Views of a Tesseract

…the breadth, and length, and depth, and height…

– Ephesians 3:18

And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length and the breadth and the height of it are equal. And he measured the wall thereof, an hundred and forty and four cubits, according to the measure of a man, that is, of the angel. And the building of the wall of it was of jasper: and the city was pure gold, like unto clear glass.

– Revelation 21:16-18

Un homme qui y consacrerait son existence arriverait peut-être à se peindre la quatrième dimension. [A man who devoted his life to it could perhaps succeed in picturing to himself the fourth dimension.]

– Henri Poincaré

This spring I have scaled the awful, sanity-threatening Unknown Kadaths of the fourth dimension in a desperate, god-provoking quest to visualize the six regular polytopes.

What is a polytope, you ask? The word polytope is the general term in the sequence whose first terms are the line segment (dimension one), the polygon (dimension two), and the polyhedron (dimension three). A regular polytope is a polytope which is "completely symmetric."

Theatetus, a contemporary of Plato, proved that there are exactly five regular polyhedra: the tetrahedron, the octahedron, the icosahedron, the cube, and the dodecahedron. They are called the Platonic solids because Plato identified each of the first four with a material element (fire, air, water, earth), and the fifth with "the delineation of the universe" [Timaeus]. Their construction is the crowning achievement of Euclid's Elements, written in about 300 BC. But the world had to wait more than two thousand years for the "discovery" of their analogues in the fourth dimension.

Fourth-dimensional geometry, thought it might seem mysterious to the uninitiated, is defined axiomatically, just like Euclid's three-dimensional geometry, and has an intuitive basis. It was first described by Ludwig Schläfli, a Swiss mathematician, in the 1850s, but his work remained relatively inaccessible and unknown. Then, between 1880 and 1900, the geometry of higher dimensions was rediscovered in nine different publications written independently of each other. The time, it seems, was ripe. It was the dawn of a new era.

Not that era. [source]

This phenomenon of numerous researchers all suddenly reaching the same conclusion at the same time, though surprising when it happens, isn't all that uncommon in the history of math, science, and technology. What's striking is the way four-dimensional geometry fired the popular imagination, which seems in some cases to have outstripped academia.

Last year I blogged about Flatland: A Romance of Many Dimensions, a strange geometrical fantasy written by the English schoolmaster Edwin A. Abbott (1838-1926) and published in 1884. In it, Abbott gives what must be the first popular description of the tesseract, or four-dimensional hypercube, by way of analogy.

In One Dimension, did not a moving Point produce a Line with TWO terminal points?

In Two Dimensions, did not a moving Line produce a Square with FOUR terminal points?

In Three Dimensions, did not a moving Square produce – did not this eye of mine behold it – that blessed Being, a Cube, with EIGHT terminal points?

And in Four Dimensions shall not a moving Cube – alas, for Analogy, and alas for the Progress of Truth, if it be not so – shall not, I say, the motion of a divine Cube result in a still more divine Organization with SIXTEEN terminal points?

Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this – if I might quote my Lord's own words – "strictly according to Analogy"?

Again, was I not taught by my Lord that as in a Line there are TWO bounding Points, and in a Square there are FOUR bounding Lines, so in a Cube there must be SIX bounding Squares? Behold once more the confirming Series, 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have 8 bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to Analogy"?

How much exactly did Abbott know of contemporary research? One imagines he must have encountered something, but I can't seem to find anything definite. A matter for further research, I suppose.

Inspired by Abbott, a high school teacher and amateur mathematician by the name of Charles Howard Hinton (1853-1907) wrote a number of "scientific romances" exploring higher dimensions. It was Hinton who coined the term tesseract, and his book A New Era of Thought, published in 1888, provides a detailed account of the hypercube's structure. It also offers a mystical interpretation of the fourth dimension, following to some extent in Abbott's footsteps, but with considerably greater gravity and self-importance.

We have been subject to a limitation of the most absurd character. Let us open our eyes and see the facts.

Now, it requires some training to open the eyes. For many years I worked at the subject without the slightest success. All was mere formalism. But by adopting the simplest means, and by a more thorough knowledge of space, the whole flashed clear.

