to as a Hubble volume, but no one believes that’s the full extent of the universe. According to the best current data, the universe appears to extend limitlessly, with straight lines literally stretching from here to eternity in every direction we can point. There’s a chance, however, that the universe is ultimately curved and bounded. But even if it is, the allowable curvature is so slight that, according to some analyses, the Hubble volume we see is just one out of at least one thousand such volumes that must exist. And a recently launched space instrument, the Planck telescope, may reveal within a few years that there are at least one million Hubble volumes out there in the cosmos, only one of which we’ll ever have access to.1 I’m trusting the astrophysicists on this one, realizing that some may quarrel with the exact numbers cited above. One fact, however, appears to be unassailable: What we see is just the tip of the iceberg. At the other extreme, microscopes, particle accelerators, and various imaging devices continue to reveal the universe on a miniature scale, illuminating a previously inaccessible world of cells, molecules, atoms, and smaller entities. By now, none of this should be all that surprising. We fully expect our telescopes to probe ever deeper into space, just as our microscopes and other tools bring more of the invisible to light. But in the last few decades—owing to developments in theoretical physics, plus some advances in geometry that I’ve been fortunate enough to participate in—there has been another realization that is even more startling: Not only is there more to the universe than we can see, but there may even be more dimensions, and possibly quite a few more than the three spatial dimensions we’re intimately acquainted with. That’s a tough proposition to swallow, because if there’s one thing we know about our world—if there’s one thing our senses have told us from our first conscious moments and first groping explorations—it’s the number of dimensions. And that number is three. Not three, give or take a dimension or so, but exactly three. Or so it seemed for the longest time. But maybe, just maybe, there are additional dimensions so small that we haven’t noticed them yet. And despite their modest size, they could be crucial in ways we could not have possibly appreciated from our entrenched, threedimensional perspective. While this may be hard to accept, we’ve learned in the past century that whenever we stray far from the realm of everyday experience, our intuition can fail us. If we travel extremely fast, special relativity tells us that time slows down, not something you’re likely to intuit from common sense. If we make an object extremely small, according to the dictates of quantum mechanics, we can’t say exactly where it is. When we do experiments to determine whether the object has ended up behind Door A or Door B, we find it’s neither here nor there, in the sense that it has no absolute position. (And it sometimes may appear to be in both places at once!) Strange things, in other words, can and will happen, and it’s possible that tiny, hidden dimensions are one of them. If this idea is true, then there might be a kind of universe in the margins—a critical chunk of real estate tucked off to the side, just beyond the reach of our senses. This would be revolutionary in two ways. The mere existence of extra dimensions—a staple of science fiction for more than a hundred years—would be startling enough on its own, surely ranking among the greatest findings in the history of physics. But such a discovery would really be a starting point rather than an end unto itself. For just as a general might obtain a clearer perspective on the battlefield by observing the proceedings from a hilltop or tower and thereby gaining the benefit of a vertical dimension, so too may our laws of physics become more apparent, and hence more readily discerned, when viewed from a higherdimensional vantage point. We’re familiar with travel in three basic directions: north or south, east or west, and up or down. (Or, equivalently, left or right, backward or forward, and up or down.) Wherever we go—whether it’s driving to the grocery store or flying to Tahiti—we move in some combination of those three independent directions. So familiar are we with these dimensions that trying to conceive of an additional dimension—and figuring out exactly where it would point—might seem impossible. For a long while, it seemed as if what you see is what you get. In fact, more than two thousand years ago, Aristotle argued as much in his treatise On the Heavens: “A magnitude if divisible one way is a line, if two ways a surface, and if three a body. Beyond these there is no other magnitude, because the three dimensions are all that there are.”2 In A.D. 150, the astronomer and mathematician Ptolemy tried to prove that four dimensions are impossible, insisting that you cannot draw four mutually perpendicular lines. A fourth perpendicular, he contended, would be “entirely without measure and without definition.”3 His argument, however, was less a rigorous proof than a reflection of our inability both to visualize and to draw in four dimensions. To a mathematician, a dimension is a “degree of freedom”—an independent way of moving in space. A fly buzzing around over our heads is free to move in any direction the skies permit. Assuming there are no obstacles, it has three degrees of freedom. Suppose that fly lands on a parking lot and gets stuck in a patch of fresh tar. While it is temporarily immobilized, the fly has zero degrees of freedom and is effectively confined to a single spot—a zerodimensional world. But this creature is persistent and, after some struggle, wrests itself free from the tar, though injuring its wing in the process. Unable to fly, it has two degrees of freedom and can roam the surface of the parking lot at will. Sensing a predator—a ravenous frog, perhaps—our hero seeks refuge in a rusted tailpipe lying in the lot. The fly thus has one degree of freedom, trapped at least for now in the onedimensional or linear world of this narrow pipe. But is that all there is? Does a fly buzzing through the air, stuck in tar, crawling on the asphalt, or making its way through a pipe include all the possibilities imaginable? Aristotle or Ptolemy would have said yes, but while this may be the case for a not terribly enterprising fly, it is not the end of the story for contemporary mathematicians, who typically find no compelling reason to stop at three dimensions. On the contrary, we believe that to truly understand a concept in geometry, such as curvature or distance, we need to understand it in all possible dimensions, from zero to n, where n can be a very big number indeed. Our grasp of that concept will be incomplete if we stop at three dimensions—the point being that if a rule or law of nature works in a space of any dimension, it’s more powerful, and seemingly more fundamental, than a statement that only applies in a particular setting. Even if the problem you’re grappling with pertains to just two or three dimensions, you might still secure helpful clues by studying it in a variety of dimensions. Let’s return to our example of the fly flitting about in threedimensional space, which has three directions in which to move, or three degrees of freedom. Yet let’s suppose another fly is moving freely in that same space; it too has three degrees of freedom, and the system as a whole has suddenly gone from three to six dimensions—with six independent ways of moving. With more flies zigzagging through the space—all moving on their own without regard to the other—the complexity of the system goes up, as does the dimensionality. One advantage in looking at higherdimensional systems is that we can divine patterns that might be impossible to perceive in a simpler setting. In the next chapter, for instance, we’ll discuss the fact that on a spherical planet, hypothetically covered by a giant ocean, all the water cannot flow in the same direction—say, from west to east—at every point. There have to be some spots where the water is not moving at all. Although this rule applies to a twodimensional surface, it can only be derived by looking at a much higherdimensional system in which all possible configurations—all possible movements of tiny bits of water on the surface—are considered. That’s why we continually push to higher dimensions to see what it might lead to and what we might learn. One thing that higher dimensions lead to is greater complexity. In topology, which classifies objects in terms of shape in the most general sense, there are just two kinds of onedimensional spaces: a line (a curve with two open ends) and a circle (a closed curve with no ends). There aren’t any other possibilities. Of course, the line could be squiggly, or the closed curve oblong, but those are questions of geometry, not topology. The difference between geometry and topology is like the difference between looking at the earth’s surface with a magnifying glass and going up in a rocket ship and surveying the planet as a whole. The choice comes down to this: Do you insist on knowing every last detail—every ridge, undulation, and crevice in the surface—or will the big picture (“a giant round ball”) suffice? Whereas geometers are often concerned with identifying the exact