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"With this delightful anthology, Frucht throws a bridge across the chasm separating the 'Two Cultures' of science and literature."—Booklist
"A marvelous colledtion of diverse talents and writing."—San Diego Union-Tribune
A wildly inventive treasury of the most artful words ever written about numbers. Mathematics and writing may seem to exist in opposite realms, but as William Frucht reveals, the world of numbers has always held a special fascination for men and women of letters. Imaginary Numbers displays the ...
"With this delightful anthology, Frucht throws a bridge across the chasm separating the 'Two Cultures' of science and literature."—Booklist
"A marvelous colledtion of diverse talents and writing."—San Diego Union-Tribune
A wildly inventive treasury of the most artful words ever written about numbers. Mathematics and writing may seem to exist in opposite realms, but as William Frucht reveals, the world of numbers has always held a special fascination for men and women of letters. Imaginary Numbers displays the fruits of this cross-fertilization by collecting the best creative writing about mathematical topics from the past hundred years. In this engaging anthology, we can explore the many ways writers have played with mathematical ideas. Delve into the fourth dimension and infinity, into fantasy and philosophy with such masters as Lewis Carroll, Edwin Abbott Abbott, Philip K. Dick, Martin Gardner, and Alan Lightman. Revel in renowned tales by Italo Calvino and Jorge Luis Borges, cult classics such as Connie Willis's "The Schwartzschild Radius,"and lesser-known gems by such visionaries as William Gibson and A. K. Dewdney. For mathematical mavens and literary lions alike, Imaginary Numbers adds up to one fascinating read.
Italo Calvino: The Form of Space.
Rudy Rucker: A New Golden Age.
Lewis Carroll: A Serpent with Corners (from A Tangled Tale).
J. A. Lindon: A Positive Reminder.
Raymond Smullyan: How Kazir Won His Wife.
Alan Lightman: 11 May 1905 (from Einstein's Dreams).
Roald Hoffman: Why Does Disorder Increase in the Same Direction of Time as That in Which the Universe Expands?
Philip K. Dick: The Golden Man.
Hilbert Schenck: The Morphology of the Kirkham Wreck.
Abner Shimony: Resolution of the Paradox: A Philosophical Puppet Play.
Piet Hein: Parallelism.
Douglas Hofstadter Prelude...(from Godel, Escher, Bach: An Eternal Golden Braid).
Stanislaw Lem: The Third Sally, or The Dragons of Probability (from The Seven Sallies of Trurl and Klapaucius).
Edwin Abbott Abbott: Concerning Irregular Figures (from Flatland).
Andrew Marvell: The Definition of Love.
A. K. Dewdney: On Fiddib Har (from The Planiverse: Computer Contact with a Two-Dimensional World).
Martin Gardner: The Church of the Fourth Dimension.
Stanislaw Lem: The Extraordinary Hotel, or the Thousand and First Journey of Ion the Quiet.
Anonymous: Ten Weary, Footsore Travelers.
Racter: From The Policeman's Beard Is Half-Constructed.
William Gibson: Burning Chrome.
Fritz Leiber: Gonna Roll the Bones.
Wislawa Szymborska: A Word on Statistics.
Tommaso Landolfi: Giovanni and His Wife.
Joe Haldeman: The Private War of Private Jacob.
Jorge Luis Borges: The Library of Babel.
Christian Bok: Enantiomorphosis (A Natural History of Mirrors).
Connie Willis: Schwarzschild Radius.
Siv Cedering: Letter from Caroline Herschel (1750-1848).
Yevgeny Zamyatin: From We.
J. G. Ballard: The Garden of Time.
The equations of the gravitational field which relate the curve of space to the distribution of matter are already becoming common knowledge.
To fall in the void as I fell: none of you knows what that means. For you, to fall means to plunge perhaps from the twenty-sixth floor of a skyscraper, or from an airplane which breaks down in flight: to fall headlong, grope in the air a moment, and then the Earth is immediately there, and you get a big bump. But I'm talking about the time when there wasn't any Earth underneath or anything else solid, not even a celestial body in the distance capable of attracting you into its orbit. You simply fell, indefinitely, for an indefinite length of time. I went down into the void, to the most absolute bottom conceivable, and once there I saw that the extreme limit must have been much, much farther below, very remote, and I went on falling, to reach it. Since there were no reference points, I had no idea whether my fall was fast or slow. Now that I think about it, there weren't even any proofs that I was really falling: perhaps I had always remained immobile in the same place, or I was moving in an upward direction; since there was no above or below these were only nominal questions and so I might just as well go on thinking I was falling, as I was naturally led to think.
