We've now covered all the hard parts for the last topology we wish to propose for the Library. The end result will once again be either a 3-torus or a 3-Klein bottle; we will just take a different, more elegant, path to achieve it. In our discussions of the tori and Klein bottles, we began with a square or a solid cube and then, respectively, identified edges or faces.5
In the next chapter, we'll look at a single floor of the Library, in part by choosing an initial hexagon and considering rings of adjacencies to it. Any ring of adjacency, combined with the hexagons contained inside, forms another shape, which is essentially hexagonal in nature (figure 47). (This is particularly clear if we look at the midpoints of the hexagons.) If we start with a hexagon and carefully identify the sides, just as when we start with a square, the resulting object is either a torus or a Klein bottle. Jeff Weeks, in The Shape of Space, pages 116—26, discusses this and provides very clear illustrations, and Weeks' detailed explanation of these issues is both elegant and relatively accessible. For a reader wanting to know more about 2- and 3-manifolds, it is an excellent reference.
There is, however, a major difference between the hexagon and the square as the object that will have its edges identified. It turns out that to embed the "hexagon with identified edges" in Euclidean 3-space, which is not its "natural" home, the hexagon must be twisted as well as stretched and bent.
Generalizing from two dimensions, it shouldn't be hard to believe that by starting with a hexagonal prism, identifying faces may yield either a 3-torus or a 3-Klein bottle (figure 48). This model of the Library has the additionally pleasing aspect that, in some difficult-to-define sense, the larger geometric structure mimics the smaller structure of the hexagon. We wrote "difficult-to-define" because as soon as the faces are identified, the hexagonal prism is subsumed into the 3-torus or 3-Klein bottle. Again, a torus may be considered a square with its edges identified. However, once the edges are identified, the edges are gone and the square is gone: all that's left is the torus. The identified edges may be drawn in for the purpose of clarifying the process, but they are no longer there. Equally possible, a torus may be a hexagon with its edges identified. In both cases, the resulting object is a torus, but the differing characters of the square and hexagon may leave a detectable, classifiable trace.
A Library modeled on the 3-torus or 3-Klein bottle could be based on either a cube or a hexagonal prism with identified faces. It's conceivable that an immortal librarian of genius, endowed with a means of measuring curvature and possessed of an infinite photographic memory, who repeatedly traversed the Library might also some day be able to guess the Library's topologic structure. Regardless, we who may now consider ourselves the architects of the Library may feel the special glow that derives from the outlining of exquisite solutions to a demanding problem.
FIVE
Geometry and Graph Theory
Ambiguity and Access
If one does not expect the unexpected one will not find it out.
—Heraclitus, Fragment 18
A library is a collection of possible futures.
—John Barth, Further Fridays
THE LIBRARY, AS EVOKED IN THE STORY, HAS inspired many artists and architects to provide a graphic or atmospheric rendition of the interior. These range from Stefano Imbert's lean and elegant drawing adorning the cover of this book, to the deliberately alienated Piranesi-like drawings of Desmazieres in the Godine Press edition of The Library of Babel, to Toca's beautifully symmetric honeycombs in Architecture and Urbanism, to Packer's bold expressionist frontispiece in the Folio edition of Labyrinths, to a host of illustrations easily found online. All of these illustrations sacrifice, to some degree, accuracy in favor of artistic effect. For example, even the cover illustration of this book locates the spiral staircase in the center of the hexagon, whereas Borges writes (emphases added):
The universe (which others call the Library) is composed of an indefinite, perhaps infinite number of hexagonal galleries. In the center of each gallery is a ventilation shaft, bounded by a low railing. From any hexagon one can see the floors above and below—one after another, endlessly. The arrangement of the galleries is always the same: Twenty bookshelves, five to each side, line four of the hexagon's six sides; their height of the bookshelves, floor to ceiling, is hardly greater than the height of a normal librarian. One of the hexagon's free sides opens onto a narrow sort of vestibule, which in turn opens onto another gallery, identical to the first—identical in fact to all. To the left and the right of the vestibule are two tiny compartments. One is for sleeping, upright; the other, for satisfying one's physical necessities. Through this space, too, there passes a spiral staircase, which winds upward and downward into the remotest distance.
As we shall see, quite a lot pivots on the ambiguity arising from the italicized phrase "One of the hexagon's free sides opens onto a narrow sort of vestibule, which in turn opens onto another gallery, identical to the first—identical in fact to all." (This passage is equally uncertain in Spanish, "Una de las caras libres da a un angosto zaguán, que desemboca en otra galería, idéntica a la primera y a todas.")
Now, each hexagon has six sides, two sides of which lead to another hexagon. If Borges meant that each one of the free sides gives upon a narrow entrance way with two miniature rooms, then it follows that in every doorway, there is a spiral staircase, rising and sinking beyond sight. Surprisingly, profound and prodigious consequences derive from this doubled staircase arrangement.