shape and curvature of some object, topologists only care about the overall shape. In that sense, topology is a holistic discipline, which stands in sharp contrast to other areas of mathematics in which advances are typically made by taking complicated objects and breaking them down into smaller and simpler pieces. As for how this ties into our discussion of dimensions, there are—as we’ve said—just two basic onedimensional shapes in topology: A straight line is identical to a wiggly line, and a circle is identical to any “loop”—oblong, squiggly, or even square—that you can imagine. The number of twodimensional spaces is similarly restricted to two basic types: either a sphere or a donut. A topologist considers any twodimensional surface without holes in it to be a sphere, and this includes everyday geometric shapes such as cubes, prisms, pyramids, and even watermelonlike objects called ellipsoids. The presence of the hole in the donut or the lack of the hole in the sphere makes all the difference in this case: No matter how much you manipulate or deform a sphere—without ripping a hole in it, that is—you’ll never wind up with a donut, and vice versa. In other words, you cannot create new holes in an object, or otherwise tear it, without changing its topology. Conversely, topologists regard two shapes as functionally equivalent if—supposing they are made out of malleable clay or PlayDoh—one shape can be molded into the other by squeezing and stretching but not ripping. A donut with one hole is technically called a torus, but a donutlike surface could have any number of holes. Twodimensional surfaces that are both compact (closed up and finite in extent) and orientable (doublesided) can be classified by the number of holes they have, which is also known as their genus. Objects that look quite different in two dimensions are considered topologically identical if they have the same genus. 1.1—In topology, there are just two kinds of onedimensional spaces that are fundamentally distinct from each other: a line and a circle. You can make a circle into all kinds of loops, but you can’t turn it into a line without cutting it. Twodimensional surfaces, which are orientable—meaning they have two sides like a beach ball, rather than just one side like a Möbius strip—can be classified by their genus, which can be thought of, in simple terms, as the number of holes. A sphere of genus 0, which has no holes, is therefore fundamentally distinct from a donut of genus 1, which has one hole. As with the circle and line, you can’t transform a sphere into a donut without cutting a hole through the middle of it. 1.2—In topology, a sphere, cube, and tetrahedron—among other shapes—are all considered equivalent because each can be fashioned from the other by bending, stretching, or pushing, without their having to be torn or cut. 1.3—Surfaces of genus 0, 1, 2, and 3; the term genus refers to the number of holes. The point made earlier about there being just two possible twodimensional shapes—a donut or a sphere—is only true if we restrict ourselves to orientable surfaces, and those are the surfaces we’ll generally be referring to in this book. A beach ball, for example, has two sides, an inside and an outside, and the same goes for a tire’s innertube. There are, however, more complicated cases—singlesided, “nonorientable” surfaces such as the Klein bottle and Möbius strip—where the foregoing is not true. In dimensions three and beyond, the number of possible shapes widens dramatically. In contemplating higherdimensional spaces, we must allow for movements in directions we can’t readily imagine. We’re not talking about heading somewhere in between north and west like northwest or even “North by Northwest.” We’re talking about heading off the grid altogether, following arrows in a coordinate system that has yet to be drawn. One of the first big breakthroughs in charting higherdimensional space came courtesy of René Descartes, the seventeenthcentury French mathematician, philosopher, scientist, and writer, though his work in geometry stands foremost for me. Among other contributions, Descartes taught us that thinking in terms of coordinates rather than pictures can be extremely productive. The labeling system he invented, which is now called the Cartesian coordinate system, united algebra and geometry. In a narrow sense, Descartes showed that by drawing x, y, and z axes that intersect in a point and are all perpendicular to each other, one can pin down any spot in threedimensional space with just three numbers—the x, y, and z coordinates. But his contribution was much broader than that, as he vastly enlarged the scope of geometry and did so in one brilliant stroke. For with his coordinate system in hand, it became possible to use algebraic equations to describe complex, higherdimensional geometric figures that are not readily visualized. Using this approach, you can think about any dimension you want—not just (x, y, z) but (a, b, c, d, e, f) or (j, k, l, m, n, o, p, q, r, s)—the dimension of a given space being the number of coordinates you need to determine the location of a given point. Armed with this system, one could contemplate higherdimensional spaces of any order—and do various calculations concerning them—without having to worry about trying to draw them. The great German mathematician Georg Friedrich Bernhard Riemann took off with this idea two centuries later and carried it far. In the 1850s, while working on the geometry of curved (nonEuclidean) spaces—a subject that will be taken up in the next chapter—Riemann realized that these spaces were not restricted in terms of the number of dimensions. He showed how distance, curvature, and other properties in such spaces could be precisely computed. And in an 1854 inaugural lecture in which he presented principles that have since come to be known as Riemannian geometry, he speculated on the dimensionality and geometry of the universe itself. While still in his twenties, Riemann also began work on a mathematical theory that attempted to tie together electricity, magnetism, light, and gravity—thereby anticipating a task that continues to occupy scientists to this day. Although Riemann helped free up space from the limitations of Euclidean flatness and three dimensions, physicists did not do much with that idea for decades. Their lack of interest may have stemmed from the absence of experimental evidence to suggest that space was curved or that any dimensions beyond three existed. What it came down to was that Riemann’s advanced mathematics had simply outpaced the physics of his era, and it would take time—another fifty years or so—for the physicists, or at least one physicist in particular, to catch up. The one who did was Albert Einstein. In developing his special theory of relativity—which was first presented in 1905 and further advanced in the years after, culminating in the general theory of relativity—Einstein drew on an idea that was also being explored by the German mathematician Hermann Minkowski, namely, that time is inextricably intertwined with the three dimensions of space, forming a new geometrical construct known as spacetime. In an unexpected turn, time itself came to be seen as the fourth dimension that Riemann had incorporated decades before in his elegant equations. Curiously, the British writer H. G. Wells had anticipated this same outcome ten years earlier in his novel The Time Machine. As explained by the Time Traveller, the main character of that book, “there are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter.”4 Minkowski said pretty much the same thing in a 1908 speech—except that in this case, he had the mathematics to back up such an outrageous claim: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”5 The rationale behind the marriage of these two concepts—if, indeed, marriages ever have a rationale—is that an object moves not only through space but through time as well. It thus takes four coordinates, three of space and one of time, to describe an event in fourdimensional spacetime (x, y, z, t). Although the idea may seem slightly intimidating, it can be expressed in extremely mundane terms. Suppose you make plans to meet somebody at a shopping mall. You note the location of the building—say it’s at the corner of First Street and Second Avenue—and decide to meet on the third floor. That takes care of your x, y, and z coordinates. Now all that remains is to fix the fourth coordinate and settle on the time. With those four pieces of information specified, your assignation is all set, barring any unforeseen circumstances that might intervene. But if you want to put it in Einstein’s terms, you shouldn’t look at it as setting the exact place for this little gettogether, while separately agreeing on the time. What you’re really spelling out is the location of this event in spacetime itself. So in a single bound, early in the twentieth century, our conception of space grew from the cozy threedimensional nook that had nurtured humankind since antiquity to the more esoteric realm of fourdimensional spacetime. This conception of spacetime formed the bedrock on which Einstein’s theory of gravity, the general theory of relativity, was soon built. But is that the end of the line, as we asked once before? Does the buck stop there, at four dimensions, or can our notion of spacetime grow further still? In 1919, a possible answer to that question arrived unexpectedly in the form of a manuscript sent to Einstein for review by a thenunknown German mathematician, Theodor Kaluza. 