It is our considered opinion that Borges simultaneously intended for the Library to have a spiral staircase in every doorway and also to present the librarians with a bewildering array of options. The "stark and depressing realization" of the librarians in the preceding section indicates the impossibility of such a conjunction, whereas this minor modification allows for enormous mutability in the floor plans and potentially a quick access to any nearby hexagon.
Math Aftermath: A Labyrinth, Not a Maze
The subject does not belong to the world; rather, it is a limit of the world.
—Ludwig Wittgenstein, Tractatus Logico-Philosophicus
The goal of this Math Aftermath is to provide grounds for believing an even stronger consequence than the "stark and depressing conclusion" that followed from the first case we deconstructed, that adjacent hexagons need not be accessible. The stronger consequence has a relatively easy proof, but is too messy to offer up in these pages due to the necessity of breaking down a number of related cases. (Despite not offering much of a framework for the proof, it is tempting to employ the standard trope appearing in works of mathematics at moments such as this: "The reader may supply the details.") The librarians in this pastiche will name it "our conjecture of extreme disconsolation," and it is:
In any Library constrained as this one,
on any given floor,
for any positive integer n less than, say, 1,000,000,000,000 = 1012,
there must necessarily be pairs of abutting hexagons, Hi and H2,
such that a librarian would need to walk through more than n distinct hexagons to travel from H1 to H2.
That is, there are many, many hexagons that possess effectively inaccessible adjacent hexagons.
(Now, if these observations were deep mathematical insights worthy of publication in an eminent journal—or even a second-rate journal—we'd probably label the "Stark and Depressing Conclusion" weak inaccessibility and "Our Conjecture ofExtreme Disconsolation" strong inaccessibility. Also, an aspect of the statement of the stronger result is worthy of comment: notice that the amount of detail accrued in the service of excluding unwanted interpretations renders it difficult to read and to understand. This seems to yield the counterintuitive notion that the more precise the transmission of an idea, the more opaque the language.)
Let us now rejoin our librarians, at the moment after their "stark and depressing conclusion."
Librarians Redux
It was as if our minds, mocking our exhausted, rooted, dispirited bodies, were set free. Almost in opposition to our wills, without fully digesting the realization, we continued to ruminate on these matters. One of us— does it matter which?—invoked a fragment in a contentious book found on a lower level, that read in part,
Imagine a narrow flexible tube, one thousand miles long, called a "garden hose," laid out flat on a gigantic floor so that the hose impeccably fills the floor. The hose may curve abruptly, swirl painfully, spiral exuberantly, loop discursively, or even run straight, but it may never cross over itself nor rise from the ground in any way. Perhaps at many points the hose makes a kind of moral cusp or treacherous eddy and the close-by exterior parts of the tube nestle next to nearby parts of the tube. Skywards down to the hose, the view of the godlike will pinpoint many spots where the hose appears as parallel strands lying next to each other. At those spots, an ant—a tiny six-legged librarian— crawling through the interior of the hose may travel a considerable distance, perhaps miles, to reach a contiguous section. Even worse, never mind the Origin of the ant: the more it crawls, the more places it finds where the walls of the hose keep it further and further away from places the godlike can see. Lament, therefore, the linear forwards-and-backwards motion of the ant inside, while the nonlinear arabesques of the exterior hose bring grace and redemption to those who can read them. Never shall the ant crawl from the interior and gaze upon the wholeness of the hose.
Paralyzed, we saw that although our limbs numbered four, and despite the fact that we weren't trapped in such a strange loop, there were striking similarities between the situation for the ants and for us. Regardless of the clever patterns taken by a godlike being laying down the garden hose, there must ever be more spots where the long, slimber structure of the loops of the hose would thwart an ant's attempt to move to any point athwart of the hose besides those immediately forwards or directly backwards. Clenching the hose into a crimp and then twisting it around in a whirlpool will produce a section where the ant could easily travel to all spots near its starting point, but then as the hose continues to be laid down, filling out the floor, circling around and again in a dizzying whorl of a world for the ant... we simply stopped talking, exhausted, looking up and down the airshafts.
Our Conjecture of Extreme Disconsolation: There are unimaginably vast numbers of pairs of adjacent hexagons such that the span of our combined lives would not suffice to travel from the one to the other.
Our earlier impotence was now seen to be a dream; our true plight lay revealed: perhaps we inhabited a section of the Library where all or most hexagons would allow us to attain only two of the six adjacent hexagons. All of those books, perhaps my or my friend's Vindication, perhaps a grammar of an ideal logic capable of straightening out the labyrinth in which we found ourselves, perhaps a fitting valediction for a carelessly dropped book mournfully hurtling down an airshaft, all these books would never be read by us.
SIX
More Combinatorics
Disorderings into Order
There is a secret element of regularity in the object which corresponds to a secret element of regularity in the subject.
—Johann Wolfgang von Goethe, Maxims and Reflections
Thinking man has a strange trait: when faced with an unsolved problem he likes to concoct a fantastic mental image, one he can never escape, even when the problem is solved and the truth revealed.
