The Unimaginable Mathematics of Borges' Library of Babel (20 page)

Expanding on this idea,
suppose we were forced to squish the 2-sphere, whose natural home is in
3-space, down into 2-space. Since we just conceived of the 2-sphere as a
collection of stacked circles combined with two poles, we may envision a
flattened planar depiction as a collection of intersecting circles with two
points signifying the north and south poles (figure 15). The related problem,
the one that's been tasking us, is how to represent the 3-sphere down-sized
into 3-space. If we think of the 3-sphere as "stacked" 2-spheres—in
the same sense that a 2-sphere is stacked 1-spheres—the analogous 3-space
representation is a collection of intersecting 2-spheres (figure 16).

For the third way of
envisioning the 3-sphere, the lower-dimensional correlate is to take a section
of the 2-sphere and flatten it out into the Euclidean plane. If our section
includes, say, the south pole, the flattened section is a disk. If our section
doesn't include either pole, the flattened section is an
annulus,
which
is a ring, a thickened circle. Note that the equator of the sphere (the dotted
circle in figure 17) is flattened to the central circle of the annulus. The
circle-slices above the equator on the sphere are smaller than the equator, but
when flattened become
larger
than the central circle of the annulus.
Similarly, the circle-slices below the equator flatten to even smaller circles
in the annulus than they were in the sphere. This process of dimensional
flattening distorts the object; necessarily, information is lost.

If we take sections of the
3-sphere, we must consider how to "flatten out" the resulting object
into 3-space. If our section includes, for example, the bottom ofthe 3-sphere,
the flattened section is, by analogy, a solid ball. If the section of the
3-sphere doesn't include the north or south poles, the "flattened"
section is a solid ball with a smaller ball removed from the center—a pitless
olive, or a tennis ball, or an empty walnut shell, or a thickened spherical
shell. In figure 18, perhaps the most counterintuitive aspect is the means by
which the middle collection of 2-spheres collates to a thickened spherical shell.
The centralmost, the largest 2-sphere, is flattened to itself. The smaller
spheres directly to the left, say, ofthe central sphere thicken it on the
inside. The smaller spheres directly to the right of the central sphere are
distorted by the flattening into
larger
spheres that thicken the
exterior of the central sphere. Again, unfortunately, the process entails that
we must lose information about the size of the spheres.

All the girders and struts of
the framework are now in place to finish assembling the topology and cosmology
of the Library. The 3-sphere is a three-dimensional manifold; at every point,
if we inhabited the 3-sphere, we would say—locally—that space was Euclidean. If
we walked what we perceived to be a straight line in any direction, we
would—possibly after countless ages—return to our starting point; the 3-sphere
can be construed as
periodic.
There are
no boundaries,
no walls
to bump into; the 3-sphere is
limitless.
Moreover, in his luminous story
"The Garden of Forking Paths," Borges has the sympathetic sinologist
Stephen Albert say, ". ..I had wondered how a book could be infinite. The
only way I could surmise was that it be a cyclical, or circular, volume, a
volume whose last page would be identical to the first, so that one might go on
indefinitely." Even though Albert rejects this line of reasoning for
"The Garden of Forking Paths," this quote, coupled with Borges'
well-known interest in Nietzsche's idea of eternal recurrence, indicates that
Borges was willing to consider cyclic or recurrent structures as tokens of, or
synonymous with, infinity.

Considered
as a three-dimensional manifold,
the center of the 3-sphere is everywhere
and nowhere.
Furthermore, if the 3-sphere is so large that, regardless of
our transport, we could never come close to circumnavigating it, it would not
be illegitimate to say that the
circumference is unattainable.
Finally,
this answers the question concerning what the hexagons "rest on." By
forming
great circles
—circles which are essentially equators of a sphere—the
hexagons all rest upon each other and ultimately themselves, and thus there is
no need for an impossible "external" foundation for the Library.
¹