1.4—As we don’t know how to draw a picture in four dimensions, this is a rather crude, conceptual rendering of fourdimensional spacetime. The basic idea of spacetime is that the three spatial dimensions of our world (represented here by the xyz coordinate axis) have essentially the same status as a fourth dimension—that being time. We think of time as a continuous variable that’s always changing, and the figure shows snapshots of the coordinate axis at various moments frozen in time: t1, t2, t3, and so forth. In this way, we’re trying to show that there are four dimensions overall: three of space plus the additional one labeled by time. In Einstein’s theory, it takes ten numbers—or ten fields—to precisely describe the workings of gravity in four dimensions. The force can be represented most succinctly by taking those ten numbers and arranging them in a fourbyfour matrix more formally known as a metric tensor—a square table of numbers that serves as a higherdimensional analogue of a ruler. In this case, the metric has sixteen entries in all, only ten of which are independent. Six of the numbers repeat because gravity, along with the other fundamental forces, is inherently symmetrical. In his paper, Kaluza had basically taken Einstein’s general theory of relativity and added an extra dimension to it by expanding the fourbyfour matrix to a fivebyfive one. By expanding spacetime to the fifth dimension, Kaluza was able to take the two forces known at the time, gravity and electromagnetism, and combine them into a single, unified force. To an observer in the fivedimensional world that Kaluza envisioned, those forces would be one and the same, which is what we mean by unification. But in a fourdimensional world, the two can’t go together; they would appear to be wholly autonomous. You could say that’s the case simply because both forces do not fit into the same fourbyfour matrix. The additional dimension, however, provides enough extra elbow room for both of them to occupy the same matrix and hence be part of the same, more allencompassing force. I may get in trouble for saying this, but I believe that only a mathematician would have been bold enough to think that higherdimensional space would afford us special insight into phenomena that we’ve so far only managed to observe in a lowerdimensional setting. I say that because mathematicians deal with extra dimensions all the time. We’re so comfortable with that notion, we don’t give it a moment’s thought. We could probably manipulate extra dimensions in our sleep without interfering with the REM phase. Even if I think that only a mathematician would have made such a leap, in this case, remarkably, it was a mathematician building on the work of a physicist, Einstein. (And another physicist, Oskar Klein, whom we’ll be discussing momentarily, soon built on that mathematician’s work.) That’s why I like to position myself at the interface between these two fields, math and physics, where a lot of interesting crosspollination occurs. I’ve hovered around that fertile zone since the 1970s and have managed to get wind of many intriguing developments as a result. But returning to Kaluza’s provocative idea, people at the time were puzzled by a question that is equally valid today. And it’s one that Kaluza undoubtedly grappled with as well: If there really is a fifth dimension—an entirely new direction to move at every point in our familiar fourdimensional world—how come nobody has seen it? The obvious explanation is that this dimension is awfully small. But where would it be? One way to get a sense of that is to imagine our fourdimensional universe as a single line that extends endlessly in both directions. The idea here is that the three spatial dimensions are either extremely big or infinitely large. We’ll also assume that time, too, can be mapped onto an infinite line—an assumption that may be questionable. At any rate, each point w on what we’ve thought of as a line actually represents a particular point (x, y, z, t) in fourdimensional spacetime. In geometry, lines are normally just length, having no breadth whatsoever. But we’re going to allow for the possibility that this line, when looked at with an exceedingly powerful magnifying glass, actually has some thickness. When seen in this light, our line is not really a line at all but rather an extremely slender cylinder or “garden hose,” to choose the standard metaphor. Now, if we slice our hose at each point w, the crosssection of that cut will be a tiny circle, which, as we’ve said, is a onedimensional curve. The circle thus represents the extra, fifth dimension that is “attached,” in a sense, to every single point in fourdimensional spacetime. 1.5—Let’s picture our infinite, fourdimensional spacetime as a line that extends endlessly in both directions. A line, by definition, has no thickness. But if we were to look at that line with a magnifying glass, as suggested in the KaluzaKlein approach, we might discover that the line has some thickness after all—that it is, in fact, harboring an extra dimension whose size is set by the diameter of the circle hidden within. A dimension with that characteristic—being curled up in a tiny circle—is technically referred to as being compact. The word compact has a fairly intuitive meaning: Physicists sometimes say that a compact object or space is something you could fit into the trunk of your car. But there’s a more precise meaning as well: If you travel in one direction long enough, it is possible to return to the same spot. Kaluza’s fivedimensional spacetime includes both extended (infinite) and compact (finite) dimensions. But if that picture were correct, why wouldn’t we notice ourselves moving around in circles in this fifth dimension? The answer to that question came in 1926 from Oskar Klein, the Swedish physicist who carried Kaluza’s idea a step further. Drawing on quantum theory, Klein actually calculated the size of the compact dimension, arriving at a number that was tiny indeed—close to the socalled Planck length, which is about as small as you can get—around 1030 cm in circumference.6 And that is how a fifth dimension could exist, yet remain forever unobservable. There is no foreseeable means by which we could see this minuscule dimension; nor could we detect movements within it. KaluzaKlein theory, as the work is now known, was truly remarkable, showing the potential of extra dimensions to demystify the secrets of nature. After sitting on Kaluza’s original paper for more than two years, Einstein wrote back saying he liked the idea “enormously.”7 In fact, he liked the idea enough to pursue KaluzaKleininspired approaches (sometimes in collaboration with the physicist Peter Bergmann) off and on over the next twenty years. But ultimately, KaluzaKlein theory was discarded. In part this was because it predicted a particle that has never been shown to exist, and in part because attempts to use the theory to compute the ratio of an electron’s mass to its charge went badly awry. Furthermore, Kaluza and Klein—as well as Einstein after them—were trying to unify only electromagnetism and gravity, as they didn’t know about the weak and strong forces, which were not well understood until the latter half of the twentieth century. So their efforts to unify all the forces were doomed to failure because the deck they were playing with was still missing a couple of important cards. But perhaps the biggest reason that KaluzaKlein theory was cast aside had to do with timing: It was introduced just as the quantum revolution was beginning to take hold. Whereas Kaluza and Klein put geometry at the center of their physical model, quantum theory is not only an ungeometric approach, but also one that directly conflicts with conventional geometry (which is the subject of Chapter 14). In the wake of the upheaval that ensued as quantum theory swept over physics in the twentieth century, and the amazingly productive period that followed, it took almost fifty years for the idea of new dimensions to be taken seriously again. General relativity, the geometrybased theory that encapsulates our current understanding of gravity, has also held up extraordinarily well since Einstein introduced it in 1915, passing every experimental test it has faced. And quantum theory beautifully describes three of the known forces: the electromagnetic, weak, and strong. Indeed, it is the most precise theory we have, and “probably the most accurately tested theory in the history of human thought,” as Harvard physicist Andrew Strominger has claimed.8 Predictions of the behavior of an electron in the presence of an electric field, for example, agree with measurements to ten decimal points. Unfortunately, these two very robust theories are totally incompatible. If you try to mix general relativity with quantum mechanics, the combination can create a horrific mess. The trouble arises from the quantum world, where things are always moving or fluctuating: The smaller the scale, the bigger those fluctuations get. The result is that on the tiniest scales, the turbulent, everchanging picture afforded by quantum mechanics is totally at odds with the smooth