A geometry for the self—non-Euclidean, of course; in fact, strictly Nevskyan, as it has to be to help me, as I make my projections from visual space to auditive space, to haptic space.
The next time I met Blasingame he was anxious to hear what I thought of his diagram. (There could be an acoustics of emotion—thus a mathematics of emotion; meanwhile, the ears of the blind do these calculations every day.)
"One drawing isn't much to go on, Jeremy. I mean, you're right, it looks like a simple projective drawing, but with some odd lines crossing it. Who knows what they mean? The whole thing might be something scribbled by a kid."
"She's not that young. Want to see more?"
"Well…" This woman he kept mentioning, some sort of Mata Hari prisoner in the Pentagon, drawing geometrical figures and refusing to speak except in riddles… naturally I was intrigued.
"Here, take these anyway. There seems to be a sort of progression."
"It would help if I could talk to this subject who's doing all these."
"Actually, I don't think so… but"—seeing my irritation—"I can bring her by, I think, if these interest you."
"I'll check them out."
"Good, good." Peculiar edge of excitement in his voice, tension, anticipation of… Frowning, I took the papers from him.
That afternoon I shuffled them into my special Xerox machine, and the stiff reproductions rolled out of it heavily ridged. I ran my hands over the raised lines and letters slowly.
Here I must confess to you that most geometrical drawings are almost useless to me. If you consider it, you will quickly see why: Most drawings are two-dimensional representations of what a three-dimensional construction looks like. This does me no good and in fact is extremely confusing. Say I feel a trapezoid on the page; is that meant to be a trapezoid, or is it rather a representation of a rectangle not coterminous with the page it lies on? Or the conventional representation of a plane? Only a description of the drawing will tell me that. Without a description I can only deduce what the figure appears to mean. Much easier to have 3-D models to explore with my hands.
But in this case, not possible. So I swept over the mishmash of ridges with both hands, redrew it with my ridging pen several times over, located the two triangles in it, and the lines connecting the two triangles' corners, and the lines made by extending the triangles' sides in one direction. I tried to make from my Taylor collection a 3-D model that accounted for the drawing—try that sometime, and understand how difficult this kind of intellectual feat can be! Projective imagination…
Certainly it seemed to be a rough sketch of Desargues' theorem.
Desargues' theorem was one of the first theorems clearly concerned with projective geometry; it was proposed by Girard Desargues in the mid-seventeenth century, in between his architectural and engineering efforts, his books on music, etc. It is a relatively simple theorem, showing that two triangles that are projections of each other generate a group of points off to one side that lie on a single line. Its chief interest is in showing the kind of elegant connections that projection so often creates.
(It is also true that this theorem is reciprocal; that is, if you postulate two triangles whose extensions of the sides meet at three collinear points, then it is possible to show that the triangles are projections of each other. As they say in the textbooks, I leave the proof of this as an exercise for the reader.)
But so what? I mean, it is a beautiful theorem, with the sort of purity characteristic of Renaissance math—but what was it doing in a drawing made by some poor prisoner of the Pentagon?
I considered this as I walked to my health club, Warren's Spa (considered it secondarily, anyway, and no doubt subconsciously; my primary concerns were the streets and the traffic. Washington's streets bear a certain resemblance to one of those confusing geometrical diagrams I described [the state streets crossing diagonally the regular gridwork, creating a variety of intersections]; happily, one doesn't have to comprehend all the city at once to walk in it. But it is easy to become lost. So as I walked I concentrated on distances, on the sounds of the streets that tended to remain constant, on smells [the dirt of the park at M and New Hampshire, the hot dog vendor on 21st and K]; meanwhile, my cane established the world directly before my feet, my sonar shades whistled rising or falling notes as objects approached or receded… It takes some work just to get from point A to point B without getting disoriented [at which point one has to grind one's teeth and ask for directions] but it can be done, it is one of those small tasks/accomplishments [one chooses which, every time] that the blind cannot escape)—still, I did consider the matter of the drawing as I walked.
