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

a
conceivable topology of the Library that best reflects the anonymous
librarian's received wisdom and secret hopes.
    Early in the
story—and many commentators have noted the connection between the italicized
phrase and Borges' essay "Pascal's Sphere"— Borges writes
     
    Let it
suffice for the moment that I repeat the classic dictum: The Library is a
sphere whose exact center is any hexagon and whose circumference is
unattainable.
     
    The final sentences of the
story invite us to reopen the question of the topology of the Library:
     
    I am perhaps
misled by old age and fear, but I suspect that the human species—the only human
species—teeters at the verge of extinction, yet that the Library—enlightened,
solitary, infinite, perfectly unmoving, armed with
precious volumes, pointless, incorruptible, and secret—will endure.
    I have just written the word "infinite." I have not
included that adjective of out of mere rhetorical habit; I hereby state that it
is not illogical to think that the world is infinite. Those who believe it to
have limits hypothesize that in some remote place or places the corridors and
stairs and hexagons may, inconceivably, end—which is absurd. And yet those who
picture the world as unlimited forget that the number of possible books is not. I will be bold enough to suggest this solution to the ancient problem: The Library is unlimited but periodic. If an eternal voyager should journey
it in any direction, he would find after untold centuries that the same volumes
are repeated in the same disorder—which, repeated, becomes order: the Order. My
solitude is cheered by that elegant hope.
     
    Collecting the properties of
the classic dictum (CD) and the Librarian's solution (LS), we obtain the
following list:
     
    1. Spherical
(CD)
    2. Center
can be anywhere—uniform symmetry (CD)
    3. Circumference
is unattainable. (CD)
    4. No
boundaries (LS)
    5.  Limitless (LS)
    6. Periodic
(LS)
     
    Is there a space that embodies
all six of these properties? If so, how can we best envision it and grasp it
with our intellect? We claim there is an excellent candidate that encompasses
these properties, if we are willing to refine our interpretations just a
smidge. In the Math Aftermath to this chapter, we discuss two other compelling
ways ofconfiguring the Library that each significantly expand our conceptions
of the possible.
    Let's begin
with the space most familiar to our intuitive geometric sense: Euclidean
three-dimensional space (henceforth, 3-space). It is a space we think of
as possessing volume, as having three axes of orientation with ourselves as the
central point; we may move forward or backwards, we may move left or right, and
we may move up or down. And, ofcourse, we may also move in combinations of
these directions. Notice that from this description, there is no fixed
preferred center point: we are our own central points.
    Indeed, one
of Descartes' deepest ideas was to specify a point—some point, any point—in
3-space and call it the origin. Three axes intersecting at the origin,
typically called the x, y, and z axes, are set with each axis at right
angles to the other two. They abstract our innate, intuitive orientation and,
with the introduction of a unit length, which naturally induces a numbering of
the axes, give rise to a coordinatization of space. Algebra can now conjoin
geometry, creating analytic geometry, and later spawn calculus.
    But there
are no distinguished points of any kind in Euclidean 3-space; in fact, the view
from any point is the same as from any other point. There are no walls, no
boundaries, and no limits. It seems at the end of the story the librarian
envisioned this kind of space, partitioned into hexagons, filled with books,
extending infinitely throughout the totality of 3-space. The books' shelving
pattern repeats endlessly along each of the three axes, much as a symmetric
wallpaper pattern does in two dimensions. While this conception of the Library
satisfies