The Unimaginable Mathematics of Borges' Library of Babel (21 page)

Now, though, it's conceivable
that generalizing from a 2-sphere might generate some disquietude: on a
2-sphere, any two distinct great circles intersect at exactly two points
(figure 19). It is not unreasonable to worry that any two distinct great
circles on the 3-sphere would also of necessity intersect in at least two
points. This might entail that all the air shafts and all the spiral staircases
would converge, say, at the north and south poles of the 3-sphere, causing a
traffic jam of epic proportions. Fortunately, this intuitively plausible
scenario doesn't happen. Perhaps the easiest way to begin to get a handle on
why this isn't a problem is to grasp that a circle is only
one
dimension
smaller than a 2-sphere. Consequently, it has special properties of
"dividing" space locally into two pieces; certainly a great circle
divides a 2-sphere into two hemispheres. However, a circle is
two
dimensions smaller than a 3-sphere and hence has no such special division
property in the 3-sphere. Imagine a circle floating in the center of the
room—space flows through it and around it with aplomb.

If the
Library is the universe, and the universe is a 3-sphere, then
the Library is
a sphere whose exact center is any hexagon and whose circumference is
unattainable; moreover, it is limitless and periodic.
That is, the
3-spherical Library satisfies both the classic dictum and the librarian's
cherished hope.

Math Aftermath: Flat Out Disoriented

The reverse side also has a
reverse side.

—Japanese
proverb

Donuts. Is there anything
they can't do?

—Homer
Simpson,
The Simpsons

The enemy of my enemy is my
friend.

—Ancient
proverb

This Math Aftermath comes with
a travel advisory of sorts for the potential explorer. In some sense—at least,
in the author's sense—the material herein represents the mathematical zenith of
the book: it's an extended journey into some other three-dimensional manifolds.
While we wish to encourage the intrepid reader to forge ahead, we issue the
advisory just in case you experience the Aftermath as an overwhelming deluge of
math. If so, our advice is to jump to the next chapter until the feeling
subsides. And with that, on to the math.

If we are
willing to forego one-third of the Librarian's classic dictum that
the
Library is a sphere whose exact center is any hexagon and whose circumference
is unattainable
by yielding on the spherical nature of space, then there
are two candidates for the large-scale shape of the Library, the
3-torus
and the
3-Klein bottle,
both worthy of our time and attention.
²
The two are intimately related, for the second can be thought of
as the twisted, disoriented reassemblage of the first.

We'll
proceed as we did earlier in the chapter: first, we'll gain an understanding of
a two-dimensional object that lives in three dimensions, then we'll use that
knowledge-base to visualize a three-dimensional manifold that lives in higher
dimensions. This time, though, there will also be an intermediate step of
reconfiguring our mind's eye to allow the hope of visualizing a two-dimensional
object that lives most naturally in
four
dimensions. Finally, we'll
briefly discuss the attributes of a Library modeled on either a 3-torus or
3-Klein bottle.

We start with a familiar
object, the everyday square, and then show that by "gluing" its edges
together, various two-dimensional manifolds emerge. (Note that the square
itself is NOT a manifold. Our rule is that it must be locally Euclidean, which
we are taking to mean that if we stand at any point and take a few steps in any
direction, we perceive ourselves as being in a Euclidean space. However, if we
start at the edge of the square, we can't walk over the edge and still imagine
ourselves in 2-space, for 2-space has no boundaries.)

Begin by
marking the left and right sides with arrows pointing down, then continue by
marking the top and bottom sides with double arrows pointing towards the right
(figure 20). Now, identify the top and the bottom edges with each other, so
that the arrows continue to point in the same direction. The mathematical sense
of
identify
entails that the sides truly unify; it is as if they were
never separate entities. By contrast, the best physical approximations are
unfortunately coarse; one must glue, tape, or solder the edges together.
Manifestly, after the mathematical identification the square has become a
cylinder (figure 21).

Now identify the ends of the
cylinder so that the arrows continue to revolve in the same direction—in
3-space, this is accomplished by bending the cylinder around so that the ends
come together. When this identification is complete, the cylinder has
transformed into a
torus
: the surface of a donut, the surface of a
bagel—or, as topologists like to point out, the surface of a coffee mug (figure
22).
³
(A
statistician, Morris DeGroot, once jokingly remarked to me the literal truth
that topologists don't know their asses from a hole in the ground.) This nifty
sequence leads to the expression that
the torus is just a square with the
edges identified to preserve orientation.
The torus is a 2-manifold; every
point in it locally looks like the Euclidean plane. It has no boundary edges or
walls, and if we think of it as a space into and of itself, like Euclidean
space and the 3-sphere, the center is both everywhere and nowhere. The torus
has an additional property which is quite extraordinary: it is
flat,
which means it can be embedded in Euclidean space in such a way that a bug
walking between any two points on the torus could find a path whose distance is
precisely the same as the straight-line distance between those two points on
the square.

This should
sound implausible; after all, the torus looks quite bent and the distance on
the outer edge looks much longer than that on the inner edge. In fact, this is
true; for the purposes of the illustrations and for boosting our intuition, we
bent the cylinder until the ends met. We were purposely ambiguous and merely
wrote "can be embedded in Euclidean space," several sentences back,
rather than adding the key phrase: It must be
four-dimensional
Euclidean
space. However, it's easy to see that the cylinder is truly flat in the
geometric sense: Mark any two points on a cylinder. Now let the cylinder unroll
so that it is once again a square. Connect the dots in the square by a straight
line. Now reroll the square into a cylinder. Voila! Staying in the surface of
the cylinder, the shortest distance between two points is, at most, the same as
the distance between the two points in the unidentified square. (Why "at
most"? Because there may well be a path crossing the identified edges of
the cylinder that is even shorter than the straight-line path inherited from
the square; regardless, by the unrolling/rerolling, we are guaranteed to
achieve, at worst, the same distance on the torus as in the square.)

The twisted, nonorientable
reassemblage of the torus is called the
Klein bottle.
We form it by
starting, once again, with a square. Again, mark double arrows on the top and
bottom sides so that the arrows point in the same direction. This time, though,
we place the arrows on the left and right sides so that they point in
opposite
directions (figure 23). Again, we identify the top and bottom
sides with the arrows pointing in the same direction and thereby obtain a
cylinder. This time, however, when we try to identify the ends of the cylinder,
there is an insurmountable problem in 3-space. No matter how we twist or turn
the cylinder around, there is no way to put the ends together so that the
arrows are revolving in the same direction (figure 24). Although the picture
looks bleak—impossible, in fact—the last twisted cylinder actually provides a
ray of hope. If we rotate the orienting arrow counterclockwise around over the
top of the bottom end of the cylinder, it's still pointing in the correct
direction, and we obtain the mildly cheering picture shown in figure 25.
Twisted around like this, one opening above the other, the orientations of the
end-pieces match up: they are both counterclockwise.