Space shapes can only be symbolical of four-dimensional shapes; and if we do not deal with space shapes directly, but only treat them by symbols on the plane – as in analytical geometry – we are trying to get a perception of higher space through symbols of symbols, and the task is hopeless. But a direct study of space leads us to the knowledge of higher space. And with the knowledge of higher space there come into our ken boundless possibilities. All those things may be real, whereof saints and philosophers have dreamed.

Hinton was read by Jorge Luis Borges, and his book is mentioned "Tlön, Uqbar, Orbis Tertius."

Through his father, described by some as a religious crank, Hinton came to know the family of the late George Boole, the father of algebraic logic, whose untimely death had left his wife, Mary Everest* Boole, with their five daughters to raise. Mrs. Boole's interests ranged from mathematics to mysticism to politics; she wrote a number of pedagogical works, organized controversial discussion groups, and hobnobbed with the denizens of the fringes. Among these were the polygamy advocate James Hinton and his son Howard.

Howard married the eldest daughter, Mary Ellen Boole, in 1880, and they had four children together. A few later, he married a second woman under an assumed name, had two children with her, was convicted of bigamy, spent a few days in jail, lost his job, and moved to the United States with his (first) wife to become a university professor. He died unexpectedly in 1907, and Mary Ellen committed suicide the next year.

H. S. M. Coxeter's Regular Polytopes, published (in its second edition) in 1963, remains the main authority on its subject. I've entertained myself by constructing the various solids he describes in it.

More importantly for us, each chapter concludes with historical notes. There Coxeter discusses Alicia Boole Stott (1860-1940), another of George Boole's daughters, with whom he was personally acquainted in her later years. Curiously, though he mentions both Hinton and his book (in deprecatory terms), he says nothing about the family connection or about the fact that Stott assisted in finishing and publishing A New Era in Human Thought when Hinton left the country.

When Alice was about thirteen the five girls were reunited with their mother (whose books reveal her as one of the pioneers of modern pedagogy) in a poor, dark, dirty, and uncomfortable lodging in London. There was no possibility of education in the ordinary sense, but Mrs. Boole's friendship with James Hinton attracted to the house a continual stream of social crusaders and cranks. It was during those years that Hinton's son Howard brought a lot of small wooden cubes, and set the youngest three girls the task of memorizing the arbitrary list of Latin words by which he named them, and piling them into shapes. To Ethel, and possibly Lucy too, this was a meaningless bore; but it inspired Alice (at the age of about eighteen) to an extraordinarily intimate grasp of four-dimensional geometry. Howard Hinton wrote several books on higher space, including a considerable amount of mystical interpretation. His disciple did not care to follow him along these other lines of thought, but soon surpassed him in geometrical knowledge.

In 1890, she married an actuary and "led a life of drudgery" [Coxeter] as a wife and mother with a small income. But she continued to explore the fourth dimension as a kind of hobby, building cardboard models of three-dimensional "slices" of four-dimensional figures. Somehow her husband came across the work of the Dutch mathematician Pieter Hendrik Schoute, whose published diagrams mirrored her models. She contacted Schoute and the two began a long and fruitful collaboration. As Coxeter puts it,

Mrs. Stott's power of geometrical visualization supplemented Schoute's more orthodox methods, so they were an ideal team.

It was she who coined the term polytope.

Among other "enthusiasts" (as opposed to academicians) who contributed to four-dimensional geometry, Coxeter mentions Paul S. Donchian, an Armenian American.

His great-grandfather was a jeweller at the court of the Sultan of Turkey, and many of his other ancestors were oriental jewellers and handicraftsmen. He was born in Hartford, Connecticut, in 1895. His mathematical training ended with high school geometry and algebra, but he was always interested in scientific subjects. He inherited the rug business established by his father, and operated it for forty years. At about the age of thirty he suddenly began to experience a number of startling and challenging dreams of the previsionary type soon to be described by Dunne in 'An Experiment with Time'. In an attempt to solve the problems thus presented, he determined to make a thorough analysis of the geometry of hyper-space.

