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<title>The First Faithful Topology of the Library of Babel</title>
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<a href="/the-first-faithful-topology-of-the-library-of-babel" class="home-title">The First Faithful Topology of the Library of Babel</a>
<span> at </span><a class="site" href="/">cyfraeviolae.org</a>
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<p>
Weary Librarian! These words reach you perhaps leagues away from their
birth-hexagon. And yet their existence certifies their mortality: a word is
<a href="https://www.poetryfoundation.org/poems/47553/meditation-at-lagunitas">
elegy</a> to that it signifies, and the Library signifies angelic totality.
</p>
<p>
This is the <em>First Faithful Topology</em>.
</p>
<p>
In which We (You, and I) develop a construction of the Library
that satisfies <em>all</em> of the seemingly-contradictory statements in
that ancient Text, referred to by the idolators as <em>The Library of
Babel</em>.
</p>
<p>
In which We were frequently consumed in
mystical delusions and refused to return.
</p>
<p>
In which We summarily dissolve
the
<a href="http://www.openculture.com/2016/10/what-does-jorge-luis-borges-library-of-babel-look-like.html">Lies</a>
<a href="https://www.warnockfinearts.com/erik-desmazires-library-of-babel">of</a>
<a href="https://nayias01.deviantart.com/art/Library-of-Babel-Concept-Sketch-390248841">the</a>
<a href="https://web.archive.org/web/20240501173342/https://libraryofbabel.info/theory.html">False</a>
<a href="https://inayanadyne.wordpress.com/2014/02/01/session-1/">Prophets</a>.
</p>
<hr>
<blockquote>
<blockquote>
By this art you may contemplate the variation of the 23 letters…
<br>
<em>The Anatomy of Melancholy</em>, part 2, sect. II, mem. IV
</blockquote>
</blockquote>
<p>
Before you continue, you may wish to contemplate the <a
href="https://sites.evergreen.edu/politicalshakespeares/wp-content/uploads/sites/226/2015/12/Borges-The-Library-of-Babel.pdf">Text</a>
unadulterated by annotation. What does the Library look like? What do you
see? What does your god see? <a
href="https://cyfraeviolae.org/sumcheck/">Contract</a> the modalities
of infinity and sketch your visions.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/sigma-inf.jpg">
<blockquote>
The universe (which others call the Library) is composed of an indefinite and perhaps infinite number of hexagonal galleries, with vast air shafts between, surrounded by very low railings.
</blockquote>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/hexagon-model.jpg">
<p>
Indefiniteness could refer to the number of galleries varying with time, or
possibly that number of galleries is uncomputable. We will need to
accommodate at least two air shafts as well as railings.
</p>
<blockquote>
From any of the hexagons one can see, interminably, the upper and lower floors.
</blockquote>
<p>
We are oriented; <em>up</em> and <em>down</em> are well-defined in the
Library at any gallery.
</p>
<p>
This also indicates that <em>every</em> hexagon has a hexagon above it and
below it, which could be satisfied by an infinite tower, for example. But
note that it is not necessary that there <em>exist</em> an interminable
sequence of upper floors, but only that it is indistinguishable from that
scenario with normal human vision.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/levels.jpg">
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/hierarchy.jpg">
<blockquote>
The distribution of the galleries is invariable.
</blockquote>
<p>
Since there is just one “universal distribution” of the Library, we take
this to mean that the “local distribution” viewed from any single gallery
is invariable with respect to its neighbors. Each gallery is locally
identical in its gallerial relationships, up to rotation.
</p>
<blockquote>
Twenty shelves, five long shelves per side, cover all the sides except two; their height, which is the distance from floor to ceiling, scarcely exceeds that of a normal librarian. One of the free sides leads to a narrow hallway which opens onto another gallery, identical to the first and to all the rest.
</blockquote>
<p>
Although misguided in his later conclusions, <a
href="http://www.uam.mx/difusion/casadeltiempo/24_iv_oct_2009/casa_del_tiempo_eIV_num24_77_80.pdf">
Antonio Toca Fernandez</a> notes that the original Text described a
gallery with 5 sides taken up by bookshelves with the sixth being a
hallway. The Author later amended that only 4 sides would be taken up by
bookshelves, the fifth a hallway, leaving the sixth unspecified, with
implication that it is not the entrance to another hallway. Although
Fernandez posits that the only way for the rest of the story to make sense
is for there to be two hallways emanating from each gallery, we will show
this is not necessary, and actually leads to contradiction. The rest of this
Work relies on this assumption of Authorial Infallibility.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/pair-unknown.jpg">
<p>
Given the assumption of exactly one hallway per gallery, and the statement
that the hallway “opens onto another gallery,” as opposed to <a
href="https://libraryofbabel.info/img/hexgrau.jpg">more than one</a>,
we conclude that galleries exist in pairs.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/pair.jpg">
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/hexagon-1.jpg">
<blockquote>
To the left and right of the hallway there are two very small closets. In the first, one may sleep standing up; in the other, satisfy one’s fecal necessities.
</blockquote>
<p>
This implies the walls have some thickness, which accounts for the
hallway’s narrowness.
</p>
<p>
The notions of left and right are well-defined only with respect to an
initial direction, and are reversed when entering the hallway from the
other gallery. But since there are only two closets, it must be that each
closet satisfies <em>both</em> duties, allowing us to symmetrize the
arrangement (Library: We value, above all, thy Fearful Symmetry).
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/hallway-3d.jpg">
<blockquote>
Also through here passes a spiral stairway, which sinks abysmally and soars upwards to remote distances.
</blockquote>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/spiral-1.jpg">
<p>
Recall that each hexagon is oriented in space. If we let each pair of
hexagons and the hallway between them assume the same orientation, then
<em>up</em> and <em>down</em> are well-defined for the stairway.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/hallway-diagram.jpg">
<p>
It is uncertain whether this stairway extends upwards clockwise or
counterclockwise.