geometric picture of spacetime upon which the general theory of relativity rests. Everything in quantum mechanics is based on probabilities, and when general relativity is thrown into the quantum model, calculations often lead to infinite probabilities. When infinities pop up as a matter of course, that’s a tipoff that something is amiss in your calculations. It’s hardly an ideal state of affairs when your two most successful theories—one describing large objects such as planets and galaxies, and the other describing tiny objects such as electrons and quarks—combine to give you gibberish. Keeping them separate is not a satisfactory solution, either, because there are places, such as black holes, where the very large and very small converge, and neither theory on its own can make sense of them. “There shouldn’t be laws of physics,” Strominger maintains. “There should be just one law and it ought to be the nicest law around.”9 Such a sentiment—that the universe can and should be describable by a “unified field theory” that weaves all the forces of nature into a seamless whole—is both aesthetically appealing and tied to the notion that our universe started with an intensely hot Big Bang. At that time, all the forces would have been at the same unimaginably high energy level and would therefore act as if they were a single force. Kaluza and Klein, as well as Einstein, failed to build a theory that could capture everything we knew about physics. But now that we have more pieces of the puzzle in hand, and hopefully all the big pieces, the question remains: Might we try again and this time succeed where the great Einstein failed? That is the promise of string theory, an intriguing tough unproven approach to unification that replaces the pointlike objects of particle physics with extended (though still quite tiny) objects called strings. Like the KaluzaKlein approaches that preceded it, string theory assumes that extra dimensions beyond our everyday three (or four) are required to combine the forces of nature. Most versions of the theory hold that, altogether, ten or eleven dimensions (including time) are needed to achieve this grand synthesis. 1.6—String theory takes the old KaluzaKlein idea of one hidden “extra” dimension and expands it considerably. If we were to take a detailed look at our fourdimensional spacetime, as depicted by the line in this figure, we’d see it’s actually harboring six extra dimensions, curled up in an intricate though minuscule geometric space known as a CalabiYau manifold. (More will be said about these spaces later, as they are the principal subject of this book.) No matter where you slice this line, you will find a hidden CalabiYau, and all the CalabiYau manifolds exposed in this fashion would be identical. But it isn’t just a matter of throwing in some extra dimensions and hoping for the best. These dimensions must conform to a particular size and shape—the right one being an asofyet unsettled question—for the theory to have a chance of working. Geometry, in other words, plays a special role in string theory, and many adherents would argue that the geometry of the theory’s extra dimensions may largely determine the kind of universe we live in, dictating the properties of all the physical forces and particles we see in nature, and even those we don’t see. (Because of our focus on socalled CalabiYau manifolds and their potential role in providing the geometry for the universe’s hidden dimensions—assuming such dimension exist—this book will not explore loop quantum gravity, an alternative to string theory that does not involve extra dimensions and therefore does not rely on a compact, “internal” geometry such as CalabiYau.) We will explore string theory in depth, starting in Chapter 6. But before plunging into the complex mathematics that underlies that theory, it might be useful to establish a firmer grounding in geometry. This subject will be explored in depth, starting in Chapter 6. But before plunging into the complex mathematics that underlies that theory, it might be useful to establish a firmer grounding in geometry. (In my admittedly biased experience, that is always a useful tactic.) So we’re going to back up a few steps from the twentieth and twentyfirst centuries to review the history of this venerable field and thereby gain a sense of its place in the order of things. And as for that place, geometry has always struck me as a kind of express lane to the truth—the most direct route, you might say, of getting from where we are to where we want to be. That’s not surprising, given that a fair chunk of geometry is devoted to the latter problem—finding the distance between two points. Bear with me if the path from the mathematics of ancient Greece to the intricacies of string theory seems a bit convoluted, or tangled, at times. Sometimes, the shortest path is not a straight line, as we shall see. Two GEOMETRY IN THE NATURAL ORDER Over most of the last two and a half thousand years in the European or Western tradition, geometry has been studied because it has been held to be the most exquisite, perfect, paradigmatic truth available to us outside divine revelation. Studying geometry reveals, in some way, the deepest true essence of the physical world. —PIERS BURSILLHALL, “WHY DO WE STUDY GEOMETRY? ” What is geometry? Many think of it as simply a course they took in high school—a collection of techniques for measuring the angles between lines, calculating the area of triangles, circles, and rectangles, and perhaps establishing some measure of equivalence between disparate objects. Even with such a limited definition, there’s no doubt that geometry is a useful tool—one that architects, for instance, make use of every day. Geometry is these things, of course, and much, much more, for it actually concerns architecture in the broadest sense of the term, from the very smallest scales to the largest. And for someone like me, obsessed with understanding the size, shape, curvature, and structure of space, it is the essential tool. The word geometry, which comes from geo (“earth”) and metry (“measure”), originally meant “measuring the earth.” But we now put it in more general terms to mean measuring space, where space itself is not a particularly welldefined concept. As Georg Friedrich Bernhard Riemann once said, “geometry presupposes the concept of space, as well as assuming the basic principles for constructions in space,” while giving “only nominal definitions of these things.”1 Odd as it may sound, we find it useful to keep the concept of space rather fuzzy because it can imply many things for which we have no other terms. So there’s some convenience to that vagueness. For example, when we contemplate how many dimensions there are in space or ponder the shape of space as a whole, we might just as well be referring to the entire universe. A space could also be more narrowly defined to mean a simple geometric construct such as a point, line, plane, sphere, or donut—the sorts of figures a grade school student might draw—or it could be more abstract, more complex, and immensely more difficult to picture. Suppose, for instance, you have a bunch of points spread out in some complicated, haphazard arrangement with absolutely no way of determining the distance between them. As far as mathematicians are concerned, that space has no geometry; it’s just a random assortment of points. But once you put in some kind of measurement function, technically called a metric, which tells you how to compute the distance between any two points, then your space has suddenly become navigable. It has a welldefined geometry. The metric for a space, in other words, gives you all the information you need to divine its shape. Armed with that measurement capability, you can now determine its flatness to great precision, as well as its deviation from flatness, or curvature, which is the thing I find most interesting of all. Lest one conclude that geometry is little more than a wellcalibrated ruler—and this is no knock against the ruler, which happens to be a technology I admire—geometry is one of the main avenues available to us for probing the universe. Physics and cosmology have been, almost by definition, absolutely crucial for making sense of the universe. Geometry’s role in all this may be less obvious, but it is equally vital. I would go so far as to say that geometry not only deserves a place at the table alongside physics and cosmology, but in many ways it is the table. For you see, this entire cosmic drama—a complex dance of particles, atoms, stars, and other entities, constantly shifting, moving, interacting—is played out on a stage, inside a “space,” if you will, and it can never be truly understood without grasping the detailed features of that space. More than just a passive backdrop, space actually imbues its constituents with intrinsically vital properties. In fact, as we currently view things, matter or particles sitting (or moving) in a space are actually part of that space or, more precisely, spacetime. Geometry can impose constraints on spacetime and on physical systems in general—constraints that we can deduce purely from the principles of mathematics and logic. Consider the climate of the earth. Though it may not be obvious, the climate can be profoundly influenced by geometry—in this case by the essentially spherical shape