On 21st and H, I was pleased to smell the pretzel cart of my friend Ramon, who is also blind. His cart is the only one where the hot plate hasn't roasted several pretzels to that metallic burnt odor that all the other carts put off; Ramon prefers the clean smell of freshly baked dough, and he claims it brings him more customers, which I certainly believe. "Change only, please," he was saying to someone briskly, "there's a change machine on the other side of the cart for your convenience, thanks. Hot pretzels! Hot pretzels, one dollar!"
"Hey there, Superblink!" I called as I approached him.
"Hey yourself, Professor Superblink," he replied. (Superblink is a mildly derogatory name used by irritated sighted social service people to describe those of their blind colleagues who are aggressively or ostentatiously competent in getting around, etc., who make a display of their competence. Naturally, we have appropriated the term for our own use; sometimes it means the same thing for us—when used in the third person, usually—but in the second person, it's a term of affection.) "Want a pretzel?"
"Sure."
"You off to the gym?"
"Yeah, I'm going to throw. Next time we play you're in trouble."
"That'll be the day, when my main mark starts beating me!"
I put four quarters in his callused hand, and he gave me a pretzel. "Here's a puzzle for you," I said. "Why would someone try to convey a message by geometrical diagram?"
He laughed. "Don't ask me, that's your department!"
"But the message isn't for me."
"Are you sure about that?"
I frowned.
At the health club I greeted Warren and Amanda at the front desk. They were laughing over a headline in the tabloid newspaper Amanda was shaking; they devoured those things and pasted the best headlines all over the gym.
"What's the gem of the day?" I asked.
"How about ‘Gay Bigfoot Molests Young Boys'?" Warren suggested.
"Or ‘Woman Found Guilty of Turning Husband into Bank President,'" Amanda said, giggling. "She drugged him and did ‘bemod' to him until he went from teller to president."
Warren said, "I'll have to do that for you, eh, Amanda?"
"Make me something better than a bank president."
Warren clicked his tongue. "Entirely too many designer drugs, these days. Come on, Carlos, I'll get the range turned on." I went to the locker room and changed, and when I got to the target room Warren was just done setting it up. "Ready to go," he said cheerily as he rolled past me.
I stepped in, closed the door, and walked out to the center of the room, where a waist-high wire column was filled with baseballs. I pulled out a baseball, hefted it, felt the stitching. A baseball is a beautiful object: nicely flared curves of the seams over the surface of a perfect sphere, exactly the right weight for throwing.
I turned on the range with a flick of a switch and stepped away from the feeder, a ball in each hand. Now it was quite silent, only the slightest whirr faintly breathing through the soundproofed walls. I did what I could to reduce the sound of my own breathing, heard my heartbeat in my ears.
Then a beep behind me to my left, and low; I whirled and threw. Dull thud. "Right… low," said the machine voice from above, softly. Beep—I threw again: "Right… high," it said louder, meaning I had missed by more. "Shit," I said as I got another two balls. "Bad start."
&
nbsp; Beep—a hard throw to my left—clang! "Yeah!" There is very little in life more satisfying than the bell-like clanging of the target circle when hit square. It rings at about middle C with several overtones, like a small, thick church bell hit with a hammer. The sound of success.
Seven more throws, four more hits. "Five for ten," the machine voice said. "Average strike time, one point three five seconds. Fastest strike time, point eight four seconds."
Ramon sometimes hit the target in half a second or less, but I needed to hear the full beep to keep my average up. I set up for another round, pushed the button, got quiet; beep throw, beep throw, working to shift my feet faster, to follow through, to use the information from my misses to correct for the next time the target was near the floor, or the ceiling, or behind me (my weakness is the low ones; I can't seem to throw down accurately). And as I warmed up I threw harder and harder… just throwing a baseball as hard as you can is a joy in itself. And then to set that bell ringing! Clang! It chimes every cell of you.
But when I quit and took a shower, and stood before my locker and reached in to free my shirt from a snag on the top of the door, my fingers brushed a small metal wire stuck to an upper inside corner, where the door would usually conceal it from both me and my sighted companions; it came away when I pulled on it. Fingering the short length I couldn't be certain what it was, but I had my suspicions, so I took it to my friend James Gold, who works in acoustics in the engineering department, and had him take a confidential look at it.