On the other hand, we may read the italicized phrase in a different way. If Borges meant that exactly one of the free sides gives upon a narrow entrance way with two miniature rooms, then it follows that the Library contains pairs of hexagons, joined by small rooms and spiral staircases. Although the difference may seem slight, in this variation of the Library, not only are the plumbing and spiral staircase construction costs cut in half, it actually turns out to be the case that the librarians can lead very different kinds of lives than in the first scenario.
Before we illustrate these two possibilities, in the service of verisimilitude let us sensibly estimate the dimensions of one hexagon and then sketch it. In his "Autobiographical Essay," appearing in The Aleph and Other Stories, pages 243—44, Borges notes that
My Kafkian story "The Library of Babel" was meant as a nightmare version or amplification of that municipal library [the Miguel Cane Municipal Library], and certain details in the text have no particular meaning. Clever critics have worried over those ciphers and generously endowed them with mystic significance.
We were fortunate to be able to visit the Miguel Cane Municipal Library in Buenos Aires, and also both the old and new National Libraries of Argentina. The first three measurements below come from the Miguel Cane Municipal Library, while the fourth comes from a narrow and steep marble spiral staircase in the old National Library.
Length of bookshelf: 3 meters (large double-sided bookcase)
Depth of bookshelf: 0.3 m
Height of bookcase: ~ 2.21 m
Diameter of spiral staircase: ~ 1m
Miniature room for standing sleeping: ~ 0.5 m by 0.5 m
Miniature room for relief of physical necessities: ~ 0.5 m by 0.5 m
Walking space between staircase and walls: ~ 0.5 m
Thus the approximate length of each hexagon's side needs to be three meters, which corresponds nicely to the actual size of the original bookcases at the Miguel Cane Municipal Library. Based on the size of the (presumably square) miniature rooms, the thickness of the walls of each hexagon should be approximately 0.5 meters, although it appears the builders could get by with walls 0.25 meters thick. See figure 49 for the layout of a hexagon.
Another pertinent item is what Beatriz Sarlo notes on page 71 of Jorge Luis Borges: A Writer on the Edge, "As Borges himself declared in an interview, his first spatial idea for the Library of Babel was to describe it as an infinite combination of circles, but he was annoyed with the idea that the circles, when put in a total structure, would hav
e vacant spaces." From the description in the story and especially this quote from Borges, we conclude, as have a number of other commentators, that the Library resembles a honeycomb with no interstices.
Figures 50 and 51 are a pair of illustrations putting together the hexagons; for the first, we show four conjoined hexagons modeled with a spiral staircase appearing in every doorway (figure 50). Based on the second interpretation, figure 51 shows a linear arrangement of hexagons designed with the staircases and two small rooms located in every other doorway.
Now, we'll make three general observations. Then we'll first assume that each and every doorway has a spiral staircase and see what consequences ensue. After that, we'll examine what happens when alternating doorways are pierced by a spiral staircase.
Three General Observations
The most important fact is deceptively hidden in Borges' simple phrasing, "Twenty shelves—five long shelves per side—cover all sides except two..." This obviously means that each hexagon has two doors, but when combined with the snug nesting of the honeycombed hexagons, it means that a librarian disdaining the stairs may only move forward into a new hexagon or double back to the previously visited hexagon. As in a labyrinth, a librarian remaining on the same floor has one path only to tread.
The next observation is that it's conceivable the floor plan of a level of the Library may look like the preceding illustrations, in which the paths run straight through the hexagons. However, it is consistent with the text—and the atmosphere of the story—that the corridors weave and spiral around symmetrically or chaotically (figure 52). (In the succeeding pictures, in the interests of graphic clarity, we omit the miniature rooms, the very low fence, and the spiral staircases, and shrink the enormous ventilation shafts to small black dots.)
The last of the three observations is practical rather than structural or theoretical. It also provides a nice example of "thinking like a mathematician." If we use the shovel, pick, and whisk of our analytical imagination to pare away the obscuring accretions of reality, we reveal the artifacts of our ideas, which provide the wherewithal to build a theory. In this case, we collapse each hexagon to a point, represent the passageways by lines connecting the dots, and throw away the walls and bookcases of the hexagons (figure 53). With these notions and simplifications in hand, let us journey to the first of the two Libraries.
In Every Doorway, There Is a Spiral Staircase
Assume that every doorway is intersected by a spiral staircase, regardless of pattern of floor plan. This prevalence of spiral staircases led to an inkling, then a hunch that became a surmise, which we ultimately formulated as a conjecture and subsequently proved. We approach this theorem as a variation of the locked-room detective story, and hope that H. Bustos Domecq, Borges and Bioy Casares' fictional anti-detective, would admire it.
We are librarians talking in a hexagon about the significance of the 25 orthographic symbols that comprise the markings in the books when, from an adjacent hexagon, we hear raised voices shouting muffled words that are difficult to comprehend; only the rage is clear. We hear thuds, now, as the violence escalates. We look at each other, shocked, and peer through the doorways into two of the six hexagons adjacent to ours. As far as we can see through the portals, the nearby hexagons are empty. Without any discussion, acting on impulses born of common humanity, we each dart through one of the doorways leading out of our hexagon. As we both scan the exit passages of the hexagons we've just run into, the same thought, remarkably, simultaneously enters our minds:
Will I, or my friend, necessarily be able to reach the adjacent hexagon in time to prevent a murder?