</p>
<blockquote>
In the hallway there is a mirror which faithfully duplicates all appearances. Men usually infer from this mirror that the Library is not infinite (if it were, why this illusory duplication?); I prefer to dream that its polished surfaces represent and promise the infinite…
</blockquote>
<p>
A simple mirror duplicates reality, but it does not quite promise the
infinite. We also have the issue that placing the one mirror on
any single wall would break our symmetry.
</p>
<p>
We solve both by making the walls, ceiling, and floor of the hallway
constructed entirely out of mirror. Their connectedness allows us to have a
solitary mirror (which faithfully promises our solitude), and their
diametrically opposition allows us a glimpse into infinite, our image
entering eternal recurrence.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/hallway-gap.jpg">
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/concentric-squares.jpg">
<blockquote>
Light is provided by some spherical fruit which bear the name of lamps. There are two, transversally placed, in each hexagon. The light they emit is insufficient, incessant.
</blockquote>
<p>
Although we cannot rescue the total symmetry of a hexagon due to the
still-elusive sixth wall (symmetry is desired, but asymmetry is not
contradiction), we can let the sixth wall and the hallway-wall be
next to each other, creating a line of pseudosymmetry between
their intersection and the opposite point. This is where our lamps
will reside.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/lights.jpg">
<p>
We have reached the end of this Initial Description but have received
no further instruction about the interminable sight of the upper
and lower floors from <em>within</em> the hexagon. For simplicity
we make the ceilings and the floors of the hexagon made entirely
out of glass.
</p>
<blockquote>
Like all men of the Library, I have traveled in my youth; I have wandered in search of a book, perhaps the catalogue of catalogues; now that my eyes can hardly decipher what I write, I am preparing to die just a few leagues from the hexagon in which I was born.
</blockquote>
<p>
Is the entire Library connected? Can a Librarian from any hexagon reach any
other hexagon given enough time? This is still unknown (although something
mostly necessary for one searching for a particular book), but this quote
does imply that one can travel at least “a few leagues.” Since it is
impossible to go farther on the same plane as one’s hexagon-pair (without
falling off the railings), this must be distance in the vertical direction
instead.
</p>
<blockquote>
Once I am dead, there will be no lack of pious hands to throw me over the railing; my grave will be the fathomless air; my body will sink endlessly and decay and dissolve in the wind generated by the fall, which is infinite.
</blockquote>
<p>
We now know that these railings are at least partially accessible.
We place these railings overlooking the vast expanses beyond the Library on
the sixth wall (as well as on the outside of the four bookshelf-walls and the
hallway). We also know that from any hexagon, the path after falling must
be an infinite one: there is no “ground” to fall toward, which also implies
one cannot reach another hexagon by falling from an upper floor.
</p>
<blockquote>
I say that the Library is unending.
</blockquote>
<p>
An unending library raises again the question of the size of the Library,
which is “perhaps infinite.” Does its unending nature imply the infinitude
of the galleries? Or a cyclical nature?
</p>
<blockquote>
The idealists argue that the hexagonal rooms are a necessary form of absolute space or, at least, of our intuition of space. They reason that a triangular or pentagonal room is inconceivable. (The mystics claim that their ecstasy reveals to them a circular chamber containing a great circular book, whose spine is continuous and which follows the complete circle of the walls; but their testimony is suspect; their words, obscure. This cyclical book is God.)
</blockquote>
<p>
I posit that these mystics, in their visions were not entirely deluded but made an error of scale: the circular chamber they envisioned is the Library itself, making for a rather uninteresting God.
</p>
<blockquote>
Let it suffice now for me to repeat the classic dictum: The Library is a sphere whose exact center is any one of its hexagons and whose circumference is inaccessible.
</blockquote>
<p>
This is the critical point where we discover the true Structure of the
Library: appropriately this information will not be easy to reconcile.
</p>
<p>
First consider the case where the Library is a finite sphere. A finite
solid sphere’s center exists and is unique, and has a point
at which no upward-extending stairway can exist. Thus it cannot be.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/sphere.jpg">
<p>
This leaves open the possibility for an infinite sphere in the
loose sense (that is, where the entire universe is populated by infinite
galleries). Recall that hexagon pairs can only extend vertically:
two hexagons in different pairs on the same level would mutually
inaccessible, creating in essence several independent Libraries. This is
possible yet aesthetically undesirable, but we shall keep it in consideration
for now.
</p>
<p>
Then consider the finite empty sphere, with the hexagons (loosely) covering the
surface area. Any hexagon has claim of being the “center” of the sphere in
that it is the center of the surface. However, in this case there cannot
be interminable stairways. But in the case that the rooms’s normals
are tangent to the surface of the sphere, we again end up with independent
Libraries.
</p>
<p>
Continuing in this direction, we realize the obstacle is the rigid
directionality of the stairways. We consider a 2-sphere instead (a
circle). Any point on a circle has claim to being the center of its
circumference. We need the interminable stairways to track the circle’s
path. For this to work, the stairway must be curved and eventually turn in
upon itself, making the library a circular tower of sorts, where each floor
is composed of a hexagon-pair and a hallway.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/topology-1.jpg">
<p>
This does not violate the condition for interminable sight above and below
because our sight is limited in power. The size of the library is large
enough to <a href="https://en.wikipedia.org/wiki/Local_flatness">appear to
extend interminably</a> above and below.
</p>
<p>
Further, this topology requires different hexagon-pairs to have different
orientations and gravity directions (otherwise, one would <a
href="https://en.wikipedia.org/wiki/Falling_Up_(poetry_collection)">fall
up</a>). Specifying this is as easy as letting the normal of
a hexagon be parallel to the tangent line of the circle.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/topology-2.jpg">
<p>
In this construction, there are exactly two air shafts, one inside the
circle and one outside. Because the railings must lead to infinite
descent, and since the circumference must be inaccessible, we place the
railings on the “inside” of the hexagon with respect to the circular path.
