Age eleven was when Jessy took off on her own. Now she made her own sheets; she no longer needed a mathematical companion, though a resourceful helper taught her to calculate areas. Megan drew diagrams and Jessy solved the problems handily; she understood their connection to the real world. But the world that was most real to her was not that of our everyday bread-and-butter problems. With the necessary notations and operations in place, numbers, not words, became Jessy’s primary expressive instrument.
We can’t know how great a part circumstances played in Jessy’s annus mirabilis. For we were not at home in her familiar house full of built-in activities; as once before, when she was four, her father was on sabbatical. We were living ten floors up in a small apartment outside Paris. No toys — not that Jessy played with toys much — nothing but paper, pencils, and a typewriter. In the cold, cloudy spring, only the activities we could think up. Prodigiously inventive, Megan lured Jessy into reading, typing questions to which Jessy typed out answers in a progressive dialogue. She typed out, in words, all the numbers from i to 100; it turned out she could spell better than we knew. The typewriter was good for numbers; with Megan she converted fractions to percents and solved simple equations. But Jessy could not be constantly accompanied. Alone in her own world she pursued different calculations.
Clocks became fascinating when she learned that the French numbered time not in twelve hours but in twenty-four. She drew a ten-hour clock, a twelve-hour clock, a fourteen-hour clock, sixteen-, eighteen-, twenty-four-, and thirty-six-hour clocks. She converted hours to minutes, minutes to seconds; surviving sheets record that 3600 seconds = 60 minutes = 1 hour. Carefully she drew in each second. Time was now something to play with. Fractional conversions became so rapid as to seem intuitive: 49 hours = 2 1/24 days. Soon she was mapping space as well as time: 7 1/2 inches = 5/8 foot.
And hour after hour she multiplied and divided. There were no calculators then. I couldn’t keep all the sheets of paper she consumed, and Jessy didn’t want them. Once they were done, she’d internalized her discoveries, 51 X 51, 52 X 52, 53 X 53, and on and on. Even I could see what she was up to: determining squares, then cubes, then higher and higher powers. And what can be multiplied can be divided; her long division became more and more bizarre as she searched out larger primes and identified more factors. She liked the number 60; it was handy for clocks. I kept the sheet that said it was divisible twelve ways. But that was just the beginning. Months later another sheet recorded that 26082 was divisible by 1, 2, 3, 6, 7, 9, 14, 18, 21, 23, 27, 42, 46, 54, 63, 69, 81, 126, 138, 161, 162, 189, 207, 322, 378, 414, 483, 567, 621, 966, 1134, 1242, 1449, 1863, 2898. There the numbers stopped. Did she finish the series somewhere else?
Other sheets explored the factors of 13041 and 19380. Fractions too could be factored: 1 1/2 is divisible by 1/100, 1/50, 3/100, 1/20, 3/50, 1/10, 3/20, 1/4, 1/2, 3/4, 1 1/2. Though numbers were generally a solitary pastime, now and then we looked over them together. Jessy had written 2, 2, 2, 3, 5, 5, 7, 7. «If you multiply together you get 29400», she told me. On another sheet appeared 678586773483121410. When her father remarked it was not a prime, Jessy explained that it was «lots of primes multiplied together». Other numerical pastimes were less challenging. Jessy was happy counting to 10, 000 by 100’s, to 15, 000 by 150 s. Not very interesting, perhaps, but very autistic. Years later we met an Australian boy, far more advanced than Jessy, who spent his free time counting to a million, and I have seen sheets of primes and multiplications by Joseph Sullivan, one of the models for the Dustin Hoffman character in the movie Rain Man, that could easily be taken for Jessy’s.
* * *
I can follow Jessy’s math up to a point; primes and cubes and prime factors are not in fact all that complicated. But I know my limits. I quote again from Ron Ellis and Lola Bogyo, glad to reiterate what we owe to them and to all those who accompanied Jessy beyond what we could manage. «Many hands make light work» is one of Jessy’s favorite proverbs.
By age 13 she could list, on request, all the prime numbers from 1 to 1000 and beyond…. Two prime integers.. stood out among the rest: 7.. and 3… [Jessy] endlessly explored the composites and combinations of these numbers. She discovered, among many other things, that the delightful formal symmetry of the integer 10, 001 could be generated unexpectedly by multiplying 73 X 137. This formal symmetry was used to produce integers possessing a duplicating structure, as in the following examples:
10001 X 137 = 1370137
10001 X 7003 = 70037003
10001 x 7337 = 73377337
[Jessy] correctly inferred that she could generate formal duplication without directly using the integer 10001 but by embedding its factors in her calculations:
37 x 37 = 1369
1369 X 73 = 99937
99937 X 137 = 13691369 (formal duplication)
[Jessy] also found that she could encrypt selected integers within her duplicating structures by the formation of composites and then retrieve the original digits in a delight-fully altered form by further manipulations:
13691369 X 53 = 725642557 (5? is encrypted) 725642557 -5-37= 19611961 (a formal duplication)
19611961 37 = 530053 (53 appears in duplicate!)
Wherever there were patterns, however complex or subtle, [Jessy] discovered them; wherever instances adhered to some underlying rule, that rule was induced. She explored her world of numbers until it had become predictable and ordered.[24]
It took a mathematician to unravel the system behind this series of numbers Jessy produced at age twelve.