of our host planet. If we resided on a twodimensional torus, or donut, instead, life—as well as our climate—would be substantially different. On a sphere, winds can’t blow in the same direction (say, east), nor can the ocean’s waters all flow in the same direction (as mentioned in the final chapter). There will inevitably be places—such as at the north and south poles—where wind or current direction no longer points east, where the whole notion of “east” disappears, and all movement grinds to a halt. This is not the case on the surface of a singleholed donut, where there are no such impasses and everything can flow in the same direction without ever hitting a snag. (That difference would surely affect global circulation patterns, but if you want to know the exact climatological implications—and get a seasonal comparison between spherical and toroidal living—you’d better ask a meteorologist.) The scope of geometry is even broader still. In concert with Einstein’s theory of general relativity, for example, geometry has shown that the mass and energy of the universe are positive and hence that spacetime, the fourdimensional realm we inhabit, is stable. The principles of geometry also tell us that somewhere in the universe, there must be strange places known as singularities—thought to lie, for instance, in the center of black holes—where densities approach infinity and known physics breaks down. In string theory, to take another example, the geometry of weird sixdimensional spaces called CalabiYau manifolds—where much important physics supposedly takes place—may explain why we have the assortment of elementary particles we do, dictating not only their masses but the forces between them. The study of these higherdimensional spaces, moreover, has offered possible insights into why gravity appears to be so much weaker than the other forces of nature, while also providing clues about the mechanisms behind the inflationary expansion of the early universe and the dark energy that’s now driving the cosmos apart. So it’s not just idle boasting when I say that geometry has been an invaluable tool for unlocking the universe’s secrets, right up there with physics and cosmology. Moreover, with the advances in mathematics that we’ll be describing here, along with progress in observational cosmology and the advent of string theory, which is attempting a grand synthesis that has never before been realized, all three of these disciplines seem to be converging at the same time. As a result, human knowledge now stands poised and raring to go, on the threshold of remarkable insights, with geometry, in many ways, leading the charge. It’s important to bear in mind that whatever we do in geometry, and wherever we go, we never start from scratch. We’re always drawing on what came before—be it conjectures (which are unproven hypotheses), proofs, theorems, or axioms—building from a foundation that, in many cases, was laid down thousands of years before. In that sense, geometry, along with other sciences, is like an elaborate construction project. The foundation is laid down first, and if it’s built correctly—placed on firm ground, so to speak—it will last, as will the structures built on top of it, provided they too are engineered according to sound principles. That, in essence, is the beauty and strength of my elected calling. When it comes to mathematics, we always expect a completely true statement. A mathematical theorem is an exact statement that will remain an eternal truth and is independent of space, time, people’s opinions, and authority. This quality sets it apart from empirical science, where you do experiments and, if a result looks good, you accept it after a satisfactory trial period. But the results are always subject to change; you can never expect a finding to be 100 percent, unalterably true. Of course, we often come across broader and better versions of a mathematical theorem that don’t invalidate the original. The foundation of the building is still sound, to continue our construction analogy; we’ve kept it intact while doing some expansion and remodeling. Sometimes we have to go farther than just remodeling, perhaps even “gutting” the interior and starting afresh. Even though the old theorems are still true, we may need entirely new developments, and a fresh batch of materials, to create the overall picture we seek to achieve. The most important theorems are usually checked and rechecked many times and in many ways, leaving essentially no chance for error. There may be problems, however, in obscure theorems that have not received such close scrutiny. When a mistake is uncovered, a room of the building—or perhaps a whole wing—might have to be torn down and reassembled. Meanwhile, the rest of the structure—a sturdy edifice that has stood the test of time—remains unaffected. One of the great architects of geometry is Pythagoras, with the wellknown formula attributed to him being one of the sturdiest edifices ever erected in mathematics. The Pythagorean theorem, as it’s called, states that for a right triangle (a triangle, that is, with one 90degree angle), the length of the longest side (or hypotenuse) squared equals the sum of the squares of the two shorter sides. Or as schoolchildren, former and present, may recall: a2 + b2 = c2. It’s a simple, yet very powerful statement that amazingly is as relevant now as it was when formulated some 2,500 years ago. The theorem is not just restricted to elementary school mathematics. Indeed, I use the theorem just about every day, almost without thinking about it, because it has become so central and so ingrained. To my mind, the Pythagorean theorem is the most important statement in geometry, as crucial for advanced, higherdimensional math—such as for working out distances in CalabiYau spaces and solving Einstein’s equations of motion—as it is for calculations on a twodimensional plane (such as the sheet of a homework assignment) or in a threedimensional grade school classroom. The theorem’s importance stems from the fact that we can use it to figure out distances between two points in spaces of any dimension. And, as I said at the outset of this chapter, geometry has a lot to do with distance, which is why this formula is central to practically everything we do. I find the theorem, moreover, to be extremely beautiful, although beauty, admittedly, is in the eye of the beholder. We tend to like things that we know—things that have become so familiar, so natural, that we take them for granted, just like the rising and setting of the sun. Then there’s the great economy of it all, just three simple letters raised to the second power, a2 + b2= c2, almost as terse as other famous laws like F = ma or E = mc2. For me, the beauty stems from the elegance of a simple statement that sits so comfortably within nature. 2.1—The Pythagorean theorem is often pictured in two dimensions in terms of a right triangle with the sum of the lengths of the sides squared equaling the length of the hypotenuse squared: a2 + b2 = c2. But, as shown here, the theorem also works in three dimensions (a2 + b2 + c2 = d2) and higher. In addition to the theorem itself, which is without a doubt a cornerstone of geometry, equally important is the fact that it was proved to be true and appears to be the first documented proof in all of mathematics. Egyptian and Babylonian mathematicians had used the relation between the sides of a right triangle and its hypotenuse long before Pythagoras was even born. But neither the Egyptians nor the Babylonians ever proved the idea, nor do they seem to have considered the abstract notion of a proof. This, according to the mathematician E. T. Bell, was where Pythagoras made his greatest contribution: Before him, geometry had been largely a collection of rules of thumb empirically arrived at without any clear indication of the mutual connections of the rules. Proof is now so commonly taken for granted as the very spirit of mathematics that we find it difficult to imagine the primitive thing which must have preceded mathematical reasoning.2 Well, maybe Pythagoras is responsible for the proof, though you might have noticed I said the theorem was “attributed” to him, as if there were some doubt as to the authorship. There is. Pythagoras was a cultlike figure, and many of the contributions of his mathcrazed disciples, the socalled Pythagoreans, were attributed to him retroactively. So it’s possible that the proof of the Pythagorean theorem originated with one of his followers a generation or two later. Odds are we’ll never know: Pythagoras lived primarily in the sixth century B.C. and left behind little, if anything, in the way of written records. Fortunately, that’s not the case with Euclid, one of the most famous geometers of all time and the man most responsible for turning geometry into a precise, rigorous discipline. In stark contrast to Pythagoras, Euclid left behind reams of documents, the most illustrious of them being The Elements (published around 300 B.C.)