"El universo (que otros llaman la Biblioteca) se compone de un número indefinido, y tal vez infinito, de galerías hexagonales, con vastos pozos de ventilación en el medio, cercados por barandas bajísimas. Desde cualquier hexágono se ven los pisos inferiores y superiores: interminablemente. La distribución de las galerías es invariable. Veinte anaqueles, a cinco largos anaqueles por lado, cubren todos los lados menos dos; su altura, que es la de los pisos, excede apenas la de un bibliotecario normal. Una de las caras libres da a un angosto zagu n, que desemboca en otra galería, idéntica a la primera y a todas. A izquierda y a derecha del zaguan hay dos gabinetes minúsculos. Uno permite dormir de pie; otro, satisfacer las necesidades finales. Por ahí pasa la escalera espiral, que se abisma y se eleva hacia lo remoto. ... 'La Biblioteca es una esfera cuyo centro cabal es cualquier hexágono, cuya circunferencia es inaccesible.'"
Freely translated (by me):
"The universe (that others call the Library) is composed of an indefinite, and perhaps infinite, number of hexagonal galleries, with vast ventilation shafts in their centers surrounded by low railings. From each hexagon one can see the lower and higher floors--without end. The arrangement of the galleries is fixed. Twenty shelves, with five long shelves per side, cover all the sides except two; their height, that is that of the level itself, is scarely more than that of the average librarian. One of the free faces opens on to a narrow vestibule, that leads to another gallery, identical to the first and to all the others. To the left and to the right of the vestibule there are two tiny rooms. One permits sleeping standing up; the other, satisfying the "final necessities" [i.e., a latrine]. Through it also passes a spiral staircase, that goes down into the abyss and up to the remotest levels. ... 'The Library is a sphere whose precise center is any hexagon, and whose circumference is inaccessible.'"
(The latter is an obvious reference to Blaise Pascal's description/definition of Nature/the Universe: "It is a infinite sphere, the center of which is everywhere, the circumference nowhere." [Pensées, 1670] But Borges himself wrote [in the essay "Pascal's Sphere", 1951], that Pascal started to write "effroyable"--"a frightful sphere....)
When attempting to determine the topology of the Library, one thing immediately strikes the reader: Borges has failed to account for one of the six sides of each hexagon. Four have shelves, one has a vestibule, but what of the sixth? Well, a moment's thought will lead one to the conclusion that it too must lead to another hexagon, since only one exit in each hexagon would result in an infinite number of two-hexagon columns, each completely cut off from any hexagon outside that column. The assumption, however, seems to be that every hexagon is accessible from every other hexagon, and for this, two doors in each hexagon are required. (Well, not quite--see below for another way to account for this.)
But is this interpretation sufficient? Yes, although the resulting topology does not appear to be what Borges envisioned. For every hexagon to be accessible from every other hexagon, if only (a maximum of) two exits are allowed per hexagon, then it appears that the layout must be in effect a spiral. Choose a hexagon as the starting point. Exit into any adjoining hexagon, then circle the first one clockwise, creating doorways, until one would be re-entering a hexagon already visited. At that point, choose the wall to the left of the one you would have chosen and go through that one, then circle again clockwise around all those hexagons already visited, and so on. This will allow you access to every hexagon eventually, and by use of the staircases, to every hexagon on every level.
The problem with this is that it in effect makes each floor of the Library a single infinitely long room with one fixed end. This does not appear to be how Borges wanted the reader to picture the Library. And indeed, the necessity to select a starting hexagon-- which will have only one exit instead of two--violates both the statement that all hexagons are identical and that any hexagon may be considered the center of the Library. (There will be more on this later.)
(As an aside, one might marginally improve the connectivity by alternating clockwise and counter-clockwise traversals on alternating floors, but that does not change the linear layout of a given floor.) (There will be more on this also later.)
Now, I have assumed that the connections to other hexagons are "dimensionless"--basically an opening in a wall. In a Usenet posting from 1984, Donn Seeley started with different assumptions:
"[Let] us assume that the Library fills space; it extends to an arbitrarily large distance in all directions in three dimensions (or more?). Let's assume that the second 'free side' of a hexagon opens onto another gallery directly, without passing through a hall with a staircase. Without this assumption it would be difficult to establish an arrangement compatible with the first assumption. Next, let's assume that given sufficient time, it is possible to travel from any hexagon to any other; this is implied but not stated in the course of the story. Finally, to make tiling convenient, let's assume that the halls which contain stairwells are hexagonal in shape and the same size as the book hexagons. We can explain the narrowness of the corridor by the fact that the bedrooms and bathrooms and stairs take up most of the floor space. We can even put the stairs in the same position as the central ventilation shaft of the book hexagons (they were pre- fabricated!)."
As you can see, Seeley makes one major change in his assumptions from mine: he assumes that the vestibule/closets/staircase area forms a hexagon of its own, the same size as a book-filled hexagon. But as I noted (in e-mail) at the time, I do not think that Borges's description warrants that assumption.
For one thing, the description of the vestibule is that it leads to another gallery (not that it is another gallery), and that the staircase passes through the vestibule itself. The second objection is that having galleries which have the closets, staircases, and multiple exits of their own, but no books, violates the statement that each gallery is identical to all the others.
He then additionally postulates that the sixth side of the hexagon as a "simple" door into an adjoining (book-filled) hexagon. While it does eliminate the need for multiple sleeping rooms and latrines accessible from each hexagon (thus allowing for more book-filled hexagons), and does provide some explanation as to why Borges did not describe the sixth side, the asymmetry of the layout is unsettling.
