The Unimaginable Mathematics of Borges' Library of Babel

by William Goldbloom Bloch

  A faintly analogous situation occurs with the familiar circle. In the plane, there's a distinct inside and outside—look again at figure 27. As discussed earlier in the chapter, in 3-space the circle has nothing easily definable as an "inside" or an "outside." It certainly does not cut 3-space into two eternally separate pieces, as does, for example, a 2-sphere. The correspondence between the Klein bottle's and the circle's lack of an inside and an outside hinges on dimensionality. A circle is a one-dimensional object that can live in two dimensions. If the circle is in the plane, in 2-space, then the "dimensional difference"—technically, the codimension—is equal to one:

  2 - 1 = dimension(2-space) - dimension(circle) = 1.

  On the other hand, if the circle is in 3-space, the codimension is equal to two:

  3 - 1 = dimension(3-space) - dimension(circle) = 2.

  Similarly, if the Klein bottle, a two-dimensional object, is in 4-space, the codimension is once again equal to two. A codimension greater than one implies that the object can't separate the space into two distinct pieces; thus, there can be no inside or outside.

  Summarizing, the Klein bottle is an example of a one-sided 2-manifold with no boundary (By contrast, the Mobius band, another delightfully disorienting object, has a boundary, an edge.) All the boundaryless 2-manifolds familiar to us from our sensual life in 3-space have an inside and an outside—think of a sphere, a torus, the surface of a pretzel, or the surface of any familiar object. They all cut space into two distinct pieces.

  The Klein bottle does not separate space—it has one side only, and there is no way of distinguishing between the inside and the outside. Moreover, in four-dimensional Euclidean space, the Klein bottle is geometrically flat for the same reasons as the torus: pick any two points on it, reverse the identifications back to a square, then draw the straight line that connects the two points. An unimaginable construct, to say the least.

  Inside Outsights

  There is another disorientation involving the Klein bottle. For this one, we imagine, taking a cue from Flatland, that we live a two-dimensional existence wholly contained within the surface of the Klein bottle. Outside and inside are meaningless words to us: the Klein bottle is our entire universe. Befitting our new planar existence, let us take a new form, that of a flag rather than an arrow. The flag that we are curves and bends with the Klein bottle as we move around; again, it is—we are—wholly contained within the universe that is the Klein bottle.

  Again, this time the black flag is part of the surface of the Klein bottle, not perpendicular to the surface like the arrows in the previous illustration. See figure 30. Now move the flag counterclockwise along a path that exploits the one-sided nature of the Klein bottle, the same path as in the previous section. As in figure 29, we change the shade of both the path and the flag as they proceed from the "outside" of the Klein bottle through the "hole" to the "inside." Note that throughout our journey, the black flag points in the direction of motion.

  Observe that although the flag begins its journey with the pole pointing in one direction on the surface of the Klein bottle, after it has slid around to the starting point the pole, still contained in the universe that is the Klein bottle, is now pointing in the other direction. Perhaps, jaded by the one-sided oddity of the Klein bottle, this isn't a big surprise—after all, it is easy to imagine slithering around on the floor and ending up with our feet located at their initial spot and with our head pointed in the opposite direction than at the start.

  More disorienting, though, the black flag has come back a mirror reflection of itself. There is no obvious intuitive analogue for us. Any journey you take, transformative though it may be, will not result in your coming back as a mirror-reflected image of yourself. You may, for example, feel a shadow of your former self, or half the person you used to be, or find your partner besieged by 50 suitors; regardless, it will not be the case that your heart is now, from everyone else's perspective, on the right-hand side of your body. Figure 31 illustrates the categorical difference between a rotation and a mirror-reflection.

  On a sphere, on a torus, or in the Euclidean plane, any journey the flag might take would result in, at worst, a rotation. There is no possible path that allows for the flag to be mirror-reflected—a journey into the fourth spatial dimension is required.