—Johann Wolfgang von Goethe, Maxims and Reflections
Either a universe that is all order, or else a farrago thrown together at random yet somehow forming a universe. But can there be some measure of order subsisting in yourself, and at the same time disorder in the greater whole?
—Marcus Aurelius, Meditations
CALCULATING THE NUMBER OF DISTINCT BOOKS in the Library, as seen in "Combinatorics: Contemplating Variations of the 23 Letters," is an example of a straightforward problem with a tidy solution. In this chapter, we do not so much solve a problem as explore how a maximally disordered and chaotic distribution of books in the Library can be seen as a Grand Pattern. This work is grounded in ancient ideas of combinatorial analysis, and although the ideas are consistent with the structure of the story, the ordering of the books we outline is incompatible with the Librarian's "elegant hope" that
If an eternal traveler should journey in any direction, he would find after untold centuries that the same volumes are repeated in the same disorder—which, repeated, becomes order: the Order.
The Order conjectured by the librarian is an iterative order; a two-dimensional analogue may help to visualize it. Think of the complete ordering of all the books as being given by the imprint of a rubber stamp. After making an initial stamp (the section of the Library that the librarian lives in), without rotating the stamp at all and without overlapping stamps, continue applying the stamp up and down, left and right, and eventually cover the piece of paper. This translates the original order in all directions, vertically and horizontally, forming a simple kind of symmetry
The Grand Pattern we propose in lieu of the Librarian's iterative Order is, in some sense, an ever-growing chain of concatenations of all possible orderings. To help envision what we mean, imagine that the Library is finite and approximately in the shape of a cube. Suppose we adjoined another Library-sized and Library-shaped building to the first one and distributed the 251,312,000 unique books in a different ordering. This surely violates the Librarian's elegant hope, for it contradicts his vision that the addition should contain the books in precisely the s
ame order as the original section. Now, suppose we continue to extend the Library by adjoining Library-sized and Library-shaped structures, each time distributing the books in a new ordering. Our endeavor now is to formalize the process, being as disorderly as possible, and at the end of a piece-by-piece construction, find an infinitely sized Library with a Grand Pattern occupying the whole of Euclidean 3-space.
Let's begin with a relatively simple question: how many distinct linear orderings are there of the three objects {▲, █, ●} such that each object appears exactly once? A few moments of work produces the following list:
1. ▲, █, ●
2. ▲, ●, █
3. █, ▲, ●
4. █, ●, ▲
5. ●, ▲, █
6. ●, █, ▲
How might we convince ourselves that the list exhausts all possibilities? Perhaps by noting that we can fill the first slot three different ways, with either ▲, █, or ●.
Once the first slot is filled, we are left with exactly two objects and two slots. Either of the two remaining objects can fill the second slot: we have two choices. Then, whichever object is left must fill the last slot. In other words, there are six different ways to fill the slots (figure 61).
Since the list has six distinct entries, we may be sure we've exhausted all possibilities. Generalizing this line of thinking, if we have four objects, there will be
4 x 3 x 2 x 1=24
distinct ways to arrange the four objects: four choices for the first slot, three for the second slot, two for the third slot, and only one object remaining to fill the last slot.
Explicitly writing out the multiplications is viable for a relatively small number of objects. However, if we wished to signify the integer corresponding to the number of different ways to order only 25 objects, we'd find it cumbersome. Fortunately, a snappy notation, that of the factorial, was developed in the early 1800s:
1! = 1
2! = 2 x 1=2
3! = 3 x 2 x 1=6
4! = 4 x 3 x 2 x 1=24
.
.
.
25! = 25 x 24 x 23 x … x 3 x 2 x 1
.
.
.
n! = n x (n - 1) x (n - 2) x … x 3 x 2 x 1.
The orthographic symbol "3!" is read and pronounced as "three factorial," where "factorial" is understood to represent the process of multiplying an initial integer by every positive integer smaller than itself.
Several observations about factorials. First, on a personal note, even after 25 years of the serious study of mathematics we still tend to read "3!" as "THREE!" (very excitedly). Second, although we won't use the answers in this work, some natural questions to ask are "How is 0! defined?" and "Can we make sense of the expressions
or
even though they are not integers?" One tempting possible answer for the latter question is to focus on the "keep multiplying by numbers reduced in size by subtracting one" aspect of the factorial, and define, for example,
Instead, in 1729, a similar yet more encompassing route was discovered by Leonhard Euler. Euler used the integral calculus to define a new function, called the gamma function, which, like the logarithm, possesses many interesting properties. One is that if a positive integer n is input to the gamma function, then is the output, meaning that the gamma function is essentially a generalization of the factorial. We test our naïve guess by inputting 13/2 to the gamma function, and find that the output is, in fact, pretty close:
At any rate, factorial notation shares a property with exponential notation: it is easy to write down unimaginably large numbers. Take, for example, the number 70!, which by virtue of the simplicity of its written form appears as though it should fall within the grasp of the human imagination. In reality, 70! is larger than 10100, and as we saw in the chapter "Combinatorics," 10100 grains of sand would completely fill 10 billion universes the size of our own.
