The Unimaginable Mathematics of Borges' Library of Babel (24 page)

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 head.)

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.