The Unimaginable Mathematics of Borges' Library of Babel (26 page)

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 have 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.

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.

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.