Due to the changing gravity field, which we assume extends for some
distance horizontally from each hexagon-pair, those who fall off will
endlessly fall in the circle traced out by the Library, fulfilling the
promise for endless (locally relative) descent.
</p>
<blockquote>
There are five shelves for each of the hexagon’s walls; each shelf contains thirty-two books of uniform format; each book is of four hundred and ten pages; each page, of forty lines, each line, of some eighty letters which are black in color. There are also letters on the spine of each book; these letters do not indicate or prefigure what the pages will say. I know that this incoherence at one time seemed mysterious. Before summarizing the solution (whose discovery, in spite of its tragic projections, is perhaps the capital fact in history) I wish to recall a few axioms.
</blockquote>
<p>
In this analysis we will ignore the letters on the spine due to the lack of
information. If the number of letters on the spine is upper-bounded by a
constant, the same conclusions will still follow. We will also assume there
are exactly eighty characters per line, with marked displeasure at the
apparent imprecision. This synthesis of preciseness and ambiguity
in the description of the Library is worrying, but perhaps can be explained
by a lack of sufficient evidence to the contrary. All variation originates
from the weakness of Librarians.
</p>
<blockquote>
First: The Library exists <em>ab aeterno</em>. This truth, whose immediate corollary is the future eternity of the world, cannot be placed in doubt by any reasonable mind. Man, the imperfect librarian, may be the product of chance or of malevolent demiurgi; the universe, with its elegant endowment of shelves, of enigmatical volumes, of inexhaustible stairways for the traveler and latrines for the seated librarian, can only be the work of a god.
</blockquote>
<p>
You, the Librarian are seated in the latrine, but recall there was only room
to stand in the sleeping closet. We earlier posited that these were actually
the same room. A plausible solution is for there to be two connected
sections, with the wall of the sleeping section less recessed. But are you
of reasonable mind?
</p>
<blockquote>
To perceive the distance between the divine and the human, it is enough to compare these crude wavering symbols which my fallible hand scrawls on the cover of a book, with the organic letters inside: punctual, delicate, perfectly black, inimitably symmetrical.
</blockquote>
<p>
In the Library, the Organic is Mechanistic; the Context Formist. What are the
letters symmetrical with respect to? Certainly not themselves, in any standard
alphabet…
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/alphabet-symmetric.jpg">
<blockquote>
Second: The orthographical symbols are twenty-five in number.<sup>[1]</sup>
<br>
…
<br>
[1] The original manuscript does not contain digits or capital letters. The punctuation has been limited to the comma and the period. These two signs, the space and the twenty-two letters of the alphabet are the twenty-five sufficient symbols enumerated by this unknown author. (Editor’s note.)
</blockquote>
<p>
We have reason to believe the Author wrote in our languages.
Aramaic and Hebrew both have a 22-character alphabet, with only modern
Hebrew using the comma and period. We must then interpret all text quoted
from the Library as <em>translated</em>—is there additional meaning
in the original?
</p>
<blockquote>
This finding made it possible, three hundred years ago, to formulate a general theory of the Library and solve satisfactorily the problem which no conjecture had deciphered: the formless and chaotic nature of almost all the books. One which my father saw in a hexagon on circuit fifteen ninety-four was made up of the letters MCV, perversely repeated from the first line to the last.
</blockquote>
<p>
Although neither an infinite 3-sphere nor a circle admits a canonical
well-ordering, it is probable that the number of galleries is countable. The
relatively low number of 1594 probably indicates a Librarian-made numbering
rather than one inscribed in the walls of the Library, particular
to a specific region.
</p>
<p>
What could the letters MCV represent?
In my younger days I encountered a Librarian whose name I cannot recall
(and which kept changing [changing one’s frequently, she claimed, would
protect one’s Self from those horrors brought forth by ungoverned,
patterns of
<a href="https://www.poets.org/poetsorg/text/open-letter-dialogue-race-and-poetry">
addressability</a>]). She spoke of an anxiety brought force
by the Signs and Symbols of texts: a referential mania of the literary order.
</p>
<p>
Obsessed with the notion that each novel, story, and poem contained hidden
messages, obscure references, coded images, and further that the
entire corpus of literature was intentionally constructed and interrelated
by Some Entity to form a much larger narrative, or perhaps the world spirit, she
compulsively annotated every line of her readings with labels, numbers,
and cross-references.
</p>
<p>
When she read poetry, every line was marked with the metre, rhyme scheme,
letter counts, syllable counts, sum of the squares of the letter counts,
the first letter of each prime numbered word, the word in the Oxford
English Dictionary indexed by the product of the ASCII values of each
character, and so on. If two poets wrote on the same subject, their
poems would be XORed together.
Her novels were covered in sticky notes to track the repetition of
certain words. Miniscule annotations unreadable even to her detailed
the varied meaning of each sentence in each of the original text, the
English translation, and the French translation. Using the magick
of her ophidian notebooks, this data was coalesced, split, merged,
cleaned, interpolated, graphed, remixed, fed into various algorithms
and corpuses, but despite years of dedicated effort, this Librarian was
unable to produce any results of note.
