# The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions - Shing-Tung Yau, Steve Nadis (2010)

### Chapter 1. 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 three-fourths 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 light-years. 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, three-dimensional 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 higher-dimensional 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.”

__His argument, however, was less a rigorous proof than a reflection of our inability both to visualize and to draw in four dimensions.__

^{3}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 zero-dimensional 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 one-dimensional 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 three-dimensional 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 higher-dimensional 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 two-dimensional surface, it can only be derived by looking at a much higher-dimensional 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 one-dimensional 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 one-dimensional 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 two-dimensional spaces is similarly restricted to two basic types: either a sphere or a donut. A topologist considers any two-dimensional surface without holes in it to be a sphere, and this includes everyday geometric shapes such as cubes, prisms, pyramids, and even watermelon-like 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 Play-Doh—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 donut-like surface could have any number of holes. Two-dimensional surfaces that are both compact (closed up and finite in extent) and orientable (double-sided) 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 one-dimensional 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.

Two-dimensional 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 two-dimensional 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—single-sided, “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 higher-dimensional 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 higher-dimensional space came courtesy of René Descartes, the seventeenth-century 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 three-dimensional 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, higher-dimensional 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 higher-dimensional 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 (non-Euclidean) 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 four-dimensional 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 get-together, 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 three-dimensional nook that had nurtured humankind since antiquity to the more esoteric realm of four-dimensional 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 then-unknown 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 four-dimensional *spacetime*. The basic idea of spacetime is that the three spatial dimensions of our world (represented here by the *x*-*y*-*z* 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: *t*_{1}, *t*_{2}, *t*_{3}, 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 four-by-four matrix more formally known as a *metric tensor*—a square table of numbers that serves as a higher-dimensional 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 four-by-four matrix to a five-by-five 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 five-dimensional world that Kaluza envisioned, those forces would be one and the same, which is what we mean by unification. But in a four-dimensional 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 four-by-four 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 all-encompassing force.

I may get in trouble for saying this, but I believe that only a mathematician would have been bold enough to think that higher-dimensional space would afford us special insight into phenomena that we’ve so far only managed to observe in a lower-dimensional 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 cross-pollination 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 four-dimensional 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 four-dimensional 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 four-dimensional 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 cross-section of that cut will be a tiny circle, which, as we’ve said, is a one-dimensional curve. The circle thus represents the extra, fifth dimension that is “attached,” in a sense, to every single point in four-dimensional spacetime.

1.5—Let’s picture our infinite, four-dimensional 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 Kaluza-Klein 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 five-dimensional 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 so-called Planck length, which is about as small as you can get—around 10^{-30} 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.

Kaluza-Klein 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 Kaluza-Klein-inspired approaches (sometimes in collaboration with the physicist Peter Bergmann) off and on over the next twenty years.

But ultimately, Kaluza-Klein 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 Kaluza-Klein 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 geometry-based 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, ever-changing 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 Kaluza-Klein 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 Kaluza-Klein idea of one hidden “extra” dimension and expands it considerably. If we were to take a detailed look at our four-dimensional 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 Calabi-Yau 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 Calabi-Yau, and all the Calabi-Yau 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 as-of-yet 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 so-called Calabi-Yau 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 Calabi-Yau.)

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 twenty-first 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.