(Given his conditions, by the way, Seeley was able to design a method of connecting all the hexagons that did not rely on a unique starting point.)
Seeley is at least more accurate than Shirley Neuman, who said at the opening of the Walter C. Koerner Library at UBC: "Four sides of each hexagon hold five rows each of identical bookshelves. One side is bounded by a low railing overlooking an airshaft.... The sixth side of the hexagon opens onto a modest hall, which leads transversely to other hexagons, and vertically by means of a spiral staircase to hexagons above and below."
This is just wrong. The airshaft is in the center of the hexagon, with the railing all the way around the shaft, not merely on one side of the hexagon. And the "modest hall" of Borges's description leads to a single other hexagon, not multiple ones.
As I noted above, though, there is another way around some of these problems. That is by simply assuming that the hexagons are actually free-standing. In other words, while the Library as a whole fills space, the hexagons do not. Instead, one may suppose that there are narrow passages that surround each hexagon and separate it from the other hexagons. Think of the interior walls of a house. We think of the rooms as filling the house, but in fact, there is some space between the walls of two adjoining rooms. If we expand this space to be wide enough to allow the librarians to walk through them, then they could access any room by using the passageways to get to the entrance of the room they wish to access. (When I worked at Bell Labs in Holmdel, there was indeed a wide enough space between two "adjacent" aisles to allow people to walk along this space to work on the wiring to the offices on either side.)
While it is true that this set of passageways (or rather, one giant inter-connected passageway) is not described in the original story, it is the sort of thing that could easily be over-looked in the description--just as you don't talk about the intra-wall space when describing your house.
I found this idea quite attractive, so it was very disappointing to read in Jorge Luis Borges; A Writer on the Edge by Beatriz Sarlo (ISBN-13 978-0-86091-635-2, ISBN-10 978-0-86091-635-9), "As Borges himself declared in an interview, his first spatial idea for the Library of Babel was to describe it as an infinite combination of circles, but he was annoyed with the idea that the circles, when put in a total structure, would have vacant spaces in between. He chose the hexagon for its perfect simplicity and its perceptive affinity to the circle." (The interview is in Borges y la Arquitectura by Christina Grau, Madrid 1989.) So the idea of intra-wall space, regrettably, has to be discarded.
(For the non-lit-theory trained reader, Sarlo has an unfortunate tendency to drop terminology like "en ab&icarat;me" ("in the abyss", typified the view when standing between two mirrors facing each other--an archetypal Borgesian image!) and "als ob" ("as if", connected to "willing suspension of disbelief").)
While I was writing the above, I discovered that there had just been a book published on that very topic: The Unimaginable Mathematics of Borges' Library of Babel by William Goldbloom Bloch (ISBN-13 978-0-19-533457-9, ISBN-10 0-19-533457-4).
Bloch covers all the mathematical aspects of the Library, not just the layout of rooms. So he has chapters on "Combinatorics: Contemplating Variations of the 23 Letters", "Information Theory: Cataloguing the Collection", "Real Analysis: The Book of Sand", and "Topology and Cosmology: The Universe (Which Others Call the Library)". But the chapter which covers the same topic as I did earlier is "Geometry and Graph Theory: Ambiguity and Access".
Before I talk about that, though, I want to mention his conclusions in "Topology and Cosmology: The Universe (Which Others Call the Library)". Bloch begins with the two sentences "The Library is a sphere whose precise center is any hexagon, and whose circumference is inaccessible," and "The Library is unlimited and periodic." These he reduces to six requirements: 1) spherical, 2) uniform symmetry, 3) circumference unobtainable, 4) no boundaries, 5) limitless, and 6) periodic. And from these he concludes that the Library must be on a 3-sphere (existing in 4-space). This is not the approach I took; I assumed the Library existed in a basically Euclidean three-dimensional space.
As for what I discussed above, Bloch has new insights. For example, I said, "For every hexagon to be accessible from every other hexagon, if only (a maximum of) two exits are allowed per hexagon, then it appears that the layout must be in effect a spiral. ... The problem with this is that it in effect makes each floor of the Library a single infinitely long room with one fixed end. This does not appear to be how Borges wanted the reader to picture the Library. And indeed, the necessity to select a starting hexagon--which will have only one exit instead of two-- violates both the statement that all hexagons are identical and that any hexagon may be considered the center of the Library." Bloch has no problem with each floor being a single long path, perhaps because his embedding of the Library in 4-space solves the problem of having a starting hexagon with only one exit.
However, I also said that "one might marginally improve the connectivity by alternating clockwise and counter-clockwise traversals on alternating floors." Since I had assumed each hexagon had two exits, each with a spiral staircase, this is just wrong. As Bloch points out, this assumption mandates that every floor of the Library is identical to every other. It turns out that if each hexagon has two exits but only one staircase, then it is possible to have different paths on each floor.
I also wanted to point out that the topic covered in "Combinatorics: Contemplating Variations of the 23 Letters" is one that Borges was not the first to address. Kurd Lasswitz's "The Universal Library" was first published in the United States in 1958 in Clifton Fadiman's classic anthology, Fantasia Mathematica (ISBN-13 978-0-387-94931-4, ISBN-10 0-387-94931-3), but had been published in Germany over half a century earlier, in 1901. And indeed Borges discusses it at length in his essay "The Total Library" (1939), and then explicitly lists it as a major inspiration for "The Library of Babel" (1941) in his introduction to his collection Ficciones. Lasswitz assumes 100 symbols rather than 25, but is also more concerned with the number of books rather than the layout of any library containing them.
(There have been other books covering the mathematical aspects of Borges's work. Alas, most of them seem to be out of print and hence very expensive.)