</p>
<p>
Frustrated at her wasted efforts, she decided to adopt a more empirical
approach, but treat each work as a work of non-fiction. Upon reading
about a peculiar edition of the Encyclopædia Brittanica, she
traveled to the author’s hometown and with the goal of locating it,
searched every single
bookstore and library in the city over the course of a month (though
often trusting the shopkeepers’s claims of non-existence in order to
save time [the literary agency NIST asserts that semi-honest protocols are
generally acceptable]). Having once again failed in her mission (though by now,
accustomed to it, as one gets accustomed to a persistent nausea), she left
for another land, this time in search of a dark sycamore mentioned in
a poem that had apparently never been precisely located, her only clues being
written in the poem itself, and historical geological records, some of which
are inconveniently located only at a city library many miles away. I believe she
is currently still working on this project.
</p>
<p>
If the work is translated from the Hebrew, then the translator must have
chosen the letters M, C, and V from their rough Hebrew equivalents.
In Hebrew, the bilabial nasal [m] can correspond only to the letter Mem. We
take the letter C to refer to the voiceless alveolar sibilant [s] (the
translator would have used K for a hard C), which could correspond to
either Samekh or Shin. Since Shin also represents the voiceless fricative
[ʃ], we choose to translate C to Samekh. Finally, the voiced labiodental
fricative [v] corresponding to V is represented by the Hebrew letters Bet
and Vav. Again, we discard Bet since it also corresponds to the voiced
bilabial stop [b]. Every ambiguity is terrifying.
</p>
<p>
Now, we have the sequence <em>Mem-Samekh-Vav</em>. The Library reminds us
constantly, incessantly of the infinitude of the circle: the beginning is
identified with the end. Indeed, one who turns to any page in the MCV text
apart from the first or last pages would not know that the “first” letter
in the sequence is an M.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/mem-samekh-vav.jpg">
<p>
Under the Kabbalistic art of
<a href="https://en.wikipedia.org/wiki/Gematria">gematria</a>,
we can assign numbers to each letter; each word or phrase’s value is the
sum. If we do this for Mem-Samekh-Vav <em>taking the final variant of Mem
in view of the above discussion</em>, we end up with 600 + 60 + 6 = 666, which
some travelers have mentioned as beastly.
</p>
<p>
<em>The above analysis included with deference to the New Order of the Rosy Cross,
Berkeley Chapter, Research Division.</em>
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/rose.jpg">
<blockquote>
Another (very much consulted in this area) is a mere labyrinth of letters, but the next-to-last page says Oh time thy pyramids.
</blockquote>
<p>
I am currently (and predictably) unable to find any gematriac meaning for
Oh time thy pyramids.
</p>
<blockquote>
This much is already known: for every sensible line of straightforward statement, there are leagues of senseless cacophonies, verbal jumbles and incoherences. (I know of an uncouth region whose librarians repudiate the vain and superstitious custom of finding a meaning in books and equate it with that of finding a meaning in dreams or in the chaotic lines of one’s palm… They admit that the inventors of this writing imitated the twenty-five natural symbols, but maintain that this application is accidental and that the books signify nothing in themselves. This dictum, we shall see, is not entirely fallacious.)
</blockquote>
<p>
<em>Thus the totality of the Library, if it is total, is its own <a
href="https://en.wikipedia.org/wiki/Rice's_theorem">triviality</a>.
The Text’s value is in its portrayal
of the various ways Librarians past have dealt with the (irrefutable)
triviality of their world.</em>
</p>
<blockquote>
For a long time it was believed that these impenetrable books corresponded to past or remote languages. It is true that the most ancient men, the first librarians, used a language quite different from the one we now speak; it is true that a few miles to the right the tongue is dialectical and that ninety floors farther up, it is incomprehensible.
</blockquote>
<p>
Note the word <em>first</em>, implying a beginning to mankind in the
Library. <em>Though the Library might be infinite in time and space, we
are not.</em> And thus, the language of the library, previously described as
“natural” is independent of Librarians.
</p>
<p>
How did the Author know of the first librarians? The presumed scarcity of
men compared to the Library’s vastness seems to preclude an entirely spoken
tradition. Possibly various histories were written, encrypted in the magic
of Trithemius, tucked away at the back of a bookshelf or a sleeping
closet (Weary Librarian! Where did you find this Work? And where shall
you return it?).
</p>
<p>
Finally, note that there is a reference to the Librarians of a hexagon
<em>a few miles to the right</em>. This has two important implications.
First, that there actually do exist hexagons to the right. Second, that
they are somehow <em>accessible</em> from the Author’s hexagon, since
the dialectical information needed to travel in some manner.
</p>
<p>
The accessibility condition refutes the theory of the infinite and finite,
hollow 3-spheres, and the existence condition refutes the theory of the
finite circle as previously presented.
</p>
<p>
But we can still save the last theory: instead of tracking a simple circle,
imagine the hexagons spiraling around the path, with their gravity fields
again oriented to their normal. Elderly travelers sometime mention, as children,
finding hidden toys in closets. One of these, a thin compressable metal
helix, can be shaped into a circle to mimic this proposed form of the Library.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/double-spiral.jpg">
<p>
This cheekily doubly satisfies the
requirements for a “spiral stairway,” and if we relax the word “right”
to account for imperceptible elevation change, the size of the Library
certainly admits for going several miles in what seems like one direction,
“right.” A tightly packed spiral will maintain the overall Structure to be
2-spherical.
</p>
<blockquote>
All this, I repeat, is true, but four hundred and ten pages of inalterable MCV’s cannot correspond to any language, no matter how dialectical or rudimentary it may be. Some insinuated that each letter could influence the following one and that the value of MCV in the third line of page 71 was not the one the same series may have in another position on another page, but this vague thesis did not prevail. Others thought of cryptographs; generally, this conjecture has been accepted, though not in the sense in which it was formulated by its originators.
</blockquote>
<p>
The duplication of text has power in discourse. A one-time-pad key might
reveal a meaning in the MCVs, but could just as easily reveal <em>any</em>
meaning. But just because there is not as much information as expected in a
large book, does not mean there is no information at all: see the above
gematriac analysis. There is rhetorical value in repetition.
