The Unimaginable Mathematics of Borges' Library of Babel (19 page)

A note
regarding the gravity of the situation. If the universe and the Library are
synonymous, and if we make the reasonable assumption that the universe is
neither expanding nor contracting, it follows that the natural gravitational
field would be identically zero everywhere. Even though there are unimaginable
amounts of matter in the universe/Library, its homogeneous distribution entails
that the gravitational effect from any one direction would be canceled out by
precisely the same effect from the opposite direction. Since the builders of a
Library must be, at least from our perspective, omnipotent, their talents
surely must include the ability of imposing a useful constant gravitational
field on the Library.

Euclidean
3-space embodies some of the qualities of interest in our quest to understand
the large-scale structure of the Library. We need to limn two more ideas, one
mathematical, one mystical, before we can describe the form of a Library that
reconciles the characteristics of the classic dictum and the Librarian's
solution.

The
mathematical idea is relatively recent—it comes from the early part of the
twentieth century. For the purposes of this book, we'll say that a
manifold
is a space that is
locally Euclidean
but that on a global scale may be
non-Euclidean.
Perhaps the simplest possible example is that of a sphere,
or globe, or surface of a cantaloupe, or of the earth, balloon, soccer ball;
take your pick. Locally, assuming that we are so small we can't detect the
curvature, each micropatch of a sphere is, in essence, a two-dimensional
Euclidean plane
(2-space).
One need think only of the steppes of Central
Asia, the corn belt of the United States, the Sahara desert, or any large, calm
body of water to engage vivid testimony on this point. Globally, despite the
essential flatness of each little patch, we find non-Euclidean behavior: if we
begin at a point, pick a direction, and continue moving in that direction, we
circumscribe the sphere and return to our starting point. This can't occur in
2-space, where we perforce travel forever in one direction and can't ever come
close to a previously visited point.

Again, a
manifold is
locally
Euclidean. If we start at any point in space, look
around and take a few steps in any direction, do we think we are in Euclidean
space? If the answer is yes, then we are in a manifold. If we continue walking,
and some unusual phenomenon occurs, such as returning to our starting point,
then we realize we are in a nontrivial manifold; that is, one with global
non-Euclidean properties. Our universe, for example, seems to be a manifold,
although interesting questions arise at black holes. Certainly one cannot
imagine standing at a black hole and taking a step in any direction! Researchers
are trying to devise methods of determining the global structure; a readable
introduction to this area of research can be found in Luminet et al.

The mystical
idea is relatively ancient—I leave it to a Borgesian intellect to trace its
roots and agelong echoes. Let's begin in a familiar place, our own universe. If
we talk about an object in our universe— for example, a desk or chair—we view
it as embedded in a larger space. Consequently, we often use our relative
coordinate system to refer to objects, as when we say "It's on my
right," or "Over there! Directly behind you, to the left," or
"Scratch my back.. . lower.. . lower... to the right. . . now up .. .
that's it!" Over the millennia, primarily as navigation aids, we've
settled upon somewhat less arbitrary reference points, such as the North Star,
the magnetic North Pole, and the true North Pole. The point is, though, that
these references, these origins, are all within our universe. "Outside the
universe" is a phrase beyond normal comprehension. Some theories place our
universe in a larger matrix, such as a superheated gas cloud containing an
infinite number of inaccessible universes, or in a higher-dimensional space, or
in a multiplicitous welter of bifurcating universes. However, these theories
raise the question,

What
is outside of the larger universe?

Really, now, though,
"What is outside of the universe?" The answer is no thing; nothing;
non-space; indescribability; un-thing-ed-ness; Void beyond vacuum: all these
non-things are the "outside" of our universe.

These two
ideas, the mathematical and the mystical, are woven together in this question
and its answer.

Where
is the center of a sphere?

If the sphere is considered as
an everyday object embedded in our universe, the answer may take a form such as
"at the intersection of two diameters," or, pointing at it
dramatically, saying with particular emphasis, "There! In the middle, in
the interior!" See figure 9. If, though, we consider the surface of the
sphere as a manifold, as a space in itself and of itself, then the question and
answer are subtler. As in the case of our universe, as if we were points
residing in the sphere itself, there is no legitimate referral to a point
outside the universe of the surface of the sphere. There is only the sphere;
every thing else is no thing. Where is the center of a sphere? Considered as a
manifold, then, the answer is

Everywhere
and nowhere.