  From 2-Manifolds to 3-Manifolds

  Let's generalize these 2-manifolds, the torus and the Klein bottle, to their three-dimensional equivalents, the 3-torus and the 3-Klein bottle. To do so, we start with a solid cube, instead of a square, and identify opposite sides, and mimicking what we did in two dimensions, we'll begin by creating a 3-torus. Again, we'll take advantage of working in three dimensions to bend the cube to identify the sides; the natural space for the identifications is 51^-dimensional Euclidean space: in six dimensions, the 3-torus is flat. If we are willing to have a curved, distorted representation akin to the 2-torus in three dimensions, a "mere" four Euclidean dimensions suffices to hold the 3-torus. For the 3-torus, arrows are insufficient to specify an orientation of a face of the cube, but spirals will serve. (Think about why this should be so.)

  Figure 32 shows the initial solid block inscribed with appropriate spirals. This time, we bend the sides of the cube around, identifying the left and right faces while taking care that the spirals being glued together spin in harmony (figure 33). After this first identification, we are confronted with a millstone, whose top and bottom must be identified. (The inside and outside are, of course, also identified. We'll discuss that afterwards.) Turn the millstone sideways—and shrink it—to make it easier to visualize this step (figure 34).Proceed by identifying the visible gray ring on the right-hand side of the millstone with the hidden gray ring on the left-hand side (figure 35).

  Now, we are presented with a donut that has a smaller donut drilled out of the middle of its interior; a donut waiting for a filling, as it were. A donut with a non-donut inside. The surface of the exterior donut is a 2-torus, while the surface of the interior non-donut is also a 2-torus. These two tori correspond to the front and back square faces of the initial cube, and they sprang into being when, in the process of identifying the other four faces of the cube, only the edges of these squares were identified. Now, the (invisible) interior 2-torus must be identified with the (visible) exterior 2-torus. By tugging them both into the fourth dimension, where they no longer divide space into an "inside" and "outside," they may be glued together, producing the 3-torus.

  Before moving on, let's look at one more way to visualize a 3-torus. Once again, we'll proceed by analogy with the eminently imaginable 2-torus. If we take a 2-torus and intersect it with a plane (as in figure 36), the result is a circle (figure 36). Another way to see this is to take a slice of the square that becomes the 2-torus (figure 37). If we think of a 2-torus in this way and flatten it out onto the plane, we may represent it as a circle of circles (figure 38).

  If we take a three-dimensional slice of a 3-torus, we get a 2-torus. One way to see that is to look at a slice of the solid cube we started with (figure 39). Consequently, each slice of the 3-torus is "flattened" out into 3-space as a 2-torus. Since the two sides of the cube are identified, we get a circle of 2-tori (figure 40).

  Let us now consider the 3-torus as a model for the universe that is the Library. Since it is a 3-manifold, the center of the 3-torus is everywhere and nowhere, so the exact center is any hexagon.

  Next, there is a sense in which the 3-torus has sorts of circumferences, which arise in the following ways. Imagine we're at the center of the cube, facing "out of the page." If we move to the exact center of the wall on our left, when we reach it, due to the fact that it is identified with the right-hand wall, the "left wall" is simultaneously the "right wall," which is actually no wall, but rather an unrestricted passage back to the other side of the initial cube. So if we continue to move, we'll end up back where we started. (For that matter, if it is a small 3-torus, if we turn our head and look to either left or right, we'll see the back of our hea
d.)

  Similarly, if we moved up or down from the center of the initial cube, we'd again end up back at the center of the cube. Finally, if we moved forward or backwards, the same phenomenon would occur, which means that in a small 3-torus, looking in any direction means looking at the back of our head.

  In a 3-sphere, if we head off straight in any direction and stay straight, we'll eventually circumnavigate the sphere along a great circle. In a 3-torus, if we head off straight in a particular direction and stay straight, depending on the angle we set out we will eventually either end up exactly where we started or else come arbitrarily close to our initial point. If the Library is a 3-torus, by dint of its enormity, again all of its circumferences are unattainable by a librarian. Moreover, since there are no boundaries to it, the 3-torus is limitless. Because journeying straight in any direction would eventually return us to where we began, the 3-torus is periodic. Therefore the 3-torus satisfies two of the three conditions of the classic dictum and all three of the conditions of the librarian's solution; the only condition is misses is that it isn't a sphere.