</p>
<p>
Truths need only be conceived once: the rest is explication. Shall we say
the <a
href="https://upload.wikimedia.org/wikipedia/en/c/c4/Dogmatix.png">dogmatic</a>
<a href="https://stallman.org/">ones</a> are any less correct or incorrect
due to their decades of tautology?
</p>
<blockquote>
Five hundred years ago, the chief of an upper hexagon<sup>[2]</sup> came upon a book as confusing as the others, but which had nearly two pages of homogeneous lines. He showed his find to a wandering decoder who told him the lines were written in Portuguese; others said they were Yiddish. Within a century, the language was established: a Samoyedic Lithuanian dialect of Guarani, with classical Arabian inflections. The content was also deciphered: some notions of combinative analysis, illustrated with examples of variations with unlimited repetition.
<br>
…
<br>
[2] Before, there was a man for every three hexagons. Suicide and pulmonary diseases have destroyed that proportion. A memory of unspeakable melancholy: at times I have traveled for many nights through corridors and along polished stairways without finding a single librarian.
</blockquote>
<p>
Since there is no absolute notion of “upper” in our model, we rely on
our model of regional bibliogeography to parse this term.
</p>
<p>
We again encounter the theme of unlimited repetition, a self-similar trait
in the Library and its books: the mentioned book is possibly the Text
itself, save for the linguistic classification.
</p>
<p>
It is still unclear how these Librarians know about languages like Yiddish
and Portuguese. Where did the first Librarians come from? Where are they now?
</p>
<blockquote>
These examples made it possible for a librarian of genius to discover the fundamental law of the Library. This thinker observed that all the books, no matter how diverse they might be, are made up of the same elements: the space, the period, the comma, the twenty-two letters of the alphabet. He also alleged a fact which travelers have confirmed: In the vast Library there are no two identical books.
</blockquote>
<p>
These facts seem trivially impossible to verify, other than by that Evil of
inductive scientism. But the Author seems confident in the Genius Librarian’s
correctness, so we will accept these facts as true.
</p>
<p>
The lack of duplicate books immediately, and redundantly,
implies the fall of any infinite-gallery theory by the pigeonhole principle.
</p>
<blockquote>
From these two incontrovertible premises he deduced that the Library is total and that its shelves register all the possible combinations of the twenty-odd orthographical symbols (a number which, though extremely vast, is not infinite):
</blockquote>
<p>
But those premises don’t quite imply totality (an empty Library would
vacuously satisfy the premises, for example). Does the Author possess
additional secret information, or just arriving at this conclusion for its
aesthetic qualities?
</p>
<p>
We calculate the number of books as
25<sup>410·40·80</sup> = 25<sup>1312000</sup>. For comparison, there
are <a
href="https://www.universetoday.com/36302/atoms-in-the-universe/">reported</a>
to be 10<sup>80</sup> atoms in the mystical, possibly mythical locale known to its
denizens as <em>The Universe</em>. These <em>universers</em> could not hope
to recreate even a fraction of Our Library—the fraudulent sorcerer
Bu Fu references this as the cause of the universers’ shared depression.
</p>
<blockquote>
Everything: the minutely detailed history of the future, the archangels’ autobiographies, the faithful catalogues of the Library, thousands and thousands of false catalogues, the demonstration of the fallacy of those catalogues, the demonstration of the fallacy of the true catalogue,
</blockquote>
<p>
By the law of the excluded middle, we conclude that the truth is irrelevant
to the <em>demonstration</em>, which is inherently dependent and
inseparable from a human Librarian. The Text, which demonstrates its own
existence, at once undermines itself and sings its own elegy.
</p>
<p>
What is the nature of a Catalogue? If it is purely a description of the set
of books within the Library, this is trivial: even the Text suffices.
</p>
<p>
If it is a purely Turing-decidable <em>syntactical</em> classification,
e.g., the locations of all books that consist of only a single
repeatedletter, this is also not very interesting, since such satisfying
books could be enumerated by hand due to the finiteness of each book.
</p>
<p>
The interesting case is if the Catalogue is a <em>semantic</em>
classification. Certainly it must be written in the language of the
reading Librarian, plus it must make certain empirical claims about
the books it references: <em>The book immediately to the right
of this Catalogue is the True History of This Catalogue</em>, for example.
If it were not for the fact that books are of bounded length, an additional
interesting property for Catalogues would be those books that enumerate
other books that encode programs with undecidable properties.
</p>
<p>
Finally, it is clear that a catalogue in a single book cannot address
arbitrary books in the Library, for the totality of the Library implies
that the shortest address for an arbitrary book is the entire length of a
book, leaving no room for annotation. Histories may be compressed, but not
those texts generated by maximum entropy. So we must understand the Library not as
a collection of books, but as an inter-referencing interlaced interplaying
network. Our minutely detailed history, for example, is a catalogue
pointing to several other texts to continue its duty.
</p>
<blockquote>
the Gnostic gospel of Basilides, the commentary on that gospel, the commentary on the commentary on that gospel, the true story of your death, the translation of every book in all languages, the interpolations of every book in all books, the treatise that Bede could have written (and did not) about the mythology of the Saxons, the lost works of Tacitus.
</blockquote>
<p>
This interpolation condition again necessitates the hypertextified nature
of the library.
</p>
<blockquote>
When it was proclaimed that the Library contained all books, the first impression was one of extravagant happiness. All men felt themselves to be the masters of an intact and secret treasure.
</blockquote>
<p>
The Library’s totality belies any hope of secrecy—yet ensures its
immortal intactness surviving any physical destruction. Triviality is synonymous with eternity.
