The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  The three-dimensional sphere (3-sphere) provoked many advances in topology over the past century, and due to the recently solved Poincaré conjecture it remains a vibrant research topic. The 3-sphere is a generalization of the 1-sphere—a circle—and the 2-sphere. Euclid might write something such as, "Given a point p in Jour-dimensional Euclidean space, a sphere with center p is the collection of all points a specified uniform distance away from p." An analytic geometric equation for the standard unit 3-sphere is w2 + x2 + y2 + z2 =1. (Again, to see how this equation captures "sphericality," please consult the appendix "Dissecting the 3-Sphere.") An analogous topological construction for the 3-sphere is difficult to envision, but by pushing the limits of our understanding, we may learn much.

  Take a solid ball—a baseball, or an apple, or a cherry, or a cannonball—and, while leaving the interior of the ball uncompressed, crimp the entire boundary sphere upwards, and then simply contract the boundary sphere to one point (figure 11). That's it. At least the difficulty is easy to understand; for the construction of the 2-sphere, we took a two-dimensional object, the disk, and had to bend it into the third dimension before we could contract the boundary at all. Starting with a solid ball in three dimensions, we must "bend" the ball into the fourth dimension before we can contract the boundary (figure 12). At this juncture, the mathematics becomes unimaginable; the best to be hoped for is that by meditating on the lower-dimensional examples accessible to our imagination, we may be able to conjure the memory of the trace of a once-sensed intuition. Still, by proceeding with analogies to the 2-sphere, we'll use a trio of methods to begin to visualize the 3-sphere.

  If we take a two-dimensional Euclidean slice of a 2-sphere, the resulting geometric object is either a point—at the north and south poles—or a 1-sphere (figure 13). Using a mild updating of an idea from Flatland, if we make a movie of the slice moving from the north pole to the south pole, a viewer would see a point that grows into a unit circle, which then shrinks back down to a point (figure 14, left). In a similar fashion, if we take a three-dimensional Euclidean slice of a 3-sphere, the resulting geometric object is either a point—at the "top" or "bottom"—or a 2-sphere. If we make a movie of the slice moving from the top to the bottom, the viewer would see a point that grows into a unit sphere, which then shrinks back down to a point (figure 14, right). (Again, for those who find equations more convincing than pictures, we provide an analytical proof of this in the appendix "Dissecting the 3-Sphere.")

  Expanding on this idea, suppose we were forced to squish the 2-sphere, whose natural home is in 3-space, down into 2-space. Since we just conceived of the 2-sphere as a collection of stacked circles combined with two poles, we may envision a flattened planar depiction as a collection of intersecting circles with two points signifying the north and south poles (figure 15). The related problem, the one that's been tasking us, is how to represent the 3-sphere down-sized into 3-space. If we think of the 3-sphere as "stacked" 2-spheres—in the same sense that a 2-sphere is stacked 1-spheres—the analogous 3-space representation is a collection of intersecting 2-spheres (figure 16).

  For the third way of envisioning the 3-sphere, the lower-dimensional correlate is to take a section of the 2-sphere and flatten it out into the Euclidean plane. If our section includes, say, the south pole, the flattened section is a disk. If our section doesn't include either pole, the flattened section is an annulus, which is a ring, a thickened circle. Note that the equator of the sphere (the dotted circle in figure 17) is flattened to the central circle of the annulus. The circle-slices above the equator on the sphere are smaller than the equator, but when flattened become larger than the central circle of the annulus. Similarly, the circle-slices below the equator flatten to even smaller circles in the annulus than they were in the sphere. This process of dimensional flattening distorts the object; necessarily, information is lost.