"It's a little remote microphone, all right," he said, and then joked, "Who's bugging you, Carlos?"
He got serious when I asked him where I could get a system like that for myself.
John Metcalf—"Blind Jack of Knaresborough"—(1717–1810). At six he lost his sight through smallpox, at nine he could get on pretty well unaided, at fourteen he announced his intention of disregarding his affliction thenceforward and of behaving in every respect as a normal human being. It is true that immediately on this brave resolve he fell into a gravel pit and received a serious hurt while escaping, under pursuit, from an orchard he was robbing… fortunately this did not affect his self-reliance. At twenty he had made a reputation as a pugilist. (!)
Ernest Bramah, Introduction,
The Eyes of Max Carrados
When I was young I loved to read Bramah's stories about Max Carrados, the blind detective. Carrados could hear, smell, and feel with incredible sensitivity, and his ingenious deductions were never short of brilliant; he was fearless in a pinch; also, he was rich and had a mansion, and a secretary, manservant, and chauffeur who acted as his eyes. All great stuff for the imaginative young reader, as certainly I was. I read every book I could get my hands on; the voice of my reading machine was more familiar to me than any human voice that I knew. Between that reading and my mathematical work, I could have easily withdrawn from the world of my own experience into Cutsforth's "verbal unreality," and babbled on like Helen Keller about the shapes of clouds and the colors of flowers and the like. The world become nothing but a series of texts; sounds kind of like deconstructionism, doesn't it? And of course at an older age I was enamored of the deconstructionists of the last century. The world as text: Husserl's The Origin of Geometry is twenty-two pages long, Derrida's Introduction to the Origin of Geometry is 153 pages long; you can see why it would have appealed to me. If, as the deconstructionists seemed to say, the world is nothing but a collection of texts, and I can read, then I am not missing anything by being blind, am I?
The young can be very stubborn, very stupid.
"All right, Jeremy," I said. "Let me meet this mysterious subject of yours who draws all this stuff."
"You want to?" he said, trying to conceal his excitement.
"Sure," I replied. "I'm not going to find out any more about all this until I do." My own subtext, yes; but I am better at hiding such things than Jeremy is.
"What have you found out? Do the diagrams mean anything to you?"
"Not much. You know me, Jeremy, drawings are my weakness. I'd rather have her do it in models, or writing, or verbally. You'll have to bring her by if you want me to continue."
"Well, okay. I'll see what I can do. She's not much help, though. You'll find out." But he was pleased.
One time in high school I was walking out of the gym after P.E., and I heard one of my coaches (one of the best teachers I have ever had) in his office, speaking to someone (he must have had his back to me)—he said, "You know, it's not the physical handicaps that will be the problem for most of these kids. It's the emotional problems that tend to come with the handicaps that will be the real burden."
I was in my office listening to my reading machine. Its flat, uninflected mechanical voice (almost unintelligible to some of my colleagues) had over the years become a sort of helpless, stupid friend. I called it George, and was always programming into it another pronunciation rule to try to aid its poor speech, but to no avail; George always found new ways to butcher the language. I put the book facedown on the glass; "Finding first line," croaked George, as the scanner inside the machine thumped around. Then it read from Roberto Torretti, quoting and discussing Ernst Mach. (Hear this spoken in the most stilted, awkward, syllable-by-syllable mispronunciation that you can imagine.)
"‘Our notions of space are rooted in our physiological constitution'" (George raises his voice in pitch to indicate italics, which also slow him down considerably). "‘Geometric concepts are the product of the idealization of physical experiences of space.' Physiological space is quite different from the infinite, isotropic, metric space of classical geometry and physics. It can, at most, be structured as a topological space. When viewed in this way, it naturally falls into several components: visual or optic space, tactile or haptic space, auditive space, etc. Optic space is anisotropic, finite, limited. Haptic space or ‘the space of our skin corresponds to a two-dimensional, finite, unlimited (closed) Riemannian space.' This is nonsense, for R-spaces are metric, while haptic space is not. I take it that Mach means to say that the latter can naturally be regarded as a two-dimensional compact connected topological space. Mach does not emphasize enough the disconnectedness of haptic from optic space—"
There came four quick knocks at my door. I pressed the button on George that stopped him and said, "Come in!"
