The Unimaginable Mathematics of Borges' Library of Babel by William Goldbloom Bloch Page B

points 2, 4, 5, and 6, it also induces a vertiginous disorientation
born of trying to imagine a thing extending away forever. For example, if the
Library goes down forever, what do the hexagons rest on? More hexagons?
Rather remarkably, the architectural, model of the Library that we propose
provides a satisfying answer to this question.
    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