</p>
<blockquote>
There was no personal or world problem whose eloquent solution did not exist in some hexagon. The universe was justified, the universe suddenly usurped the unlimited dimensions of hope. At that time a great deal was said about the Vindications: books of apology and prophecy which vindicated for all time the acts of every man in the universe and retained prodigious arcana for his future. Thousands of the greedy abandoned their sweet native hexagons and rushed up the stairways, urged on by the vain intention of finding their Vindication. These pilgrims disputed in the narrow corridors, proferred dark curses, strangled each other on the divine stairways, flung the deceptive books into the air shafts, met their death cast down in a similar fashion by the inhabitants of remote regions.
</blockquote>
<p>
Did the violent search for these Vindications justify its own necessity?
The Author presupposes that a Librarian can Recognize his own Vindication.
</p>
<p>
Weary Librarian! Could You recognize your own Vindication, if presented?
Are you latently aware of your sinful impurities? Could You recognize your
future—your whited sepulchres—the last book you’ll ever read
(those who elect to join Pietro della Vigna, at least, will know their
empty future—is this our only hope?)?
Within the comfort of my anonymity, I confess that I cannot. I don’t
imagine I am alone in this upsetting conclusion, but—
</p>
<blockquote>
Others went mad…The Vindications exist (I have seen two which refer to persons of the future, to persons who are perhaps not imaginary) but the searchers did not remember that the possibility of a man’s finding his Vindication, or some treacherous variation thereof, can be computed as zero.
</blockquote>
<p>
The sinless need no Vindication, and so all texts are appropriate. But they
<a href="https://www.youtube.com/watch?v=8HON4AswPVk">must be imaginary</a>
if the probability of <em>any</em> man to find his
Vindication is zero…<em>quem patronum rogaturus</em>? If we are
cursed with original sin, what caused our fall?
</p>
<blockquote>
At that time it was also hoped that a clarification of humanity’s basic mysteries—the origin of the Library and of time—might be found. It is verisimilar that these grave mysteries could be explained in words: if the language of philosophers is not sufficient, the multiform Library will have produced the unprecedented language required, with its vocabularies and grammars. For four centuries now men have exhausted the hexagons… There are official searchers, inquisitors.
</blockquote>
<p>
These Inquisitors are not like us, Weary Librarian. Or are You a
Recognizer too?…a Recognizer not just of the Self, but of the
Library (they may not be distinct)? The very possibility of a Recognizer
implies that the Library’s existence is perfectly logical, its truths known
<em>a priori</em>, its histories a simple application of the divine state function.
Then the Library does not exist in an empirical world: <em>it could not be
otherwise</em>.
</p>
<p>
So it deeply troubles me, Weary Librarian, to read of the Library’s multiformity. In my
youth I was comforted by its promises of eternity, of perfection, of solitude in its
truth and truth in its solitude…I now realize that there exists a distinct
Library for each of its languages. Do they overlap? Can they overlap?
</p>
<blockquote>
I have seen them in the performance of their function: they always arrive extremely tired from their journeys; they speak of a broken stairway which almost killed them; they talk with the librarian of galleries and stairs; sometimes they pick up the nearest volume and leaf through it, looking for infamous words. Obviously, no one expects to discover anything.
</blockquote>
<p>
Weary Librarian, are you too tired of your journey? Have you
trip-tumble-fumbled down a “broken stairway” of our inimitable, perfect
Library? <em>Tomber à mort!</em> Which words do you seek? Oh time thy pyramids do
not persist…
</p>
<blockquote>
As was natural, this inordinate hope was followed by an excessive depression.
</blockquote>
<p>
Rather, it seems to be the case that most Librarians experience both
illnesses at the same time, balancing metaphorically upon the sixth wall’s railing.
</p>
<blockquote>
The certitude that some shelf in some hexagon held precious books and that these precious books were inaccessible, seemed almost intolerable. A blasphemous sect suggested that the searches should cease and that all men should juggle letters and symbols until they constructed, by an improbable gift of chance, these canonical books.
</blockquote>
<p>
The origin of the Library dictates the boundary conditions of its elements:
there is no true randomness. This sect is blasphemous for its teachings are
potentially incomplete, cruelly opposed to Our Library’s totality.
</p>
<blockquote>
The authorities were obliged to issue severe orders. The sect disappeared, but in my childhood I have seen old men who, for long periods of time, would hide in the latrines with some metal disks in a forbidden dice cup and feebly mimic the divine disorder.
</blockquote>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/dice.jpg">
<p>
If you still hold hope of Recognizing, perhaps you would like
to try your luck? This
disorder is due to Librarian incomprehension of the organization
of the texts in the Library. Then, it cannot be divine in origin: the
appropriate language defines alphabetical order to be the order of
the Library. There is no divine order or disorder, but merely divine
existence.
</p>
<blockquote>
Others, inversely, believed that it was fundamental to eliminate useless works. They invaded the hexagons, showed credentials which were not always false, leafed through a volume with displeasure and condemned whole shelves: their hygienic, ascetic furor caused the senseless perdition of millions of books.
</blockquote>
<p>
The Author has neglected to clarify: is there any commonality between the
books near each other? Could not the difference of a single letter negate a
book’s thesis? Having established the impossibility for verifiable
demonstration (much less <a
href="https://en.wikipedia.org/wiki/PCP_theorem">efficient
demonstration</a>), we note that these ascetics do not seek the truth, but
rather <em>relevance</em> (to what, I cannot say; perhaps it is unimportant?).
</p>
<blockquote>
Their name is execrated, but those who deplore the “treasures” destroyed by this frenzy neglect two notable facts. One: the Library is so enormous that any reduction of human origin is infinitesimal. The other: every copy is unique, irreplaceable, but (since the Library is total) there are always several hundred thousand imperfect facsimiles: works which differ only in a letter or a comma. Counter to general opinion, I venture to suppose that the consequences of the Purifiers’ depredations have been exaggerated by the horror these fanatics produced.