Donchian built delicate three-dimensional models of four-dimensional polytopes which were displayed at expositions in Chicago and Pittsburgh, several pictures of which appear in Coxeter's book.

I built a wire-solder model of the hypercube many years ago, using what I suppose are the same principles, though I didn't know it at the time. It remains in good shape, but it's in my parents' possession, and I don't have a picture of it handy.

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These days I'm working on a set of 3D printer files reproducing Stott's model of the 120-cell, a polytope composed of 120 dodecahedral cells. From her 1900 paper "On Certain Series of Sections  of the Regular Four-dimensional Hypersolids," I've created the virtual constructions from which I'll derive the vertex coordinates.

The following image represents a series of slices slices cut by hyperplanes parallel to a dodecahedral cell, starting with the cell itself (at the center of the image) and ending with the "equatorial" slice midway up the polytope (at the outside of the image). In my file the layers are numbered from VIII to XIV, in accord with the partial nets illustrated in her paper shown above.

And here is part of the "net" from which the 120-cell can be "folded." The "equatorial" layer of dodecahedra (not shown) fits in the interstices, with one for each edge of the dodecahedral cell forming the "base." A second set identical to the one shown then "caps" the 120-cell above the equator.

However, I find that I'm not the first to attempt reconstructing Stott's fascinating models. Well, I'll do the 600-cell as well, and that will be impressive. Here is my projection of the 600-cell to the plane.

I hope to recreate it in string art, the use of which in teaching children was pioneered by Mary Everest Boole.

Here are some of my printed polyhedra, which I built myself in Blender: we have a compound of five tetrahedra, a compound of five cubes, a compound of five tetrahedra (edges only), a great dodecahedron, and four rhombic dodecahedra, but no hypersolids yet. (Chessboard chosen advisedly: see below.)

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The geometry of the fourth dimension has made appearances in a number of imaginative works. Aside from Flatland, the earliest instance is probably The Time Machine by H. G. Wells, published in 1898. Unfortunately, he makes the rather common mistake of conflating temporal extension with a fourth spacial dimension.

"Well, I do not mind telling you I have been at work upon this geometry of Four Dimensions for some time. Some of my results are curious. For instance, here is a portrait of a man at eight years old, another at fifteen, another at seventeen, another at twenty-three, and so on. All these are evidently sections, as it were, Three-Dimensional representations of his Four-Dimensioned being, which is a fixed and unalterable thing.

"Scientific people," proceeded the Time Traveller, after the pause required for the proper assimilation of this, "know very well that Time is only a kind of Space. Here is a popular scientific diagram, a weather record. This line I trace with my finger shows the movement of the barometer. Yesterday it was so high, yesterday night it fell, then this morning it rose again, and so gently upward to here. Surely the mercury did not trace this line in any of the dimensions of Space generally recognized? But certainly it traced such a line, and that line, therefore, we must conclude was along the Time-Dimension."

H. P. Lovecraft gives a much better account of the fourth dimension in "The Dreams in the Witch House," published in 1933 and described by its Weird Tales tagline as "a story of mathematics, witchcraft and Walpurgis Night, in which the horror creeps and grows." Whatever you think of Lovecraft as a writer, one thing you can say is this: he knows when to be explicit and when to be vague and ominous. It serves him well here.

Toward the end of March he began to pick up in his mathematics, though the other studies bothered him increasingly. He was getting an intuitive knack for solving Riemannian equations, and astonished Professor Upham by his comprehension of fourth-dimensional and other problems which had floored all the rest of the class. One afternoon there was a discussion of possible freakish curvatures in space, and of theoretical points of approach or even contact between our part of the cosmos and various other regions as distant as the farthest stars or the transgalactic gulfs themselves – or even as fabulously remote as the tentatively conceivable cosmic units beyond the whole Einsteinian space-time continuum. Gilman's handling of this theme filled everyone with admiration, even though some of his hypothetical illustrations caused an increase in the always plentiful gossip about his nervous and solitary eccentricity. What made the students shake their heads was his sober theory that a man might – given mathematical knowledge admittedly beyond all likelihood of human acquirement – step deliberately from the earth to any other celestial body which might lie at one of an infinity of specific points in the cosmic pattern.