Assuming then that one was falling, everyone fell with the same speed and rate of acceleration; in fact we were always more or less on the same level: I, Ursula H'x, Lieutenant Fenimore. I didn't take my eyes off Ursula H'x: she was very beautiful to see, and in falling she had an easy, relaxed attitude. I hoped I would be able sometimes to catch her eye, but as she fell, Ursula H'x was always intent on filing and polishing her nails or running her comb through her long, smooth hair, and she never glanced to-ward me. Nor toward Lieutenant Fenimore, I must say, though he did everything he could to attract her attention.
Once I caught him-- he thought I couldn't see him-- as he was making some signals to Ursula H'x: first he struck his two index fingers, outstretched, one against the other, then he made a rotating gesture with one hand, then he pointed down. I mean, he seemed to hint at an understanding with her, an appointment for later on, in some place down there, where they were to meet. All nonsense, I knew perfectly well: there were no meetings possible among us, because our falls were parallel and the same distance always remained between us. But the mere fact that Lieutenant Fenimore had got such ideas into his head-- and tried to put them into the head of Ursula H'x-- was enough to get on my nerves, even though she paid no attention to him, indeed she made a slight blurting sound with her lips, directed-- I felt there was no doubt-- at him. (Ursula H'x fell, revolving with lazy movements as if she were turning in her bed and it was hard to say whether her gestures were directed at someone else or whether she was playing for her own benefit, as was her habit.)
I too, naturally, dreamed only of meeting Ursula H'x, but since, in my fall, I was following a straight line absolutely parallel to the one she followed, it seemed inappropriate to reveal such an unattainable desire. Of course, if I chose to be an optimist, there was always the possibility that, if our two parallels continued to infinity, the moment would come when they would touch. This eventuality gave me some hope; indeed, it kept me in a state of constant excitement. I don't mind telling you I had dreamed so much of a meeting of our parallels, in great detail, that it was now a part of my experience, as if I had actually lived it. Everything would happen suddenly, with simplicity and naturalness: after the long separate journey, unable to move an inch closer to each other, after having felt her as an alien being for so long, a prisoner of her parallel route, then the consistency of space, instead of being impalpable as it had always been, would become more taut and, at the same time, looser, a condensing of the void which would seem to come not from outside but from within us, and would press me and Ursula H'x together (I had only to shut my eyes to see her come forward, in an attitude I recognized as hers even if it was different from all her habitual attitudes: her arms stretched down, along her sides, twisting her wrists as if she were stretching and at the same time writhing and leaning forward), and then the invisible line I was following would become a single line, occupied by a mingling of her and me where her soft and secret nature would be penetrated or rather would enfold and, I would say, almost absorb the part of myself that till then had been suffering at being alone and separate and barren.
Even the most beautiful dreams can suddenly turn into nightmares, and it then occurred to me that the meeting point of our two parallels might also be the point at which all parallels existing in space eventually meet, and so it would mark not only my meeting with Ursula H'x but also-- dreadful prospect-- a meeting with Lieutenant Fenimore. At the very moment when Ursula H'x would cease to be alien to me, another alien with his thin black mustache would share our intimacies in an inextricable way: this thought was enough to plunge me into the most tormented jealous hallucinations: I heard the cry that our meeting-- hers and mine-- tore from us melt in a spasmodically joyous unison and then-- I was aghast at the presentiment-- from that sound burst her piercing cry as she was violated-- so, in my resentful bias, I imagined-- from behind, and at the same time the Lieutenant's vulgar shout of triumph, but perhaps-- and here my jealousy be-came delirium-- these cries of theirs, hers and his-- might also not be so different or so dissonant, they might also achieve a unison, be joined in a single cry of downright pleasure, distinct from the sobbing, desperate moan that would burst from my lips.
In this alternation of hopes and apprehensions I continued to fall, constantly peering into the depths of space to see if anything heralded an immediate or future change in our condition. A couple of times I managed to glimpse a universe, but it was far away and seemed very tiny, well off to the right or to the left; I barely had time to make out a certain number of galaxies like shining little dots collected into superimposed masses which revolved with a faint buzz, when everything would vanish as it had appeared, upwards or to one side, so that I began to suspect it had only been a momentary glare in my eyes.
"There! Look! There's a universe! Look over there! There's something!" I shouted to Ursula H'x, motioning in that direction; but, tongue between her teeth, she was busy caressing the smooth, taut skin of her legs, looking for those very rare and almost invisible excess hairs she could uproot with a sharp tug of her pincerlike nails, and the only sign she had heard my call might be the way she stretched one leg upwards, as if to exploit-- you would have said-- for her methodical inspection the dim light reflected from that distant firmament.