—a thirteenvolume treatise, of which eight volumes are devoted to the geometry of two and three dimensions. The Elements has been called one of the most influential textbooks ever penned, “a work of beauty whose impact rivaled that of the Bible.”3 In his celebrated tome, Euclid laid the groundwork not just for geometry but also for all of mathematics, which depends inextricably on a manner of reasoning we now call Euclidean: Starting with clearly defined terms and a set of explicitly stated axioms, or postulates (the two words being synonymous), one can then employ cool logic to prove theorems that, in turn, can be used to prove other assertions. Euclid did just that, proving more than four hundred theorems in all, thereby encapsulating virtually all of the geometric knowledge of his era. Stanford mathematician Robert Osserman explained the enduring appeal of Euclid’s manifesto this way: “First there is the sense of certainty—that in a world full of irrational beliefs and shaky speculations, the statements found in The Elements were proven true beyond a shadow of a doubt.” Edna St. Vincent Millay expressed similar appreciation in her poem “Euclid Alone Has Looked on Beauty Bare.”4 The next crucial contribution for the purposes of our narrative—with no slight intended to the many worthy mathematicians whose contributions are being overlooked—comes from René Descartes. As discussed in the previous chapter, Descartes greatly enlarged the scope of geometry by introducing a coordinate system that enabled mathematicians to think about spaces of any dimension and to bring algebra to bear on geometric problems. Before he rewrote the field, geometry was pretty much limited to straight lines, circles, and conic sections—the shapes and curves, such as parabolas and hyperbolas, that you get by slicing an infinitely long cone at different angles. With a coordinate system in place, we could suddenly describe very complicated figures, which we otherwise would not know how to draw, by means of equations. Take the equation xn+ yn= 1, for example. Using Cartesian coordinates, one can solve the equation and trace out a curve. Before we had a coordinate system, we didn’t know how to draw such a figure. Where we had been stuck before, Descartes offered us a way to proceed. And that way became even stronger when, about fifty years after Descartes shared his ideas on analytic geometry, Isaac Newton and Gottfried Leibniz invented calculus. Over the coming decades and centuries, the tools of calculus were eventually incorporated in geometry by mathematicians like Leonhard Euler, Joseph Lagrange, Gaspard Monge, and perhaps most notably Carl Friedrich Gauss, under whose guidance the field of differential geometry finally came of age in the 1820s. The approach used Descartes’ system of coordinates to describe surfaces that could then be analyzed in detail by applying the techniques of differential calculus—differentiation being a technique for finding the slope of any smooth curve. The development of differential geometry, which has continued to evolve since Gauss’s era, was a major achievement. With the tools of calculus in their grasp, geometers could characterize the properties of curves and surfaces with far greater clarity than had been possible before. Geometers obtain such information through differentiation or, equivalently, by taking derivatives, which measure how functions change in response to changing inputs. One can think of a function as an algorithm or formula that takes a number as an input and produces a number as an output: y = x2 is an example, where values for x go in and values for y come out. A function is consistent: If you feed it the same input, it will always produce the same output; if you put 2 in our example, you will always get 4. A derivative is how we describe the changes in output given incremental changes in input; the value of the derivative reflects the sensitivity of the output to slight changes in the input. The derivative is not just some abstract notion; it’s an actual number that can be computed and tells us the slope of a curve, or of a surface, at a given point. In the above example, for instance, we can determine the derivative at a point (x = 2) on our function, which in this case happens to be a parabola. If we move a little bit away from that point to, say, x = 2.001, what happens to the output, y? Here, y (if computed to three decimal points) turns out to be 4.004. The derivative here is the ratio of the change in output (0.004) to the change in input (0.001), which is just 4. And that is, in fact, the exact derivative of this function at x = 2, which is another way of saying it’s the slope of the curve (a parabola) there, too. The calculations, of course, can get much more involved than the foregoing when we pick more complicated functions and move into higher dimensions. But returning, for a moment, to the same example, we obtained the derivative of y = x2 from the ratio of the change in y to the change in x because the derivative of this function tells us its slope, or steepness, at a given point—with the slope being a direct measure of how y changes with respect to x. To picture this another way, let’s consider a ball on a surface. If we nudge it to the side a tiny bit, how will that affect its height? If the surface is more or less flat, there will be little variation in height. But if the ball is on the edge of a steep grade, the change in height is more substantial. Derivatives can thus reveal the slope of the surface in the immediate vicinity of the ball. Of course, there’s no reason to limit ourselves to just a single spot on the surface. By taking derivatives that reveal variations in the geometry (or shape) at different points on the surface, we can calculate the precise curvature of the object as a whole. Although the slope at any given point provides local information regarding only the “neighborhood” around that point, we can pool the information gathered at different points to deduce a general function that describes the slope of the object at any point. Then, by means of integration, which is a way of adding and averaging in calculus, we can deduce the function that describes the object as a whole. In so doing, one can learn about the structure of the entire object. This is, in fact, the central idea of differential geometry—namely, that you can obtain a global picture of an entire surface, or manifold, strictly from local information, drawn from derivatives, that reveals the geometry (or metric) at each point. Gauss made many other noteworthy contributions in math and physics in addition to his work on differential geometry. Perhaps the most significant contribution for our purposes relates to his startling proposition that objects within a space aren’t the only things that can be curved; space itself can be curved. Gauss’s view directly challenged the Euclidean concept of flat space—a notion that applied not only to the intuitively flat twodimensional plane but also to threedimensional space, where flatness means (among other things) that on very large scales, parallel lines never cross and the sum of the angles of a triangle always add up to 180 degrees. 2.2—One can compute the area bounded by a curve by means of a calculus technique, integration, which divides the bounded regions into infinitesimally thin rectangles and adds up the area of all the rectangles. As the rectangles become narrower and narrower, the approximation gets better and better. Taken to the limit of the infinitesimally small, the approximation becomes as good as you can get. These principles, which are essential features of Euclidean geometry, do not hold in curved spaces. Take a spherical space like the surface of a globe. When viewed from the equator, the longitudinal lines appear to be parallel because they are both perpendicular to the equator. But if you follow them in either direction, they eventually converge at the north and south poles. That doesn’t happen in (flat) Euclidean space—such as on a Mercator projection map—where two lines that are perpendicular to the same line are truly parallel and never intersect. 2.3—On a surface with positive curvature such as a sphere, the sum of the angles of a triangle is greater than 180 degrees, and lines that appear to be parallel (such as longitudinal lines) can intersect (at the north and south poles, for instance). On a flat planar surface (of zero curvature), which is the principal setting of Euclidean geometry, the sum of the angles of a triangle equals 180 degrees, and parallel lines never intersect. On a surface with negative curvature such as a saddle, the sum of the angles of a triangle is less than 180 degrees, and seemingly parallel lines diverge. In nonEuclidean space, the angles of a triangle can either add up to more than 180 degrees or to less than 180 degrees depending on how space is curved. If it is positively curved like a sphere, the angles of a triangle always add up to more than 180 degrees. Conversely, if the space has negative curvature, like the middle part of a horse’s saddle, the angles of a triangle always add up to less than 180 degrees. One can obtain a measure of a space’s curvature by determining the extent to which the angles of a triangle add up to more than, less than, or equal to 180 degrees. Gauss also advanced the concept of intrinsic geometry—the idea that an object or surface has its own curvature (the socalled Gauss curvature) that is independent of how it may be sitting in space. Let’s start, for example, with a piece of paper. You’d expect its overall curvature to be zero, and it is. But now let’s roll that sheet up into a cylinder. A twodimensional surface like