  If we take sections of the 3-sphere, we must consider how to "flatten out" the resulting object into 3-space. If our section includes, for example, the bottom ofthe 3-sphere, the flattened section is, by analogy, a solid ball. If the section of the 3-sphere doesn't include the north or south poles, the "flattened" section is a solid ball with a smaller ball removed from the center—a pitless olive, or a tennis ball, or an empty walnut shell, or a thickened spherical shell. In figure 18, perhaps the most counterintuitive aspect is the means by which the middle collection of 2-spheres collates to a thickened spherical shell. The centralmost, the largest 2-sphere, is flattened to itself. The smaller spheres directly to the left, say, ofthe central sphere thicken it on the inside. The smaller spheres directly to the right of the central sphere are distorted by the flattening into larger spheres that thicken the exterior of the central sphere. Again, unfortunately, the process entails that we must lose information about the size of the spheres.

  All the girders and struts of the framework are now in place to finish assembling the topology and cosmology of the Library. The 3-sphere is a three-dimensional manifold; at every point, if we inhabited the 3-sphere, we would say—locally—that space was Euclidean. If we walked what we perceived to be a straight line in any direction, we would—possibly after countless ages—return to our starting point; the 3-sphere can be construed as periodic. There are no boundaries, no walls to bump into; the 3-sphere is limitless. Moreover, in his luminous story "The Garden of Forking Paths," Borges has the sympathetic sinologist Stephen Albert say, ". ..I had wondered how a book could be infinite. The only way I could surmise was that it be a cyclical, or circular, volume, a volume whose last page would be identical to the first, so that one might go on indefinitely." Even though Albert rejects this line of reasoning for "The Garden of Forking Paths," this quote, coupled with Borges' well-known interest in Nietzsche's idea of eternal recurrence, indicates that Borges was willing to consider cyclic or recurrent structures as tokens of, or synonymous with, infinity.

  Considered as a three-dimensional manifold, the center of the 3-sphere is everywhere and nowhere. Furthermore, if the 3-sphere is so large that, regardless of our transport, we could never come close to circumnavigating it, it would not be illegitimate to say that the circumference is unattainable. Finally, this answers the question concerning what the hexagons "rest on." By forming great circles—circles which are essentially equators of a sphere—the hexagons all rest upon each other and ultimately themselves, and thus there is no need for an impossible "external" foundation for the Library.1

  Now, though, it's conceivable that generalizing from a 2-sphere might generate some disquietude: on a 2-sphere, any two distinct great circles intersect at exactly two points (figure 19). It is not unreasonable to worry that any two distinct great circles on the 3-sphere would also of necessity intersect in at least two points. This might entail that all the air shafts and all the spiral staircases would converge, say, at the north and south poles of the 3-sphere, causing a traffic jam of epic proportions. Fortunately, this intuitively plausible scenario doesn't happen. Perhaps the easiest way to begin to get a handle on why this isn't a problem is to grasp that a circle is only one dimension smaller than a 2-sphere. Consequently, it has special properties of "dividing" space locally into two pieces; certainly a great circle divides a 2-sphere into two hemispheres. However, a circle is two dimensions smaller than a 3-sphere and hence has no such special division property in the 3-sphere. Imagine a circle floating in the center of the room—space flows through it and around it with aplomb.

  If the Library is the universe, and the universe is a 3-sphere, then the Library is a sphere whose exact center is any hexagon and whose circumference is unattainable; moreover, it is limitless and periodic. That is, the 3-spherical Library satisfies both the classic dictum and the librarian's cherished hope.

  Math Aftermath: Flat Out Disoriented

  The reverse side also has a reverse side.

  —Japanese proverb

  Donuts. Is there anything they can't do?

  —Homer Simpson, The Simpsons

  The enemy o
f my enemy is my friend.

  —Ancient proverb

  This Math Aftermath comes with a travel advisory of sorts for the potential explorer. In some sense—at least, in the author's sense—the material herein represents the mathematical zenith of the book: it's an extended journey into some other three-dimensional manifolds. While we wish to encourage the intrepid reader to forge ahead, we issue the advisory just in case you experience the Aftermath as an overwhelming deluge of math. If so, our advice is to jump to the next chapter until the feeling subsides. And with that, on to the math.