Reminds me of my own college days! Ha ha, actually, it doesn't. I spent an entire year of my life working a problem of 10- and 26-dimensional geometry, got stuck on a minus sign for most of its duration, and finally had to give up and start a new problem. My dissertation advisor may very well have wondered about my nervous and solitary eccentricity, and my fellow students may have shaken their heads at my theories, but not for the reasons Gilman found himself the source of such disturbance…

[source]

The net of a tesseract figures in Robert A. Heinlein's 1941 story "And He Built a Crooked House," in which an architect builds a house in the shape of the three-dimensional "net" of a tesseract (from which the polytope can be "folded" much as a cube is folded from a two-dimensional cruciform net); an earthquake causes it to collapse into an actual tesseract from which other worlds can be reached. The story was anthologized in Fantasia Mathematica in 1958.

Madeleine L'Engle's A Wrinkle in Time, which contains the most well-known tesseract (and verbs the word as tesser), was published five years later, in 1963. I wonder if L'Engle got her idea (which is rather garbled) from the Heinlein story?

Meg sighed. "Just explain it to me."

"Okay," Charles said. "What is the first dimension?"

"Well, a line."

"Okay.  And the second dimension?"

"Well, you'd square the line. A flat square would be in the second dimension."

"And the third?"

"Well, you'd square the second dimension. Then the square wouldn't be flat any more. It would have a bottom, and sides, and a top."

"And the fourth?"

"Well, I guess if you want to put it into mathematical terms, you'd square the square. But you can't take a pencil and draw it the way you can the first three. I know it's got something to do with Einstein and time. I guess maybe you could call the fourth dimension Time."

"That's right," Charles said. "Good girl. Okay, then, for the fifth dimension you'd square the fourth, wouldn't you?"

"I guess so."

"Well the fifth dimension's a tesseract. You add that to the other four dimensions and you can travel through space without having to go the long way around. In other words, to put it into Euclid, or old-fashioned plane geometry, a straight line is not the shortest distance between two points."

Terrible! Just imagine an inhabitant of Flatland speaking like that: "The third dimension is Time. The fourth dimension's a cube. You add that to the three dimensions and you can travel through the plane without having to go the long way around. In other words, to put it into linear terms, a straight line is not the shortest distance between two points." Ugh! A novel is not a math textbook, it is true, but, for me, it's harder to overlook such nonsense than scientific speculation. There's nothing like 1 + 1 = 3 to break the suspension of disbelief. (Not that it's a bad book mind you.)

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[source]

Higher-dimensional geometry appears in art as well. Cubism is an oft-cited example, but geometry figures more directly in Salvador Dali's Crucifixion (Corpus Hypercubus), which depicts Christ crucified on the net of a tesseract. Just as the net permits us to approach what lies beyond our comprehension, in God, so does the Incarnation provides a "picture" of God comprehensible to humankind. That's how the picture usually seems to be interpreted.

Well, somehow this has turned into one of those posts of mine in which I draw connections between whatever unrelated topics I happen to be interested in. Here it's higher-dimensional geometry, science fiction and fantasy, the early twentieth century, and art. I do still want to describe the catalog of regular polytopes, but that will have to wait for a subsequent post.

* The mountain was named after her uncle. Quite a dynasty!

El Libro de Arena

Let's discuss Jorge Luis Borges' short story "The Book of Sand" ("El Libro de Arena"). First, a synopsis. (You can also find an English translation here.)

The story opens with a few sentences about points, lines, and planes. More on this later. A man (the narrator — Borges?) receives a visit from a traveling Bible salesman. He expresses disinterest in buying a Bible, as he owns a Wycliff Bible, a copy of Cypriano de Valera's translation, Luther's translation, and the Latin Vulgate. The salesman — a man of indistinct features, dressed in gray, with a gray valise — then shows him the Book of Sand (so-named because sand has neither beginning nor end).