I don't have to tell you the contempt Lieutenant Fenimore displayed toward what I might have discovered on those occasions: he gave a shrug-- shaking his epaulettes, his bandoleer, and the decorations with which he was pointlessly arrayed-- and turned in the other direction, snickering. Unless he was the one (when he was sure I was looking elsewhere) who tried to arouse Ursula's curiosity (and then it was my turn to laugh, seeing that her only response was to revolve in a kind of somersault, turning her behind to him: a gesture no doubt disrespectful but lovely to see, so that, after rejoicing in my rival's humiliation, I caught myself envying him this, as a privilege), indicating a labile point fleeing through space, shouting: "There! There! A universe! This big! I saw it! It's a universe!"
I won't say he was lying: statements of that sort, as far as I know, were as likely to be true as false. It was a proved fact that, every now and then, we skirted a universe (or else a universe skirted us), but it wasn't clear whether these were a number of universes scattered through space or whether it was always the same universe we kept passing, revolving in a mysterious trajectory, or whether there was no universe at all and what we thought we saw was the mirage of a universe which perhaps had once existed and whose image continued to rebound from the walls of space like the rebounding of an echo. But it could also be that the universes had always been there, dense around us, and had no idea of moving, and we weren't moving, either, and everything was arrested forever, without time, in a darkness punctuated only by rapid flashes when something or someone managed for a moment to free himself from that sluggish timelessness and indicate the semblance of a movement.
All these hypotheses were equally worth considering, but they interested me only insofar as they concerned our fall and the possibility of touching Ursula H'x. In other words, nobody really knew anything. So why did that pompous Fenimore sometimes assume a superior manner, as if he were certain of things? He had realized that when he wanted to infuriate me the surest system was to pretend to a long-standing familiarity with Ursula H'x. At a certain point Ursula took to swaying as she came down, her knees together, shifting the weight of her body this way and that, as if wavering in an ever-broader zigzag: just to break the monotony of that endless fall. And the Lieutenant then also started swaying, trying to pick up her rhythm, as if he were following the same invisible track, or rather as if he were dancing to the sound of the same music, audible only to the two of them, which he even pretended to whistle, putting into it, on his own, a kind of unspoken understanding, as if alluding to a private joke among old boozing companions. It was all a bluff-- I knew that, of course-- but still it gave me the idea that a meeting between Ursula H'x and Lieutenant Fenimore might already have taken place, who knows how long ago, at the beginning of their trajectories, and this suspicion gnawed at me painfully, as if I had been the victim of an injustice. On reflecting, however, I reasoned that if Ursula and the Lieutenant had once occupied the same point in space, this meant that their respective lines of fall had since been moving apart and presumably were still moving apart. Now, in this slow but constant removal from the Lieutenant, it was more than likely that Ursula was coming closer to me; so the Lieutenant had little to boast of in his past conjunctions: I was the one at whom the future smiled.
The process of reasoning that led me to this conclusion was not enough to reassure me at heart: the possibility that Ursula H'x had already met the Lieutenant was in itself a wrong which, if it had been done to me, could no longer be redeemed. I must add that past and future were vague terms for me, and I couldn't make much distinction between them: my memory didn't extend beyond the interminable present of our parallel fall, and what might have been before, since it couldn't be remembered, belonged to the same imaginary world as the future, and was confounded with the future. So I could also suppose that if two parallels had ever set out from the same point, these were the lines that Ursula H'x and I were following (in this case it was nostalgia for a lost oneness that fed my eager desire to meet her); however, I was reluctant to believe in this hypothesis, because it might imply a progressive separation and perhaps her future arrival in the braid-festooned arms of Lieutenant Fenimore, but chiefly because I couldn't get out of the present except to imagine a different present, and none of the rest counted.
Perhaps this was the secret: to identify oneself so completely with one's own state of fall that one could realize the line followed in falling wasn't what it seemed but another, or rather to succeed in changing that line in the only way it could be changed, namely, by making it become what it had really always been. It wasn't through concentrating on myself that this idea came to me, though, but through observing, with my loving eye, how beautiful Ursula H'x was even when seen from behind, and noting, as we passed in sight of a very distant system of constellations, an arching of her back and a kind of twitch of her behind, but not so much the behind itself as an external sliding that seemed to rub past the behind and cause a not unpleasant reaction from the behind itself. This fleeting impression was enough to make me see our situation in a new way: if it was true that space with something inside is different from empty space be-cause the matter causes a curving or a tautness which makes all the lines contained in space curve or tauten, then the line each of us was following was straight in the only way a straight line can be straight: namely, deformed to the extent that the limpid harmony of the general void is deformed by the clutter of matter, in other words, twisting all around this bump or pimple or excrescence which is the universe in the midst of space.