  If we are willing to forego one-third of the Librarian's classic dictum that the Library is a sphere whose exact center is any hexagon and whose circumference is unattainable by yielding on the spherical nature of space, then there are two candidates for the large-scale shape of the Library, the 3-torus and the 3-Klein bottle, both worthy of our time and attention.2The two are intimately related, for the second can be thought of as the twisted, disoriented reassemblage of the first.

  We'll proceed as we did earlier in the chapter: first, we'll gain an understanding of a two-dimensional object that lives in three dimensions, then we'll use that knowledge-base to visualize a three-dimensional manifold that lives in higher dimensions. This time, though, there will also be an intermediate step of reconfiguring our mind's eye to allow the hope of visualizing a two-dimensional object that lives most naturally in four dimensions. Finally, we'll briefly discuss the attributes of a Library modeled on either a 3-torus or 3-Klein bottle.

  From the Plane to the Torus

  We start with a familiar object, the everyday square, and then show that by "gluing" its edges together, various two-dimensional manifolds emerge. (Note that the square itself is NOT a manifold. Our rule is that it must be locally Euclidean, which we are taking to mean that if we stand at any point and take a few steps in any direction, we perceive ourselves as being in a Euclidean space. However, if we start at the edge of the square, we can't walk over the edge and still imagine ourselves in 2-space, for 2-space has no boundaries.)

  Begin by marking the left and right sides with arrows pointing down, then continue by marking the top and bottom sides with double arrows pointing towards the right (figure 20). Now, identify the top and the bottom edges with each other, so that the arrows continue to point in the same direction. The mathematical sense of identify entails that the sides truly unify; it is as if they were never separate entities. By contrast, the best physical approximations are unfortunately coarse; one must glue, tape, or solder the edges together. Manifestly, after the mathematical identification the square has become a cylinder (figure 21).

  Now identify the ends of the cylinder so that the arrows continue to revolve in the same direction—in 3-space, this is accomplished by bending the cylinder around so that the ends come together. When this identification is complete, the cylinder has transformed into a torus: the surface of a donut, the surface of a bagel—or, as topologists like to point out, the surface of a coffee mug (figure 22).3 (A statistician, Morris DeGroot, once jokingly remarked to me the literal truth that topologists don't know their asses from a hole in the ground.) This nifty sequence leads to the expression that the torus is just a square with the edges identified to preserve orientation. The torus is a 2-manifold; every point in it locally looks like the Euclidean plane. It has no boundary edges or walls, and if we think of it as a space into and of itself, like Euclidean space and the 3-sphere, the center is both everywhere and nowhere. The torus has an additional property which is quite extraordinary: it is flat, which means it can be embedded in Euclidean space in such a way that a bug walking between any two points on the torus could find a path whose distance is precisely the same as the straight-line distance between those two points on the square.

  This should sound implausible; after all, the torus looks quite bent and the distance on the outer edge looks much longer than that on the inner edge. In fact, this is true; for the purposes of the illustrations and for boosting our intuition, we bent the cylinder until the ends met. We were purposely ambiguous and merely wrote "can be embedded in Euclidean space," several sentences back, rather than adding the key phrase: It must be four-dimensional Euclidean space. However, it's easy to see that the cylinder is truly flat in the geometric sense: Mark any two points on a cylinder. Now let the cylinder unroll so that it is once again a square. Connect the dots in the square by a straight line. Now reroll the square into a cylinder. Voila! Staying in the surface of the cylinder, the shortest distance between two points is, at most, the same as the distance between the two points in the unidentified square. (Why "at most"? Because there may well be a path crossing the identified edges of the cylinder that is even shorter than the straight-line path inherited from the square; regardless, by the unrolling/rerolling, we are guaranteed to achieve, at worst, the same distance on the torus as in the square.)