  It is also geometrically flat, in the same sense as the 2-torus, which might be a desirable quality for the Library. Although a large enough sphere, such as the earth, will appear flat, a sphere is always curved. Considering the Library as a 3-torus embedded in 6-space, there'd be absolutely no way, locally, for the librarians to determine that they are living in a 3-torus as opposed to living in Euclidean 3-space.

  This leads to some highly speculative questions. What if the hypothetical Builder(s) of the Library wished to test the librarians? If the Library was a 3-sphere and the librarians grew tremendously technologically advanced—more than us—they might develop a method to measure the local curvature of space. If they discovered that the curvature was nonzero, they'd know that the librarian's solution at the end of the story was false: Euclidean 3-space has no curvature. If, on the other hand, they found the curvature to be zero, they would have to face the bitter realization that once again, they didn't possess enough information to decode the topology of the Library.

  The Library includes mirrors. Borges draws our attention to this via the following passage, part of the description of the particulars of the makeup of the Library:

  In the vestibule there is a mirror, which faithfully duplicates appearances. Men often infer from this mirror that the Library is not infinite—if it were, what need would there be for that illusory replication? I prefer to dream that burnished surfaces are a figuration and promise of the infinite. ...

  After the development of our final 3-manifold, we'll submit a fanciful explanation accounting for the presence of the mirrors.

  Begin by reversing the spin of one of the orienting spirals, and next identify the opposite faces of the initial cube as we did creating the 3-torus (figure 41). The outcome will be a three-dimensional Klein bottle, which we'll call the 3-Klein bottle. As with the 3-torus, we first endeavor to identify the left and right faces of the solid cube; this time, though, we are unable to accomplish the first step in three dimensions. Look closely at the left-hand "bent-square" in figure 42. The spiral on both the left-hand square and the right-hand square are turning clockwise. Thus, if we naively try to put the two squares together as we did in creating the 3-torus, the orientations do not align. Rotating either of the squares will not affect this problem, as the mere fact of the rotation will not impinge upon the spiral's clockwise orientation. (Think of it this way: imagine walking up to your reflection in the mirror and attempting to touch your right hand with its reflection. Easy to do. However, if your identical twin walked up to you and you both held out your right hands in the same fashion, your hands wouldn't align or touch. This is why the spirals need to be mirror-reflected, that is, flowing in opposite directions, for the sides to identify.)

  As with the Klein bottle, bending and twisting the cube up and around allows the spirals to be in the same alignment when placed one over the other. Again, as with the Klein bottle, the oriented squares cannot be joined in 3-space. To do so, we must again bend part of the cube "up" into the fourth dimension, precisely the same as we did with the 2-Klein bottle. (Unfortunately, due to the solidity of the interior of the cube, this is beyond our ability to effectively illustrate: rather than a simple circle of self-intersection, we'd be confronted with a truncated solid pyramid of self-intersection contained in the interior of the original cube. The top square of the solid truncated pyramid would be where the top light-gray square "entered" the original cube, and the bottom of the solid truncated pyramid would be the joined pair of light-gray squares facing us in the front.) And then we must still perform other identifications!

  The 3-Klein bottle can also be embedded so that it is flat; furthermore, it enjoys many of the other properties of the 3-torus as well. It is therefore a reasonable candidate for the topology of the Library.

  However, if an intrepid nomadic civilization of librarians or a band of immortal librarians managed to walk a loop that took them through the identified disorienting faces, they would find that they would appear normal to themselves, but when they returned to where they began, the Library would be seen as if reflected in a mirror. The Library wouldn't have changed; rather, it is the librarians' perspectives that would have been turned inside out—in fact, it's an interesting question whether or not such mirror-reversed people with mirror-reversed enzymes would be able to eat our food and digest it to extract nutritional value. If we were to ask them to raise their right hand, they would raise their left hand (from our perspective), while truthfully swearing (from their perspective) that they were raising their right hand. This is exactly parallel to the mirror-reflection of the black flag in figure 30 and in figure 31 that occurs after a complete circuit through the disorienting identification.