The narrator opens it. Each page is numbered with an Arabic numeral; the left-hand page might be numbered, say, 40514, or some eight-digit number, the left, 999, or a number raised to the ninth power. There are crude illustrations as well. "Study the page well," the salesman says, "for you'll never see it again." The narrator notes the page number, closes the book, then tries to find it again. He is unable to. He also discovers that he can't reach the beginning of the book. No matter how close he tries to open it to the cover, there are always pages between his thumb and the board.

In the end he purchases the book in exchange for his retirement fund and his Wycliff Bible. He hides it behind his volumes of the Arabian Nights. As he investigates the book over the coming months he concludes it to be monstrous. Rather than burning it — he's afraid that the smoke from an infinite book would suffocate the whole earth — he hides it on a shelf at the National Library.

A strange, enigmatic tale in its basic outline. Possibly I've left out some of the most significant details, but I've included those that strike me. Though a writer by night and a painter on Sundays, I'm a mathematician by day, and it's the mathematical aspects that I wish to speak about.

The opening seems to me to be the key to the whole thing, for in many ways the Book of Sand might be likened to a line segment of finite length, e.g., the segment from zero to one on the number line. I mean the segment between these two points, excluding zero and one themselves. For such a segment has neither beginning nor end. No matter how close to zero you choose a point to be, there will always be infinitely many numbers between zero and your point. Notable, too, is the fact that the narrator can never again find a page he's visited. This is true of any infinite set. Choose an element at random, then choose a second. The probability that the choices will be the same is precisely nil.

So, the book resembles the line segment to some extent. But there are many infinite sets within the line segment. We could, for instance, take all the points that correspond to fractions. For a number between zero and one to be a fraction we have to be able to write it as a/b, where a and b and whole numbers and a is smaller than b. Though infinite, it is possible to denumerate this set by taking them in the order

1/2    1/3    2/3    1/4    3/4    1/5    2/5    3/5    4/5    1/6    5/6    1/7

so that 1/2 is the first, 1/3 the second, 2/3 the third, and so on. We're skipping the fractions that can be reduced and thus have already appeared in the list. Because this set of rational numbers (so-called because they are ratios of whole numbers) is denumerable (can be numbered off without leaving any out), it seems to me to resemble the pages of the book. The page numbers would correspond to the labels affixed to the rational numbers. For instance, the page that occurs exactly two-fifths of the way through the book would be labeled as page 7, because 2/5 appears seventh in our list. Going against this interpretation is the fact that facing pages have different numbers, and that the narrator can turn the pages one at a time; for between each pair of rational numbers are infinitely many rational numbers. It isn't possible to find two that are right next to each other, so to speak. For the book to be exactly like the rationals, recto and verso would need to bear the same number (and thus be identified), and the pages would have to stick when the narrator turned them, much as the "first" pages cling to the front board, so that he can never quite turn a single one.

Another possibility is that the pages correspond to the points in the entire continuum from zero to one; but this seems to be ruled out by the pagination, because the continuum, as Georg Cantor showed, is not denumerable. It is "more infinite" than the infinite set of rational numbers. (Incidentally, Cantor's proof made use of the decimal system, which originated in India; the Bible salesman had bought the Book of Sand from an Indian untouchable.) What we noted about page-turning above also goes against this interpretation.

There are of course many other infinite sets on this line segment that could be considered.

It's interesting, by the bye, that the book is called the Book of Sand; for the number of grains of sand is in fact finite. Though counterintuitive to some, the fact was demonstrated by Archimedes in a little book called The Sand-Reckoner, to which I feel almost certain Borges must have referred in at least one of his stories.

In this book, Archimedes obtains an estimate for the size of the material universe, which was very, very large (it is a myth that the ancients believed in a small cosmos and a flat earth), and, using another estimate for the density of packed sand in terms of grains per unit volume, computes an upper bound for the number of grains that could possibly be contained in the cosmos, were it packed solid with sand. He arrives at the number 10⁶³, which is very large, but still, of course, finite. The square of this number would be larger than a googol, if that puts it in perspective for you.