  From the Plane to the Klein Bottle

  The twisted, nonorientable reassemblage of the torus is called the Klein bottle. We form it by starting, once again, with a square. Again, mark double arrows on the top and bottom sides so that the arrows point in the same direction. This time, though, we place the arrows on the left and right sides so that they point in opposite directions (figure 23). Again, we identify the top and bottom sides with the arrows pointing in the same direction and thereby obtain a cylinder. This time, however, when we try to identify the ends of the cylinder, there is an insurmountable problem in 3-space. No matter how we twist or turn the cylinder around, there is no way to put the ends together so that the arrows are revolving in the same direction (figure 24). Although the picture looks bleak—impossible, in fact—the last twisted cylinder actually provides a ray of hope. If we rotate the orienting arrow counterclockwise around over the top of the bottom end of the cylinder, it's still pointing in the correct direction, and we obtain the mildly cheering picture shown in figure 25. Twisted around like this, one opening above the other, the orientations of the end-pieces match up: they are both counterclockwise.

  Because of the impossibility of aligning the cylinder ends, the Klein bottle cannot live in three dimensions; it requires at least four. A way of representing it in three dimensions is depicted in figure 26, but it requires a self-intersection. You can actually do this nicely by starting with a large enough piece of paper, marking the sides, taping the top and bottom edges together to make a cylinder, and then cutting a hole in the side to pass one end of the cylinder through. This is an excellent way to see how to allow the orienting arrows to point in the same direction. Perhaps this analogy will help explain why allowing the Klein bottle to be in four dimensions effaces the self-intersection. Suppose we confined ourselves to the two-dimensional Euclidean plane and were interested in joining a point inside a circle to a point outside the circle by a line (figure 27). Regardless of devious twists, turns, or serpentine path, it's pretty obvious that any curve joining the two points must intersect the circle somewhere.

  The only possible way to connect the points without intersecting the circle is to venture into the third dimension, pulling a path out of the plane (figure 28). Similarly, one may eliminate the self-intersection of the Klein bottle by simply pulling the offending part of the cylinder into the fourth dimension.

  The "twisted" portion of the initial description of the Klein bottle comes from the fact that one could change the order of the construction by first identifying the left and right sides of the square before identifying the top and bottom. To identify the left and right sides, one must twist the square—and in so doing, create a Mobius band. If we did that, though, at this juncture it is very difficult to visualize how to glue the top and bottom together to make the Klein bottle, because the top and bottom have been merged into one entity, a circle seemingly doubled on itself.

  The "disoriented" portion stems from technical considerations and is manifest in two related, but distinct, ways. We'll cover them both below.

  Outside Insights

  Suppose we decide to
walk a counterclockwise path on what appears to be the outside of the Klein bottle. In figure 29, the black arrow pointing out of the surface into space will represent our position as we start to walk, feet on the Klein bottle, head in the clouds.

  Now some weighty philosophic problems naturally arise from even this innocent beginning. Euclid's plane and all 2-manifolds, including the Klein bottle, are "infinitely thin," much like the pages of the Book of Sand. Is the Euclidean plane therefore transparent? Does the plane, or any 2-manifold, possess a distinct "top" and "bottom"? (Borges makes playful use of these questions in his story "The Disk.") The mathematical perspective is that a path in the Euclidean plane or on a 2-manifold is simultaneously visible from both sides, and as such, it might be useful to imagine the Euclidean plane as a thin and supple sheet of transparent plastic. Then, any line painted on, for example, the top of the plastic is essentially visually indistinguishable from its image as seen through the plastic from below.

  As we begin our walk along the surface, our feet naturally remain on the surface, while our heads naturally are "outside" in 3-space. Next, follow the path to where the two ends of the cylinder are identified. (This looks like the "hole" at the front of the Klein bottle in figure 26.) Note that as we enter the "hole", the arrow and the path both are faded to suggest that we are now inside the Klein bottle, and that our heads are now pointing "inside" rather than "outside." Keep moving "inside" the Klein bottle through the self-intersection—which isn't really there— until we've circled around to our starting point (figure 29). The arrow representing us was initially pointing "out" and now it is pointing "in." The Klein bottle, which has neither holes nor boundaries, also has no inside or outside in the sense that we intuitively understand these terms— a disorienting revelation indeed.