  If the Library appeared as reflected in a mirror to the inverted librarians, there are some things that would appear different. However, by making only a few changes to the structure of the Library, we can disorient the librarians so that if they should manage to make such a loop, they wouldn't easily detect that they've been mirror-inverted.

  The first problem revolves around the spiral staircases. They might all be subject to a rule such as "walking clockwise means going down" (figure 43). When the librarians cross through the disorienting face, they will find that the rule has become "walking clockwise means going up." The easy way to remedy this staircase asymmetry is to "insist" the builders of the Library randomly designate different spiral staircases to go up or down when traversed clockwise. Similarly, the sleeping compartments, the lavatory closets, and the mirrors must be randomly distributed on left and right sides of the entrances.

  Another, and perhaps the most important, visual asymmetry is that the orthographic symbols will be mirror-reversed. For an example, see figure 44. An elegant way to avoid this asymmetry is to specify an alphabet whose orthographic symbols are invariant under left-right flips; typically, this is called bilateral symmetry. Here are 25 invariant Roman letters and symbols from a standard computer keyboard.

  A H I M O T U V W X Y 8 ‘ “ -

  = + · : * ^ | ! . (blank space)

  (A sharp-eyed reader will note that some of the letters in this font, such as A, M, U, V, W, X, and Y, aren't precisely bilaterally symmetric. These letters need only minor modifications to become bilaterally symmetric.) Fourteen other symbols, readily available, are also invariant under mirror-inversions:

  _ † ˚ ∞ ± ∏ Ω ¡ ÷ ♦ ‡ ˘

  Furthermore, there are pairs of symbols that when flipped produce each other:

  ( ) [ ] { } < >

  It wouldn't take long to create an aesthetically pleasing set of 25 symbols with the desired mirror-reversal invariance.

  Along similar lines, almost all book titles are printed on the spine so that if the book is laying flat on a table and the front cover is visible, the title can be read: the tops of the letters abut the front cover of the book. Let's call this "top-front" labeling. Try vertically holding the sp
ine of a top-front book to a mirror. Not only are the letters mirror-reversed, but the title now appears as a "top-back" label on the spine (figure 45). A solution to this problem is to simply write the titles of the books vertically down the spines (figure 46). This way, even with a mirror-reversal, a librarian wouldn't notice anything amiss.4

  Moreover, it's ironic that a band of immortal librarians who circumnavigated the 3-Klein bottle Library and returned to their originating hexagon wouldn't recognize any of the books. Although the titles would all be the same, the contents would all look different. Open a book to the first page while looking in a mirror: it appears that the book is open to the last page, not the first. So for the books in a particular hexagon to read the same to a mirror-reversed librarian, the hexagon would need to consist of books that were 410-page palindromes! However, if they were intrepid enough to complete a second circumambulation of the Library, they'd experience a mirror-reversal a second time and then everything would look the same as when they started out.

  Suppose that the constructors of the Library incorporated these design changes to the physical structure and the orthographic symbols. If all the librarians migrated, the Library would not look mirror-reversed to them, even after passing through the disorienting identified faces. However, if they split into two groups and one group managed to circumnavigate the Library, the descendants of the nomadic group would return and discover—from their perspective—a foreign group of librarians who didn't know left from right. Although it's more likely that any such disparity would be attributed to language differences, a librarian of genius might realize the significance of the invariance of the orthographic symbols when reflected in the mirrors. Such a mathematically minded librarian might then deduce that the Library is a nonorientable 3-manifold, and a 3-Klein bottle would surface as the most likely candidate. Such a librarian would know more about the topology of the Library than we know about our own universe.