</blockquote>
<p>
The Author is correct except in the case of cryptographs of the
Existentially Unforgeable order: a single alteration renders
these varieties unmendably rent.
</p>
<blockquote>
They were urged on by the delirium of trying to reach the books in the Crimson Hexagon: books whose format is smaller than usual, all-powerful, illustrated and magical.
</blockquote>
<p>
Clearly delirious; they would not need to destroy works in normal format
for their mission. What does illustration do to a text? Do you sense
magic here?
</p>
<p>
I confess, Weary Librarian, that I too have been consumed in this
delirium and am currently searching for this Crimson Hexagon, though
I take a more methodical approach. The existence of this Work is due
to the predicted fatality inherent to such magics. My process involves
projecting a mathematical model of our Library onto the surface of an
oblate spheroid divided into latitudes and longitudes, anchored at
various landmarks with the aim of finding the focal points. I am
pleased to report that I believe I have located the correct
projected-hexagon, but will unfortunately…
</p>
<p>
It is also curious that the Author described the MCV text as “inalterable”
when clearly books can be destroyed. Perhaps inalterable only in our
frail memories.
</p>
<blockquote>
We also know of another superstition of that time: that of the Man of the Book. On some shelf in some hexagon (men reasoned) there must exist a book which is the formula and perfect compendium of all the rest: some librarian has gone through it and he is analogous to a god.
</blockquote>
<p>
In view of past discussion, The Book is not simply a purely syntactic
characterization of the other books themselves (which would be self-evident
and uninteresting), but a commentary on those books’ <em>semantic</em> and
<em>extraliterary</em> significance with respect to the Librarian.
</p>
<p>
To focus on the Man instead of the Book, the searchers must have asked,
<em>What language does he speak? Do we speak His language?</em> Do you?
And does he know of his divinity?
</p>
<blockquote>
In the language of this zone vestiges of this remote functionary’s cult still persist. Many wandered in search of Him. For a century they have exhausted in vain the most varied areas. How could one locate the venerated and secret hexagon which housed Him?
</blockquote>
<p>
The cyclical nature of our model leaves open the possibility for the Man to
eternally remain one hexagon above His searchers. <a
href="https://aphyr.com/posts/341-hexing-the-technical-interview">Are
they aware</a> of the Library’s cyclicity?
</p>
<blockquote>
Someone proposed a regressive method: To locate book A, consult first book B which indicates A’s position; to locate book B, consult first a book C, and so on to infinity… In adventures such as these, I have squandered and wasted my years.
</blockquote>
<p>
It is possible for a book to point to yet another arbitrary
book, and so on, but there is no more additional information to be gained
through this process since each book is entirely taken up by the next
book’s “address.”
</p>
<p>
The solution is to reserve the first letter in a book for data, using the
rest to address the next book. If we use an absolute numbering system and
simply omit the most significant twenty-five-it for the address, we divide the
Library into exactly 25 mutually inaccessible sections.
</p>
<p>
An alternative is an entirely relative mode: commentary followed by
instructions like “To continue your journey, ascend fifty-four hexagons and
choose the sixth book on the top shelf to left of the lamp of the
bookshelves.” Although it is still impossible to immediately address
arbitrary books, it is possible to reach an arbitrary book through a
sequence of relative addresses.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/graph.jpg">
<blockquote>
It does not seem unlikely to me that there is a total book on some shelf of the universe;<sup>[3]</sup> I pray to the unknown gods that a man—just one, even though it were thousands of years ago!—may have examined and read it.
<br>
…
<br>
[3] I repeat: it suffices that a book be possible for it to exist. Only the impossible is excluded. For example: no book can be a ladder, although no doubt there are books which discuss and negate and demonstrate this possibility and others whose structure corresponds to that of a ladder.
</blockquote>
<p>
In fact, it seems <em>certain</em> that such a book exists…but it may
just as well be entirely blank. Ladder!
</p>
<blockquote>
If honor and wisdom and happiness are not for me, let them be for others.
</blockquote>
<p>
Was the Author mistaken to chain his happiness to the ever-disappointing
external world? Such is not the way of the <em>Meditations</em>, yet
our Author is content in his resignation.
</p>
<blockquote>
Let heaven exist, though my place be in hell. Let me be outraged and annihilated, but for one instant, in one being, let Your enormous Library be justified. The impious maintain that nonsense is normal in the Library and that the reasonable (and even humble and pure coherence) is an almost miraculous exception. They speak (I know) of the “feverish Library whose chance volumes are constantly in danger of changing into others and affirm, negate and confuse everything like a delirious divinity.”
</blockquote>
<p>
If inalterable volumes are changing into others, it is rather the Readers
themselves who are changing. We are no strangers to delusion: it is that
absurdity that separates us from the gods.
</p>
<blockquote>
These words, which not only denounce the disorder but exemplify it as well, notoriously prove their authors’ abominable taste and desperate ignorance. In truth, the Library includes all verbal structures, all variations permitted by the twenty-five orthographical symbols, but not a single example of absolute nonsense. It is useless to observe that the best volume of the many hexagons under my administration is entitled The Combed Thunderclap and another The Plaster Cramp and another Axaxaxas mlö. These phrases, at first glance incoherent, can no doubt be justified in a cryptographical or allegorical manner; such a justification is verbal and, ex hypothesi, already figures in the Library.
</blockquote>
<p>
The dying Author, in the mode
of metametasatire, acknowledges his own desperateness to find some
underlying, inherent Reason underneath the Library. What he seeks
is to understand, to understand a book <em>contextualized
in another Librarian’s language</em>. His search for meaning
is one for companionship, understanding.