The fact that the narrator fears polluting the whole earth with the smoke of his book makes me think of a certain nineteenth-century mathematical controversy, of which Cantor, mentioned above, was at the center. Cantor showed that the points on the line segment could be paired in a one-to-one fashion with the points on a unit square and, thus, with the set of points in any such space of any dimension. This pairing, though not continuous, scandalized the mathematical community, for it seemed contrary to reason that a one-dimensional space could be equally as infinite as a two-dimensional space.

Later on, Guiseppe Peano and David Hilbert devised continuous mappings from the segment onto the square; these were no longer one-to-one. The first several steps in the construction of Hilbert's mapping are shown above; the "squiggle" eventually fills the square, with no points left out. There is a three-dimensional analogue of this mapping, which fills the cube, and could, in principle, fill all of space.

So, if we imagine the pages of Borges' book to correspond to (say) the rational numbers, then the particles of its burning might not fill all the atmosphere, but, since the rationals are dense on the segment (infinitely many lie between any two points, no matter how close), we could easily imagine that its smoke might be so dense as to entirely pervade the atmosphere, with infinitely many particles in every space, no matter how small.

Cantor, as I noted, was a figure of controversy; he was denounced as a "scientific charlatan" and "corrupter of youth" by other mathematicians, and the strange curves and mappings he and others of his "school" described were shunned by many as monstrous or pathological. His investigation into the infinite also had a bearing on metaphysics and theology, for medievals like Thomas Aquinas had held that there could be no such thing as an actual existent infinity, only a potential infinity. Cantor's research seemed to some to indicate the existence of actual infinities, and thus, for various reasons, to tend toward pantheism. Cantor believed that further distinctions had to be made, and that Thomas wouldn't have objected to his ideas, had he been able to explain them; he corresponded with philosophers, theologians, and cardinals of the church, and even addressed a pamphlet to Pope Leo XIII, whose encyclical Aeterni Patris had advocated a renewed interest in scholastic philosophy.

But to return to the "monstrous" Book of Sand, whose serial infinity inspires in the narrator the same horror Cantor's theories had struck in his contemporaries. The narrator traded for it a copy of scripture that was a historical and literary treasure in itself as well as his savings for retirement. Might not this be a statement about modern man, who has traded his patrimony and future for a kind of gorgon's head of infinity? How fitting, too, that the narrator hides the book in a library, making it one page in the monstrous Book of Sand that is the modern glut of information, our "Library of Babel."

The Libraries of Faerie, Babel, and Urth

There was a time, perhaps ten or fifteen years ago, when I often read George MacDonald. The strain of gnostic mysticism that runs through nineteenth-century arts and letters appealed to me in his novels. Phantastes was, of course, my favorite, but I read Lilith several times as well. It's an interesting point that each of these novels—written at different stages of MacDonald's life—contain a large library with occult contents, secret rooms, and ill-defined boundaries, a library in which the protagonist finds himself as an alien in a wilderness. Here Anodos describes the Palace of Faerie:

The library was a mighty hall, lighted from the roof, which was formed of something like glass, vaulted over in a single piece, and stained throughout with a great mysterious picture in gorgeous colouring. The walls were lined from floor to roof with books and books: most of them in ancient bindings, but some in strange new fashions which I had never seen, and which, were I to make the attempt, I could ill describe… Over some parts of the library, descended curtains of silk of various dyes, none of which I ever saw lifted while I was there; and I felt somehow that it would be presumptuous in me to venture to look within them.

And Mr. Vane here describes the library bequeathed to him at the beginning of Lilith:

The library, although duly considered in many alterations of the house and additions to it, had nevertheless, like an encroaching state, absorbed one room after another until it occupied the greater part of the ground floor. Its chief room was large, and the walls of it were covered with books almost to the ceiling; the rooms into which it overflowed were of various sizes and shapes, and communicated in modes as various—by doors, by open arches, by short passages, by steps up and steps down.