Continuing to run into connecting hexagons through the unique entrance doorway and running out through the unique exit passage, we each assure ourselves that the sounds truly came through the wall from a hexagon abutting the one in which we were talking. (For example, the sounds did not float down the airshaft or up a spiral staircase.)
After running until exhausted, you perceive an omnipresent and ominous silence overwhelming the intermittent gasps of your fragile breathing. Defeated, trembling, you reverse direction: your exit doorways become entrance passages and vice versa. There is no chance that you will become lost. The hope you cherish, that which gives you strength enough to trudge back to the starting-point hexagonal gallery, is that I was able to reach the adjacent hexagon in time to temper the dispute.
My crestfallen look, lit only by my eyes eagerly seeking to read your face, is enough to tell you: I failed, also. We sit uncomfortably by the air shaft and each recount the particulars of our fruitless runs; there's not much to say, "I ran in, I ran out; I ran in, I ran out; I ran in, I ran out; I ran in, I ran out;..."
It is irrelevant which one of us first exclaimed, "The stairs! The spiral stairs! Maybe we could have gone up, over, and down to the hexagon. Or down, over, and up? Or down, down, over, up, over, over, up, up, over, and back down?" It is equally irrelevant which combination of whose thoughts destroyed this diaphanous, infinitely permutable, insight:
Every hexagon has an airshaft through the center—it is easy enough to look up and down our airshaft and see what we always see: stacked hexagons. Moreover, each hexagon has two sides singled out by the presence of a spiral staircase (figure 54). Looking up and down the spiral staircases of our hexagon confirms what we already know, that the hexagons above and below must have the staircases in the same sides as the one we are in. In other words, the hexagons above and below are, in this regard, exact clones of our hexagon. Extending this reasoning, for each and every hexagon on our floor, the hexagons above and below it are exact clones. That means that the labyrinthine paths we ran are precisely the same above and below—the stairs and shafts dictate this.
Each floor plan is inevitably, invariably, precisely the same as every other floor plan. There is no advantage gained by taking a set of stairs up or down.
We may, therefore, restrict our investigation to the floor we are on. We might have been lucky; it could have been the case that our starting-point hexagon comprised a part of the floor plan that looked like figure 55. In such a case, one of us would have reached the hexagon in seconds. On the other hand, we were unlucky, so unlucky that we couldn't say how unlucky we were (figure 56). We didn't hazard a guess as to how many hexagons we would have to pass through to reach the other librarians; such a presumptive act would be tantamount to heresy. For that matter, after thinking about it, we daren't even say if the other librarians were really fighting; perhaps they belonged to an entirely different civilization—perhaps an entirely different species.
Our Stark and Depressing Realization: Our question was decisively answered. We would not necessarily be able to reach the adjacent hexagon in time to prevent a murder.
Our conception of the Library's structure was so perturbed by these cascades of devastating insights that it didn't even occur to us until later that the floor plan could plausibly contain eternally inaccessible closed loops, such as the three in figure 57.
The unexpected, against our desires, found us and found us wanting. Without the barrier of even a single door, the adjacent hexagon—the source of noise, confusion, and probable violence—was locked away from us forever.
(We were very startled to realize this; indeed, it only became clear while flying to Buenos Aires when we sketched out the library floor plan. Our readings of "The Library of Babel" always left us with the impression that regardless of which hexagon "we" were "in," we could reach any nearby hexagon in a short period of time. The Realization of this section, in fact, a minor lemma in the field of graph theory, is a prime example of the unimagined math of the story. The Math Aftermath, "A Labyrinth, Not a Maze," at the end of this chapter contains an extension of this story providing a sense of why a stronger result about the inaccessibility of adjacent hexagons must be true.)
There Is One Spiral Staircase Per Pair of Hexagons
Now let's examine the Library by interpreting the first paragraph of the story to mean that only one pa
ssageway in each hexagon is perforated by a spiral staircase (as in figure 51). Once again, the hexagons are forced to be stacked by the existence of the ventilation shafts, but in this case, only one wall, one doorway, is specified by the spiral staircase. This entails that although the hexagons' sides are aligned, they are no longer cloned. Hexagons above and below each other must share one entranceway, but the second may branch out in a different direction (as in figure 58).
Thus, a pair of floors may have labyrinth patterns such as these depicted in figure 59. These illustrations include the spiral staircases, because the presence of a staircase induces a connecting passage between hexagons on all other floors of the Library. Combining the floor plans produces the pleasingly symmetric picture of figure 60. Most importantly, this means that a librarian may reach any adjacent hexagon by traveling through only two additional hexagons and two flights of stairs, one up, one down. The sense of a bewildering array of choices is omnipresent and factual.
The Unimaginable Mathematics of Borges' Library of Babel Page 10