The door opened. "Carlos!"
"Jeremy," I said. "How are you?"
"Fine. I've brought Mary Unser with me—you know, the one who drew—"
I stood, feeling/hearing the presence of the other in the room. And there are times (like this one) when you know the other is in some odd, undefinable way, different, or… (Our language is not made for the experience of the blind.) "I'm glad to meet you."
I have said that I can tell dark from light, and I can, though it is seldom very useful information. In this case, however, I was startled to have my attention drawn to my "sight"—for this woman was darker than other people, she was a sort of bundle of darkness in the room, her face distinctly lighter than the rest of her (or was that her face, exactly?).
A long pause. Then: "On border stand we n-dimensional space the," she said. Coming just after George's reading, I was struck by a certain similarity: the mechanical lilt from word to word; the basic incomprehension of a reading machine… Goose bumps rose on my forearms.
Her voice itself, on the other hand, had George beat hands down. Fundamentally vibrant under the odd intonation, it was a voice with a very thick timbre, a bassoon or a hurdy-gurdy of a voice, with the buzz of someone who habitually speaks partly through the sinuses; this combined with over-relaxed vocal cords, what speech pathologists call glottal fry. Usually nasal voices are not pleasant, but pitch them low enough…
She spoke again, more slowly (definitely glottal fry): "We stand on the border of n-dimensional space."
"Hey," Jeremy said. "Pretty good!" He explained: "Her word order isn't usually as… ordinary as that."
"So I gathered," I said. "Mary, what do you mean by that?"<
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"I—oh—" A kazoo squeak of distress, pain. I approached her, put out a hand. She took it as if to shake: a hand about the size of mine, narrow, strong fat muscle at base of thumb; trembling distinctly.
"I work on the geometries of topologically complex spaces," I said. "I am more likely than most to understand what you say."
"Are within never see we points us."
"That's true." But there was something wrong here, something I didn't like, though I couldn't tell exactly what it was. Had she spoken toward Jeremy? Speaking to me while she looked at him? Bundle of darkness in the dark… "But why are your sentences so disordered, Mary? Your words don't come out in the order you thought them. You must know that, since you understand us."
"Folded—oh!—" Again the double-reed squeak, and suddenly she was weeping, trembling hard; we sat her down on my visitor's couch, and Jeremy got her a glass of water while she quaked in my hands. I stroked her hair (short, loosely curled, wild) and took the opportunity for a quick phrenological check: skull regular and, as far as I could tell, undamaged; temples wide, distinct; same for eye sockets; nose a fairly ordinary pyramidal segment, no bridge to speak of; narrow cheeks, wet with tears. She reached up and took my right hand, squeezed it hard, three times fast, three times slow, all the time sobbing and sort of hiccupping words: "Pain it, station. I, oh, fold end, bright, light, space fold, oh, ohhh…"
Well, the direct question is not always the best way. Jeremy returned with a glass of water, and drinking some seemed to calm her. Jeremy said, "Perhaps we could try again later. Although…" He didn't seem very surprised.
"Sure," I said. "Listen, Mary, I'll talk to you again when you're feeling better."
After Jeremy got her out of the office and disposed of her (how? with whom?) he returned to the seventh floor.
"So what the hell happened to her?" I asked angrily. "Why is she like that?"
"We aren't completely sure," he said slowly. "Here's why. She was one of the scientists staffing Tsiolkovsky Base Five, up in the mountains on the back side of the moon, you know. She's an astronomer and cosmologist. Well—I have to ask you to keep this quiet—one day Base Five stopped all broadcasting, and when they went over to see what was wrong, they found only her, alone in the station in a sort of catatonic state. No sign of the other scientists or station crew—eighteen people gone without a trace. And nothing much different to explain what had happened, either."