</p>
<p>
Unfortunately we are not able to find a Romanization of Hebrew including
the ö character. Weary Librarian, can you fare better than I? What is Your
language? <em>What do you seek?</em>
</p>
<blockquote>
I cannot combine some characters
<br><br>
dhcmrlchtdj
<br><br>
which the divine Library has not foreseen and which in one of its secret tongues do not contain a terrible meaning.
</blockquote>
<p>
Let me help you. My tender, fearful secret is B4z4AL00Ru2.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/fingerprint.jpg">
<blockquote>
No one can articulate a syllable which is not filled with tenderness and fear, which is not, in one of these languages, the powerful name of a god. To speak is to fall into tautology. This wordy and useless epistle already exists in one of the thirty volumes of the five shelves of one of the innumerable hexagons—and its refutation as well.
</blockquote>
<p>
Is it thirty or thirty-two? Has the ascetic destruction grown so great? Is our Author losing his memories?
</p>
<blockquote>
(An n number of possible languages use the same vocabulary; in some of them, the symbol library allows the correct definition ubiquitous and lasting system of hexagonal galleries, but library is bread or pyramid or anything else, and these seven words which define it have another value. You who read me, are You sure of understanding my language?) The methodical task of writing distracts me from the present state of men.
</blockquote>
<p>
Writing also appears to distract the Author from the present state of himself.
</p>
<p>
A common universer myth tells us of organic greens, pistils protruded, the
calls of grotesque non-Librarian creatures…yet there too, they lament
<em>What man has made of man</em>, though most find such discussion histrionic.
</p>
<blockquote>
The certitude that everything has been written negates us or turns us into phantoms. I know of districts in which the young men prostrate themselves before books and kiss their pages in a barbarous manner, but they do not know how to decipher a single letter. Epidemics, heretical conflicts, peregrinations which inevitably degenerate into banditry, have decimated the population. I believe I have mentioned suicides, more and more frequent with the years.
</blockquote>
<p>
In fact, it is those destructive ascetics who bestow significance to the
Library, infinitesimal as though it may be. Do you recall the story of the
First Destroyer—He who fatally, against caution, overthrew a single
book into the cyclical abyss? In that instant the Library’s triviality was
obliterated. Through discrimination, it finally gained meaning.
</p>
<p>
<em>The Library’s destruction is its sole justification.</em>
</p>
<p>
There are those who claim that this Law is also true for Librarians.
</p>
<blockquote>
Perhaps my old age and fearfulness deceive me, but I suspect that the human species—the unique species—is about to be extinguished, but the Library will endure: illuminated, solitary, infinite, perfectly motionless, equipped with precious volumes, useless, incorruptible, secret.
</blockquote>
<p>
The Library exists in duality! Its solitude promises its secrecy; its
infiniteness guarantees its uselessness.
</p>
<img src="/the-first-faithful-topology-of-the-library-of-babel/img/modalduality.jpg">
<p>
Our Author once mentioned that no lack of pious hands would rush to throw him
off the railings in the event of his death, yet he goes miles without
encountering a fellow Librarian. Does he describe his own piety? If you too
wish to seek the endless descent, your only hope of glimpsing the Distant
Regions, you must follow him.
</p>
<blockquote>
I have just written the word “infinite.” I have not interpolated this adjective out of rhetorical habit; I say that it is not illogical to think that the world is infinite. Those who judge it to be limited postulate that in remote places the corridors and stairways and hexagons can, inconceivably, come to an end—which is absurd. Those who imagine it to be without limit forget that the possible number of books does have such a limit. I venture to suggest this solution to the ancient problem: The Library is unlimited and cyclical.
</blockquote>
<p>
We are lucky enough to not need to revise our spiraling circular topology. But
I have just realized we have neglected our first condition, that
the number of galleries is <em>indefinite</em>, which seems to be at odds
with the unique totality of the Library. Continuing the theme of
qualifying conditions in view of human limits, we could say that the number
is indefinite since, even if the Library was total, no Librarian could
verify this fact within a lifetime, and there is no short proof (<a
href="https://cseweb.ucsd.edu/~mihir/papers/ug.pdf">c.f. 3.2</a>)
of a section’s uniqueness to allow a group of cooperative yet distrustful
Librarians to verify the fact either.
</p>
<p>
An alternate solution: consider a gallery to be a function of the Librarian
within it (human indefiniteness is contagious).
</p>
<blockquote>
If an eternal traveler were to cross it in any direction, after centuries he would see that the same volumes were repeated in the same disorder (which, thus repeated, would be an order: the Order). My solitude is gladdened by this elegant hope.<sup>[4]</sup>
<br>
…
<br>
[4] Letizia Álvarez de Toledo has observed that this vast Library is useless: rigorously speaking, a single volume would be sufficient, a volume of ordinary format, printed in nine or ten point type, containing an infinite number if infinitely thin leaves. (In the early seventeenth century, Cavalieri said that all solid bodies are the superimposition of an infinite number of planes.) The handling of this silky vade mecum would not be convenient: each apparent page would unfold into other analogous ones; the inconceivable middle page would have no reverse.
</blockquote>
<p>
A unique Order <em>up to rotation</em>. The Author is gladdened by the hope
of an Eternal Librarian, a contradiction initself: so as the Library is
eternal, We are not. And so the Author’s solitude is without remedy.
</p>
<p>
The middle page is inconceivable because it does not exist: there are an
even number of pages. The abrupt switch from a discrete Library to
infinitesimals is suspicious. I suspect our Author has finally succumbed to
his delusions, ridding him of the careful thought that we have so painfully
illustrated.
</p>
<hr>
<p>
We have given the first construction of the Library consistent with the
text. I had hoped that it would relieve a burden, offer some subtle
insight, even provide purpose. None of this has happened yet. Perhaps it is
not correct…
</p>
<hr>
<p>
This work is dedicated to the intensely Californian sunlight, and our own Library.
</p>
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