The Unimaginable Mathematics of Borges' Library of Babel (25 page)

A  H 
I  M  O  T  U  V  W  X  Y  8  ‘  “  -

=  + 
·  :  *  ^  |  !  .  (blank space)

 (A sharp-eyed reader will
note that some of the letters in this font, such as A, M, U, V, W, X, and Y,
aren't precisely bilaterally symmetric. These letters need only minor
modifications to become bilaterally symmetric.) Fourteen other symbols, readily
available, are also invariant under mirror-inversions:

_ 
†  ˚  
∞  ±  
∏
  Ω
 
¡
 
÷  ♦  
‡  Â  ˘

Furthermore, there are pairs
of symbols that when flipped produce each other:

(  )  [  ]  {  }  <  >

It wouldn't take long to
create an aesthetically pleasing set of 25 symbols with the desired
mirror-reversal invariance.

Along
similar lines, almost all book titles are printed on the spine so that if the
book is laying flat on a table and the front cover is visible, the title can be
read: the
tops
of the letters abut the
front
cover of the book.
Let's call this "top-front" labeling. Try vertically holding the
spine of a top-front book to a mirror. Not only are the letters
mirror-reversed, but the title now appears as a "top-back" label on
the spine (figure 45). A solution to this problem is to simply write the titles
of the books vertically down the spines (figure 46). This way, even with a
mirror-reversal, a librarian wouldn't notice anything amiss.
⁴

Moreover,
it's ironic that a band of immortal librarians who circumnavigated the 3-Klein
bottle Library and returned to their originating hexagon
wouldn't recognize
any of the books.
Although the titles would all be the same, the contents
would all look different. Open a book to the first page while looking in a
mirror: it appears that the book is open to the
last
page, not the
first. So for the books in a particular hexagon to read the same to a
mirror-reversed librarian, the hexagon would need to consist of books that were
410-page palindromes! However, if they were intrepid enough to complete a
second circumambulation of the Library, they'd experience a mirror-reversal a
second time and then everything would look the same as when they started out.

Suppose that the constructors
of the Library incorporated these design changes to the physical structure and
the orthographic symbols. If all the librarians migrated, the Library would not
look mirror-reversed to them, even after passing through the disorienting
identified faces. However, if they split into two groups and one group managed
to circumnavigate the Library, the descendants of the nomadic group would
return and discover—from their perspective—a foreign group of librarians who
didn't know left from right. Although it's more likely that any such disparity
would be attributed to language differences, a librarian of genius might
realize the significance of the invariance of the orthographic symbols when
reflected in the mirrors. Such a mathematically minded librarian might then
deduce that the Library is a nonorientable 3-manifold, and a 3-Klein bottle would
surface as the most likely candidate. Such a librarian would know more about
the topology of the Library than we know about our own universe.

We've now covered all the hard
parts for the last topology we wish to propose for the Library. The end result
will once again be either a 3-torus or a 3-Klein bottle; we will just take a
different, more elegant, path to achieve it. In our discussions of the tori and
Klein bottles, we began with a square or a solid cube and then, respectively,
identified edges or faces.
⁵

In the next
chapter, we'll look at a single floor of the Library, in part by choosing an
initial hexagon and considering rings of adjacencies to it. Any ring of
adjacency, combined with the hexagons contained inside, forms another shape,
which is essentially hexagonal in nature (figure 47). (This is particularly
clear if we look at the midpoints of the hexagons.) If we start with a hexagon
and carefully identify the sides, just as when we start with a square, the resulting
object is either a torus or a Klein bottle. Jeff Weeks, in
The Shape of
Space,
pages 116—26, discusses this and provides very clear illustrations,
and Weeks' detailed explanation of these issues is both elegant and relatively
accessible. For a reader wanting to know more about 2- and 3-manifolds, it is
an excellent reference.

There is, however, a major
difference between the hexagon and the square as the object that will have its
edges identified. It turns out that to embed the "hexagon with identified
edges" in Euclidean 3-space, which is
not
its "natural"
home, the hexagon must be twisted as well as stretched and bent.

Generalizing
from two dimensions, it shouldn't be hard to believe that by starting with a
hexagonal prism, identifying faces may yield either a 3-torus or a 3-Klein
bottle (figure 48). This model of the Library has the additionally pleasing
aspect that, in some difficult-to-define sense, the larger geometric structure
mimics the smaller structure of the hexagon. We wrote "difficult-to-define"
because as soon as the faces are identified, the hexagonal prism is subsumed
into the 3-torus or 3-Klein bottle. Again, a torus may be considered a square
with its edges identified. However, once the edges are identified, the edges
are gone and the square is gone: all that's left is the torus. The identified
edges may be drawn in for the purpose of clarifying the process, but they are
no longer there. Equally possible, a torus may be a hexagon with its edges
identified. In both cases, the resulting object is a torus, but the differing
characters of the square and hexagon may leave a detectable, classifiable
trace.

A Library
modeled on the 3-torus or 3-Klein bottle could be based on either a cube or a
hexagonal prism with identified faces. It's conceivable that an immortal
librarian of genius, endowed with a means of measuring curvature and possessed
of an infinite photographic memory, who repeatedly traversed the Library might
also some day be able to guess the Library's topologic structure. Regardless,
we who may now consider ourselves the architects of the Library may feel the
special glow that derives from the outlining of exquisite solutions to a
demanding problem.

Geometry and Graph Theory

Ambiguity and Access

If one does not expect the
unexpected one will not find it out.

—Heraclitus,
Fragment 18

A library is a collection
of possible futures.

—John
Barth,
Further Fridays

THE LIBRARY, AS EVOKED IN THE STORY, HAS
inspired many artists and architects to provide a graphic or
atmospheric rendition of the interior. These range from Stefano Imbert's lean
and elegant drawing adorning the cover of this book, to the deliberately
alienated Piranesi-like drawings of Desmazieres in the Godine Press edition of
The Library of Babel,
to Toca's beautifully symmetric honeycombs in
Architecture and Urbanism,
to Packer's bold expressionist frontispiece in
the Folio edition of
Labyrinths,
to a host of illustrations easily found
online. All of these illustrations sacrifice, to some degree, accuracy in favor
of artistic effect. For example, even the cover illustration of this book
locates the spiral staircase in the center of the hexagon, whereas Borges
writes (emphases added):

The universe
(which others call the Library) is composed of an indefinite, perhaps infinite
number of hexagonal galleries. In the center of each gallery is a ventilation
shaft, bounded by a low railing. From any hexagon one can see the floors above
and below—one after another, endlessly. The arrangement of the galleries is
always the same: Twenty bookshelves, five to each side, line four of the
hexagon's six sides; their height of the bookshelves, floor to ceiling, is
hardly greater than the height of a normal librarian.
One of the hexagon's
free sides opens onto a narrow sort of vestibule, which in turn opens onto
another gallery, identical to the first—identical in fact to all.
To the
left and the right of the vestibule are two tiny compartments. One is for
sleeping, upright; the other, for satisfying one's physical necessities.
Through
this space, too, there passes a spiral staircase,
which winds upward and
downward into the remotest distance.