Or again, later:

I saw no raven, but the librarian—the same slender elderly man, in a rusty black coat, large in the body and long in the tails. I had seen only his back before; now for the first time I saw his face. It was so thin that it showed the shape of the bones under it, suggesting the skulls his last-claimed profession must have made him familiar with. But in truth I had never before seen a face so alive, or a look so keen or so friendly as that in his pale blue eyes, which yet had a haze about them as if they had done much weeping.
     "You knew I was not a raven!" he said with a smile.
     "I knew you were Mr. Raven," I replied; "but somehow I thought you a bird too!"
     "What made you think me a bird?"
     "You looked a raven, and I saw you dig worms out of the earth with your beak."
     "And then?"
     "Toss them in the air."
     "And then?"
     "They grew butterflies, and flew away."
     "Did you ever see a raven do that? I told you I was a sexton!"
     "Does a sexton toss worms in the air, and turn them into butterflies?"
     "Yes."
     "I never saw one do it!"
     "You saw me do it!—But I am still librarian in your house, for I never was dismissed, and never gave up the office. Now I am librarian here as well."
     "But you have just told me you were sexton here!"
     "So I am. It is much the same profession. Except you are a true sexton, books are but dead bodies to you, and a library nothing but a catacomb!"

Someone (Lewis, I suppose) says somewhere that MacDonald once worked in the library of a certain estate, and that this experience doubtless colored his stories. Certainly I can understand its making a strong impression on him. There's something almost numinous about a large, empty library. To me it's akin to the fear described by Pascal in the quote on my sidebar, though why this is I'm not certain. Perhaps it's because libraries approach the mysteries of blank infinity with their iterative Chinese-box structure and seemingly interminable strings of characters.

Borges gives greatest expression to this spectral fear in his short story "The Library of Babel." Whatever he may have meant by the fable—I tread diffidently here—it captures the unsettling, nightmarish quality of rows upon rows upon rows of books.

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. From any of the hexagons one can see, interminably, the upper and lower floors. The distribution of the galleries is invariable.

Each book in the Library consists of 410 pages with 40 lines on each page and 80 characters on each line; there are 25 characters in all, including 22 letters, the period, the comma, and the space. It is claimed that the Library contains every possible combination of letters without repetition. There are thus only finitely many rooms. However, adapting a line from Pascal, the narrator states:

The Library is a sphere whose exact center is any one of the hexagons and whose circumference is inaccessible.

This would seem to indicate a peculiar topology. The narrator expresses the opinion that the Library simply repeats. (This, however, is impossible, unless one of the rooms is different from the others. For each cell contains 20 shelves with 35 books per shelf, so that there are 2² × 5² × 7 books per cell, whereas the total number of possible combinations is 5^2,624,000, into which 2² × 5² × 7 is not divisible. Perhaps the spine characters introduce the necessary factors?) If the Library does repeat in a consistent way, then a horizontal slice would be an orientable surface, possibly of extremely high genus.

The weight and pregnant hush of libraries is handled more affectively in Wolfe's Shadow of the Torturer, when Severian descends to the stacks to find Master Ultan, the blind curator.

"We have books here bound in the hides of echidnes, krakens, and beasts so long extinct that those whose studies they are, are for the most part of the opinion that no brace of them survives unfossilized. We have books bound wholly in metals of unknown alloy, and books whose bindings are covered with thickset gems. We have books cased in perfumed woods shipped across the inconceivable gulf between creations—books doubly precious because no one on Urth can read them.
     "We have books whose papers are matted of plants from which spring curious alkaloids, so that the reader, in turning their pages, is taken unaware by bizarre fantasies and chimeric dreams. Books whose pages are not paper at all, but delicate wafers of white jade, ivory, and shell; books too whose leaves are the desiccated leaves of unknown plants. Books we have also that are not books at all to the eye: scrolls and tablets and recordings on a hundred different substances. There is a cube of crystal here—though I can no longer tell you where—no larger than the ball of your thumb that contains more books than the library itself does."

Here the library is invested with the impossible weight of ages that lie heaped upon Urth in those latter days, as the sun cools and dims.

I still have somewhat to say concerning libraries—chiefly of an autobiographical nature—but this is getting to be a long post, so I'll save it for the next. Why am I writing about all this, you ask, dear reader? Because it helps me organize my thoughts, and because the subject interests me. So, onward ho.