this, according to Gauss, has two principal curvatures running in directions that are orthogonal to each other: One curvature relates to the circle and has the value of 1/r, where r is the radius. If r is 1, then this curvature is 1. The other curvature runs along the length of the cylinder, which happens to be a straight line. The curvature of a straight line is obviously zero, since it doesn’t curve at all. The Gauss curvature of this object—or any twodimensional object—equals the product of those two curvatures, which in this case is 1 ╳ 0 = 0. So in terms of its intrinsic curvature, the cylinder is the same as the sheet of paper it can be constructed from: perfectly flat. The zero intrinsic curvature of the cylinder is a result of the fact that one can form it from a sheet of paper without any stretching or distortion. To put it another way, the distance measurements between any two points on the surface of a sheet—whether the sheet is flat on a table or rolled into a tube—remain unchanged. That means that the geometry, and hence the intrinsic curvature, of the sheet stays intact regardless of whether it’s flat or curled up. 2.4—A torus, or donutshaped, surface can be entirely “flat” (zero Gauss curvature), because it can be made, in principle, by rolling up a piece of paper into a tube or cylinder and then attaching the ends of the tube to each other. Similarly, if we could create a donut or torus by attaching the circular ends of a cylinder together—again doing so without any stretching or distortion—the torus would have the same intrinsic curvature as the cylinder, namely, zero. In practice, however, we cannot actually construct this socalled flat torus—at least not in two dimensions where folds or wrinkles will inevitably be introduced at the seams. But we can construct such an object (known as an abstract surface) in theory, and it holds just as much importance to mathematics as the objects we call real. A sphere, on the other hand, is quite different from a cylinder or flat torus. Consider, for example, the curvature of a sphere of radius r. It is defined by the equation 1/r2 and is the same everywhere on the surface of the sphere. As a result, every direction looks the same on the surface of a sphere, whereas this is obviously not the case on a cylinder or donut. And that doesn’t change, no matter how the sphere is oriented in threedimensional space, just as a small bug living on that surface is presumably oblivious to how the surface is aligned in threedimensional space; all it likely cares about, and experiences, is the geometry of its local, twodimensional abode. Gauss—in concert with Nikolai Lobachevsky and János Bolyai—made great contributions to our understanding of abstract space, particularly the twodimensional case, though he personally admitted to some confusion in this area. And ultimately, neither Gauss nor his peers were able to liberate our conception of space entirely from the Euclidean framework. He expressed his puzzlement in an 1817 letter to the astronomer Heinrich Wilhelm Matthäus Olbers: “I am becoming more and more convinced that the necessity of our geometry cannot be proved, at least by human reason and for human reason. It may be that in the next life we shall arrive at views on the nature of space that are now inaccessible to us.”5 Some answers came not in the “next life,” as Gauss had written, but in the next generation through the efforts, and sheer brilliance, of his student Georg Friedrich Bernhard Riemann. Riemann suffered from poor health and died young, but in his forty years on this planet, he helped overturn conventional notions of geometry and, in the process, overturned our picture of the universe as well. Riemann introduced a special kind of field, a set of numbers assigned to each point in space that could reveal the distance along any path connecting two points—information that could be used, in turn, to determine the extent to which that space was curved. Measuring space is simplest in one dimension. To measure a onedimensional space, such as a straight line, all we need is a ruler. In a twodimensional space, such as the floor of a grand ballroom, we’d normally take two perpendicular rulers—one called the xaxis and the other the yaxis—and work out distances between two points by creating right triangles and then using the Pythagorean theorem. Likewise, in three dimensions, we’d need three perpendicular rulers, x, y, and z. Things get more complicated and interesting, however, in curved, nonEuclidean space, where properly labeled, perpendicular rulers are no longer available. We can rely on Riemannian geometry, instead, to calculate distances in spaces like these. The approach we’ll take in computing the length of a curve, which itself is sitting on a curved manifold, will seem familiar: We break the curve down into tangent vectors of infinitesimal size and integrate over the entire curve to get the total length. The tricky part stems from the fact that in curved space, the measurement of the individual tangent vectors can change as we move from point to point on the manifold. To handle this variability, Riemann introduced a device, known as a metric tensor, that provides an algorithm for computing the length of a tangent vector at each point. In two dimensions, the metric tensor is a twobytwo matrix; in n dimensions, the metric tensor is an nbyn matrix. (It’s worth noting that this new measurement approach, despite Riemann’s great innovation, still relies heavily on the Pythagorean theorem, adapted to a nonEuclidean setting.) A space endowed with a Riemannian metric is called a Riemannian manifold. Equipped with the metric, we can measure the length of any curve in a manifold of arbitrary dimension. But we’re not limited to measuring the length of curves; we can also measure the area of a surface in that space, and a “surface” in this case is not limited to the usual two dimensions. With the invention of the metric, Riemann showed how a space that was only vaguely defined could instead be granted a welldescribed geometry, and how curvature, rather than being an imprecise concept, could be encapsulated in a precise number associated with each point in space. And this approach, he showed, could apply to spaces of all dimensions. Prior to Riemann, a curved object could only be studied from the “outside,” like surveying a mountain range from afar or gazing at the surface of Earth from a rocket ship. Up close, everything would seem flat. Riemann showed how we could still detect the fact that we were living in a curved space, even with nothing to compare that space with.6 This poses a huge question for physicists and astronomers: If Riemann was right, and that one space was all there is, without a bigger structure to fall back, it meant we had to readjust our picture of almost everything. It meant that on the largest scales, the universe need not be confined by the strictures of Euclid. Space was free to roam, free to curve, free to do whatever. It is for this very reason that astronomers and cosmologists are now making meticulous measurements in the hopes of finding out whether our universe is curved or not. Thanks to Riemann, we now know that we don’t have to go outside our universe to make these measurements, which would be a difficult feat to pull off. Instead, we should be able to figure this out from right where we’re sitting—a fact that could offer comfort to both cosmologists and couch potatoes. These, in any event, were some of the new geometric ideas circulating when Einstein began drawing together his thoughts on gravity. Early in the twentieth century, Einstein had been struggling for the better part of a decade to combine his special theory of relativity with the principles of Newtonian gravity. He suspected that the answer may lie somewhere in geometry and turned to a friend, the geometer Marcel Grossman, for assistance. Grossman, who had previously helped Einstein get through graduate coursework that he’d found uninspiring, introduced his friend to Riemann’s geometry, which was unknown to physics at the time—although the geometer did so with a warning, calling it “a terrible mess which physicists should not be involved with.”7 Riemann’s geometry was the key to solving the puzzle Einstein had been wrestling with all those years. As we saw in the previous chapter, Einstein was grappling with the idea of a curved, fourdimensional spacetime (otherwise known as our universe) that was not part of a bigger space. Fortunately for him, Riemann had already provided such a framework by defining space in exactly that way. “Einstein’s genius lay in recognizing that this body of mathematics was tailormade for implementing his new view of the gravitational force,” Brian Greene contends. “He boldly declared that the mathematics of Riemann’s geometry aligns perfectly with the physics of gravity.”8 Einstein recognized not only that spacetime could be described by Riemann’s geometry, but also that the geometry of spacetime would influence its physics. Whereas special relativity had already unified space and time through the notion of spacetime, Einstein’s subsequent theory of general relativity unified space and time with matter and gravity. This was a conceptual breakthrough. Newtonian physics had treated space as a passive background, not an active participant in the proceedings. The