Every point has the property
that locally, it looks like Euclidean space, and regardless of the direction
taken, consistently moving in any chosen direction returns us to the starting
point. No point is distinguished in any way.

One more idea is necessary to
provide a satisfying topology for the Library. The example of a manifold we
used was a two-dimensional sphere
(2-sphere).
There are a number of ways
to rigorously define a 2-sphere. Euclid might write something like, "Given
a point
p
in 3-space, a sphere with center
p
is the collection of
all points a specified uniform distance away from
p
." An analytic
geometric equation for the standard unit sphere is
x
²
+ y
²
+ z
²
= 1. (If you're interested in seeing why this equation
specifies a sphere, please turn to the appendix "Dissecting the
3-Sphere".) Here, using words and pictures, we provide a topological
construction of a 2-sphere.

Start with a
disk in the Euclidean plane and while preserving the interior of the disk
except for bending and stretching, crimp the entire boundary circle up out
of2-space, and then contract the boundary to one point. This point, the contraction
of the boundary, becomes the north pole and vanishes into the surface of the
sphere created as the process is completed (figure 10). An interesting point:
the way we've described it, and the way the picture shows this process, it
seems as though a disk is being modified over time. By contrast, though, one
should simply say, "Identify the boundary of the disk to a point."
Thus, in some sense, the creation of the sphere is a timeless step that happens
"all at once."

The three-dimensional sphere (
3-sphere
)
provoked many advances in topology over the past century, and due to the
recently solved Poincaré conjecture it remains a vibrant research topic. The
3-sphere is a generalization of the
1-sphere
—a circle—and the 2-sphere.
Euclid might write something such as, "Given a point
p
in
Jour-dimensional Euclidean space, a sphere with center
p
is the
collection of all points a specified uniform distance away from
p
."
An analytic geometric equation for the standard unit 3-sphere is w
²
+ x
²
+ y
²
+ z
²
=1. (Again, to see how this
equation captures "sphericality," please consult the appendix
"Dissecting the 3-Sphere.") An analogous topological construction for
the 3-sphere is difficult to envision, but by pushing the limits of our understanding,
we may learn much.

Take a solid
ball—a baseball, or an apple, or a cherry, or a cannonball—and, while leaving
the interior of the ball uncompressed, crimp the entire boundary sphere
upwards, and then simply contract the boundary sphere to one point (figure 11).
That's it. At least the
difficulty
is easy to understand; for the
construction of the 2-sphere, we took a two-dimensional object, the disk, and
had to bend it into the third dimension before we could contract the boundary
at all. Starting with a solid ball in three dimensions, we must
"bend" the ball into the fourth dimension before we can contract the
boundary (figure 12). At this juncture, the mathematics becomes unimaginable;
the best to be hoped for is that by meditating on the lower-dimensional examples
accessible to our imagination, we may be able to conjure the memory of the
trace of a once-sensed intuition. Still, by proceeding with analogies to the
2-sphere, we'll use a trio of methods to begin to visualize the 3-sphere.

If we take a two-dimensional
Euclidean slice of a 2-sphere, the resulting geometric object is either a
point—at the north and south poles—or a 1-sphere (figure 13). Using a mild
updating of an idea from
Flatland,
if we make a movie of the slice
moving from the north pole to the south pole, a viewer would see a point that
grows into a unit circle, which then shrinks back down to a point (figure 14,
left). In a similar fashion, if we take a three-dimensional Euclidean slice of
a 3-sphere, the resulting geometric object is either a point—at the
"top" or "bottom"—or a 2-sphere. If we make a movie of the
slice moving from the top to the bottom, the viewer would see a point that
grows into a unit sphere, which then shrinks back down to a point (figure 14,
right). (Again, for those who find equations more convincing than pictures, we
provide an analytical proof of this in the appendix "Dissecting the
3-Sphere.")