Now, in the sort of action standard for a mathematician that incurs withering scorn from engineers, we accomplish the impossible by simply asserting it as a fact: number all the book-sized slots in the bookcases in the Library from 1 to 251,312,000. (Wasn't that easy?) Via this numbering of the spaces in the bookshelves, we may use the factorial to compute the number of different ways to order the books in the Library Even though the numbering of the slots in bookcases in the Library necessarily twists and snakes through three dimensions, we can still regard it as a consecutive sequence of 251,312,000 slots laid out in a row. Put another way, regardless of how they are distributed in space, the positive integers have an intrinsic linear ordering, given by the progression 1, 2, 3, etc.
Given that each specific book fills a particular numbered slot, armed with the factorial notation we may trivially write down the number of different ways that the books in the Library may be shelved. Considering each book as a distinct object, there are
different orderings. We'd like to get a sense of the magnitude of this number; after all, a factorial as small as 70! taxes our power of visualization by easily exceeding the number of subatomic particles in our universe. Fortunately, Stirling's approximation to the factorial gives a good estimation, in the sense that we can see this gargantuan number as an exponential of 10.
Stirling's approximation applied to yields .
This says that the number of different orderings of the books in the Library is approximately a 33-million-digit number; in the context of the story, it would take about 26 volumes simply to write down the number. The upshot is that Builders may construct a finite-sized Library housing all possible orderings of the books by assembling (251,312,000)! Library-sized and -shaped buildings, filling each such building with exactly one ordering of the books.
If, though, along with the librarian, we assume that the Library is infinite in all directions, we have ample space for a more ambitious scheme than simply accounting for all orderings. We'll begin by defining a libit as a contiguous collection of accessible hexagons holding one particular ordering of the 251,312,000 distinct books of the Library—any shape of the libit is acceptable so long as Euclidean 3-space can be completely tiled by replicas of that shape. Although we are imagining a libit looking roughly cubic or almost like a hexagonal prism, here's an extreme example unlike those: a tower of stacked single hexagons sufficient to hold all the distinct books. Here's another: a giant near-hexagon of hexagons completely contained on one floor, again sufficiently large to hold all the distinct books.
Next, tile all of 3-space by clones of the libit, and from the infinite possibilities, arbitrarily choose an initial libit and also any hexagon contained within. We'll use this hexagon as a reference point, and consider it to be the origin of the Library. Starting at the origin, successively choose contiguous hexagons in an orderly fashion until the first libit is completely numbered. (It is not unreasonable to worry about how to ensure that every hexagon will be numbered. We address this issue in the chapter "Critical Points.")
Now choose an adjacent libit, and starting in the "same" hexagon as in the first libit—that is, in the hexagon in the second libit that corresponds to the origin—extend the numbering starting with 251,312,000 + 1. The numbering of the second libit will, of course, run all the way up to 2 (251,312,000). Next, repeat the process by choosing a third libit contiguous to the second one, and number the slots on the shelves as before.
Continue to iterate the procedure until the shelves in (251,312,000)! contiguous libits are numbered, and note that each of the (251,312,000)! different orderings is composed of 251,312,000 different books. Thus, for the first step of the Grand Pattern, we utilize the first
251,312,000 x (251,312,000)!
= (Number of distinct books) x (Number of distinct orderings)
slots in the infinite Library.
However, as we were filling the libits with different orderings, we were implicitly making choices regarding the possible orderings of the books. Making the Grand Pattern requires us to leap categories and consider orderings
of orders. Let's do a smaller-scale example of a Grand Pattern constructed in the Euclidean plane.
Instead of books, we'll use the three letters {a, b, c}. In Step 1, we give a straightforward ordering of the three letters, which, by analogy, is similar to using books to fill the first libit. For Step 2, we produce one list of the 3! = 6 possible orderings of the three letters, and we think of this as the set of (251,312,000)! libits described above (figure 62). Next, note that there are 6! = 720 distinct lists of six orderings. This is exactly the idea we want to communicate: that although we ran through all possible orderings in the list of six, if we think of each of the six orderings as a new unit there are 720 orderings which must be accounted for in Step 3. Put another way, first we worried about all the ways to order {a, b, c}. Now, we want all possible orderings of
{[a, b, c]; [a, c, b]; [b, a, c]; [b, c, a]; [c, a, b]; [c, b, a]}.
Figure 63 shows five distinct lists, beginning a spiral, which ultimately creates a new, larger rectangle filled out by all 720 distinct 3 x 6 rectangles. This, in turn, is an ordering of orderings of orderings and guarantees that at the next stage, Step 4, we'll be able to continue to spiral around and create an even larger rectangle.
The Unimaginable Mathematics of Borges' Library of Babel Page 11