breakthrough was all the more spectacular considering that there was no experimental motivation for this theory at all. The idea literally sprang from one person’s head (which is not to say, of course, that it could have sprung from anyone’s head). The physicist C. N. Yang called Einstein’s formulation of general relativity an act of “pure creation” that was “unique in human history . . . Einstein was not trying to seize an opportunity that had presented itself. He created the opportunity himself. And then fulfilled it on his own, through deep insight and grand design.”9 It was a remarkable achievement that might even have surprised Einstein, who hadn’t always recognized that basic physics and mathematics could be so intricately intertwined. He would conclude years later, however, that “the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.”10 Einstein’s theory of gravitation was arrived at by such a process of pure thought—realized through mathematics without any prompting from the outside world. Equipped with Riemann’s metric tensor, Einstein worked out the shape and other properties—the geometry, in other words—of his newly conceived spacetime. And the resulting synthesis of geometry and physics, culminating in the famous Einstein field equation, illustrates that gravity—the force that shapes the cosmos on the largest scales—can be regarded as a kind of illusion caused by the curvature of space and time. The metric tensor of Riemannian geometry not only described the curvature of spacetime, but also described the gravitational field in Einstein’s new theory. Thus, a massive body like the sun warps the fabric of spacetime in the same way that a large man deforms a trampoline. And, just as a small marble thrown onto the trampoline will spiral around the heavier man, ultimately falling into the dip he creates, the geometry of warped spacetime causes Earth to orbit the sun. Gravity, in other words, is geometry. The physicist John Wheeler once explained Einstein’s picture of gravity this way: “Mass grips space by telling it how to curve; space grips mass by telling it how to move.”11 Another example might help drive this point home: Suppose that two people start at different spots on the equator and set out at the same speed toward the north pole, moving along longitudinal lines. As time goes on, they get closer and closer to each other. They may think they are affected by some invisible force that’s drawing them together. But another way to think of it is that the assumed force is really a consequence of the geometry of the earth and that there’s actually no force at all. And that, in short, gives you some idea as to the force of geometry itself. The power of that example hit me with full impact when I was a firstsemester graduate student learning about general relativity for the first time. It was no secret, of course, that gravity shapes our cosmos and that gravity was, indeed, its principal architect in terms of the big picture. On smaller scales, in the confined venue of most physics apparatus, gravity is extremely feeble compared with the other forces: electromagnetic, strong, and weak. But in the grand scheme of things, gravity is pretty much all there is: It is responsible for the creation of structure in the universe, from individual stars and galaxies up to giant superclusters stretching a billion lightyears across. If Einstein was right, and it all came down to geometry, then geometry, too, was a force to be reckoned with. I was sitting in a lecture class, pondering the implications, when a series of thoughts occurred to me. I had been interested in curvature since college and sensed that in light of Einstein’s insights, it may be a key to understanding the universe, as well as an avenue through which I might make my own mark someday. Differential geometry had provided tools for describing how mass moves in a curved spacetime without explaining why spacetime is curved in the first place. Einstein had taken those same tools to explain where that curvature comes from. What had been seen as two separate questions—the shape of a space under the influence of gravity and its shape under the influence of curvature—turned out to be the same problem. Taking it a step further, the question I pondered was this: If gravity comes from mass telling space how to curve, what happens in a space that has no mass whatsoever—a space we call a vacuum? Who does the talking then? Put in other terms, does the socalled Einstein field equation for the vacuum case have a solution other than the most uninteresting one—that is, a “trivial” spacetime with no matter, no gravity, and no interaction and where absolutely nothing happens? Might there be, I mused, a “nontrivial” space that has no matter, yet whose curvature and gravity are nonzero? I wasn’t yet in any position to answer these questions. Nor did I realize that a fellow named Eugenio Calabi had posed a special case of that very question more than fifteen years before, though he had approached it from a purely mathematical standpoint and wasn’t thinking about gravity or Einstein at all. The best I could do then was to marvel, openmouthed, and wonder: “What if?” It was a surprising question for me to ask, in many ways, especially given where I’d come from—starting on a trajectory that was as likely to have taken me to the poultry trade as it was to have led me to geometry, general relativity, and string theory. I was born in mainland China in 1949, but my family moved within a year to Hong Kong. My father was a university professor with a modest salary and a wife and eight children to feed. Despite his taking three teaching jobs at three universities, his total earnings were meager, affording neither enough money nor food to go around. We grew up poor, without electricity or running water, taking our baths in a river nearby. Enrichment, however, came in other forms. Being a philosopher, my father inspired me to try to perceive the world through a more abstract lens. I remember as a young child overhearing the conversations he had with students and peers; I could feel the excitement of their words even though I couldn’t grasp their meaning. My father always encouraged me in mathematics, despite my not getting off to the most promising start. When I was five, I took an entrance exam for a topnotch public school but failed the mathematics part because I wrote 57 instead of 75 and 69 instead of 96—a mistake, I now tell myself, that’s easier to make in Chinese than in English. As a result, I was forced to go to an inferior rural school populated by a lot of rough kids who had little patience for formal education. I had to be rough to survive, so rough that I dropped out of school for a time in my preteen years and headed a gang of youths who, like me, wandered the streets looking for trouble and, more often than not, found it. Personal tragedy turned that around. My father died unexpectedly when I was fourteen, leaving our family not only griefstricken but destitute, with a slew of debts to pay off and virtually no income. As I needed to earn some money to support the family, an uncle advised me to leave school and raise ducks instead. But I had a different idea: teaching mathematics to other students. Given our financial circumstances, I knew there was just one chance for me to succeed and I placed my bets on math, double or nothing. If I didn’t do well, my whole future was done, leaving nothing to fall back on (other than fowl husbandry, perhaps) and no second chances. In situations like that, I’ve found, people tend to work harder. And though I may have my shortcomings, no one has ever accused me of being lazy. I wasn’t the best student in high school but tried to make up for that in college. While I was a reasonably good student in my first year, though by no means exceptional, things really picked up for me in the second year when Stephen Salaff, a young geometer from Berkeley, came to teach at our school, the Chinese University of Hong Kong. Through Salaff I got my first taste of what real mathematics was all about. We taught a course together on ordinary differential equations and later wrote a textbook together on that same subject. Salaff introduced me to Donald Sarason, a distinguished Berkeley mathematician who paved the way for me to come to the university as a graduate student after I had completed just three years of undergraduate work. Nothing I’d encountered in mathematics up to that point rivaled the bureaucratic challenges we overcame—with the help of S. S. Chern, the great Chinese geometer, also based at Berkeley—in order to secure my early admittance. 2.5—The geometer S. S. Chern (Photo by George M. Bergman) Arriving in California at the age of twenty, with the full range of mathematics lying before me, I had no idea of what direction to pursue. I was initially inclined toward operator algebra, one of the more abstract areas of algebra, owing to my vague sense that the more abstract a theory was, the better. Although Berkeley was strong in many branches of math, it happened to be a world center—if not the world center—for geometry at the time, and the presence of many impressive scholars like Chern began to exert an inexorable tug on me. That, coupled with a growing recognition that geometry constituted a large, rich subject ripe with possibilities, slowly lured me into the