My point of reference was always Ursula and, in fact, a certain way she had of proceeding as if twisting could make more familiar the idea that our fall was like a winding and unwinding in a sort of spiral that tightened and then loosened. However, Ursula-- if you watched her carefully-- wound first in one direction, then in the other, so the pattern we were tracing was more complicated. The universe, therefore, had to be considered not a crude swelling placed there like a turnip, but as an angular, pointed figure where every dent or bulge or facet corresponded to other cavities and projections and notchings of space and of the lines we followed. This, however, was still a schematic image, as if we were dealing with a smooth-walled solid, a compenetration of polyhedrons, a cluster of crystals; in reality the space in which we moved was all battlemented and perforated, with spires and pinnacles which spread out on every side, with cupolas and balustrades and peristyles, with rose windows, with double-and triple-arched fenestrations, and while we felt we were plunging straight down, in reality we were racing along the edge of moldings and invisible friezes, like ants who, crossing a city, follow itineraries traced not on the street cobbles but along walls and ceilings and cornices and chandeliers. Now if I say city it amounts to suggesting figures that are, in some way, regular, with right angles and symmetrical proportions, whereas instead, we should always bear in mind how space breaks up around every cherry tree and every leaf of every bough that moves in the wind, and at every indentation of the edge of every leaf, and also it forms along every vein of the leaf, and on the network of veins inside the leaf, and on the piercings made every moment by the riddling arrows of light, all printed in negative in the dough of the void, so that there is nothing now that does not leave its print, every possible print of every possible thing, and together every transformation of these prints, instant by instant, so the pimple growing on a caliph's nose or the soap bubble resting on a laundress's bosom changes the general form of space in all its dimensions.
All I had to do was understand that space was made in this way and I realized there were certain soft cavities hollowed in it as welcoming as hammocks where I could lie joined with Ursula H'x, the two of us swaying together, biting each other in turn along all our persons. The properties of space, in fact, were such that one parallel went one way, and another in another way: I for example was plunging within a tortuous cavern while Ursula H'x was being sucked along a passage communicating with that same cavern so that we found ourselves rolling together on a lawn of algae in a kind of subspatial island, writhing, she and I, in every pose, upright and capsized, until all of a sudden our two straight lines resumed their distance, the same as always, and each continued on its own as if nothing had happened.
The grain of space was porous and broken with crevasses and dunes. If I looked carefully, I could observe when Lieutenant Fenimore's course passed through the bed of a narrow, winding canyon; then I placed myself on the top of a cliff and, at just the right moment, I hurled myself down on him, careful to strike him on the cervical vertebrae with my full weight. The bottom of such precipices in the void was stony as the bed of a dried-up stream, and Lieutenant Fenimore, sinking to the ground, remained with his head stuck between two spurs of rock; I pressed one knee into his stomach, but he meanwhile was crushing my knuckles against a cactus's thorns-- or the back of a porcupine? (spikes, in any case, of the kind corresponding to certain sharp contractions of space)-- to prevent me from grabbing the pistol I had kicked from his hand. I don't know how I happened, a moment later, to find myself with my head thrust into the stifling granulosity of the strata where space gives way, crumbling like sand; I spat, blinded and dazed; Fenimore had managed to collect his pistol; a bullet whistled past my ear, ricocheting off a proliferation of the void that rose in the shape of an anthill. And I fell upon him, my hands at his throat, to strangle him, but my hands slammed against each other with a "plop!": our paths had become parallel again, and Lieutenant Fenimore and I were descending, maintaining our customary distance, ostentatiously turning our backs on each other, like two people who pretend they have never met, haven't even seen each other before.
What you might consider straight, one-dimensional lines were similar, in effect, to lines of handwriting made on a white page by a pen that shifts words and fragments of sentences from one line to another, with insertions and cross-references, in the haste to finish an exposition which has gone through successive, approximate drafts, always unsatisfactory; and so we pursued each other, Lieutenant Fenimore and I, hiding behind the loops of the l 's, especially the l 's of the word "parallel," in order to shoot and take cover from the bullets and pretend to be dead and wait, say, till Fenimore went past in order to trip him up and drag him by his feet, slamming his chin against the bottoms of the v's and the u's and the m's and the n's which, written all evenly in an italic hand, became a bumpy succession of holes in the pavement (for example, in the expression "unmeasurable universe"), leaving him stretched out in a place all trampled with erasings and x-ings, then standing up there again, stained with clotted ink, to run to-ward Ursula H'x, who was trying to act sly, slipping behind the tails of the f which trail off until they become wisps, but I could seize her by the hair and bend her against a d or a t just as I write them now, in haste, bent, so you can recline against them, then we might dig a niche for ourselves down in a g, in the g of "big," a subterranean den which can be adapted as we choose to our dimensions, being made more cozy and almost invisible or else arranged more horizontally so you can stretch out in it. Whereas naturally the same lines, rather than remain series of letters and words, can easily be drawn out in their black thread and unwound in continuous, parallel, straight lines which mean nothing beyond themselves in their constant flow, never meeting, just as we never meet in our constant fall: I, Ursula H'x, Lieutenant Fenimore, and all the others.