That was impressive enough. But it took our friend Freeman Dyson of the Institute for Advanced Study in Princeton to recognize Jessy’s most remarkable feat. The summer she turned twelve, Jessy produced sheet after sheet bearing the same strange series of fifty four-digit numbers. Or was it a series? It didn’t look like one. There is no obvious relation between 4096, 9216, 6400, and 5184. Only a mathematician’s eye would recognize that Jessy had arranged the squares of the numbers from 51 to 100 according to the number of powers of 2 they contain. 4096 is 642 or 212; it is made of nothing but 2’s. 9216 is 962; 96 is 3 X 32, 32 is 25, and so on. The odd numbers, beginning with 2601, which is the square of 51, complete the list. (And since many hands do make work lighter, it is Jessy’s father who deserves the credit for the wording of this explanation.)
* * *
To certain minds the language of mathematics is particularly attractive for its abstraction, for the beauty of pure idea, independent of human waywardness. And certainly that was one reason Jessy liked math. Yet the explanation is too simple, for as with clocks, many of Jessy’s numbers had strong linkages to the world. Some of these were emotionally neutral. Back home three of the Jessy-friends were counting calories; she recorded their weights (111, 126, and 140) and multiplied them together for the impressive total. She factored numbers derived from the number of times her superball bounced on a given day. But many were emotionally charged; like clouds and sunshine they could bring misery or delight. We had never noticed that telephone poles are numbered, but Jessy had. If she missed one there were mumbles, or worse. As once with washcloths, Jessy could be desolated by the incomplete.
SOMEBODY ATE A PIECE OF THE SALAD
ONLY 399 PIECES IN A BOWL
I HAVE A COLD
7256425570 = 52/7 X 1372837270
I CRIED WHEN SOMEBODY ATE A PIECE OF THE SALAD
Though Jessy had typed answers to Megan’s questions, these words, carefully printed in capitals, were almost her first spontaneous written communication. She was twelve. Every statement but the second was literally true, yet together they seemed meaningless. But they were not meaningless to Jessy. She had cried, she had shrieked. Then, alone in her room, she had expressed her anguish in words, and in numbers as bewildering as the cause of her distress. It was hard to forget that timeless wailing and remember her delight in the discovery that 70003 is a primer Yet we had learned that numerical desolations, like others, were temporary. Later that year Jessy, havin
g noted that 2730 was a HATE number, recorded its factors, as she had done in the past: 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 21, 26, 30, 35, 39, 42, 65, 70, 78, 91, 105, 130, 182, 195, 210, 273, 390, 455, 546, 910, 1365, 2730. But this time she added a comment: ITS CHANGED TO GOOD. And suddenly time rolled backward; I realized why, three years earlier, not yet able to write, she’d drawn so many chains. As words and numbers connected, another window opened on that strange inner world.
* * *
All that was long, long ago. Jessy’s math, though as accurate as ever, is now ordinary, real-world stuff, useful for balancing a checkbook or making out a tax form; fractional conversions are handy for baking. There are no more HATE numbers, and she has forgotten many of the primes and prime factors that she used to whisper because they were «too good». Numbers, like others of her obsessions, have, in her own phrase, «worn away», as she has entered more and more fully into the normal world of the everyday. And that itself is normal. How many of us leave an interest behind, or a skill, as the piano stands unopened or the sketchbooks gather dust?
And yet — how much there is in Jessy’s mind we don’t know about! A year ago her father and I were wondering why our home phone gave a continual busy signal; was it out of order? Jessy picked up on our conversation. (That itself is something she never used to do.) Unexpectedly she informed us that the out-of- order signal was like the busy signal but with 120 beeps per minute instead of 60. Now who knew that?
But counting may be only a habit when you’ve done it all your life. There are other indications that numbers persist, no longer emotion-filled, no longer secret, but underground. This year I notice that her social security number ends in 1421; I mention that (obviously) it’s divisible by 7. It’s even more obvious to Jessy; «Divisible by 72», she says. I recall another instance. Jessy had written 1875 in one of her old books and I asked her if she remembered anything about it. Though it was half her lifetime past, her answer was immediate. «Has a 3 in it. And 5V». And indeed, divide it by 3 and you get 625: 5 X 5 X 5 X 5! It’s no great surprise, then, to find she’s factored the year of her birth: 1958 is 2 X 979. «979», she says, «is definitely a prime».
* * *
As I go over envelopes and slips I find an old calculation. 1988 = 22 X 497, 497 = 7 X 71. Jessy is interested in the book about her, and answers willingly when I ask her about 1998. Divided by 33 it equals 74, which divided by 2 gives (old faithful!) 37. I ask about 1999. It’s early in the year and Jessy, ever truthful, says she’s not sure about 1999. 1997? But by now she’s had enough. What’s past is past: «I’m too old», she says.
Chapter 7
«The hangman hangs by the clothespin because of new politeness»
Strangeness/Secret Life — a label on an envelope in a folder crammed with other envelopes, Hypersensitivities, Obsessions, Compulsions, and the rest. As if a folder, as if even a suitcase, could contain the strangeness that suffused our family’s every day. Strange systems, strange numbers; still, I need another chapter to explore (not exhaust) the strangeness of that busy mind, the bewildering interplay between its creativity and its handicap.
The contents of the suitcase are spread all over my bedroom as I try to classify and select. Drawings of «little imitation people». «Books», hundreds of them; there were months when she made one almost every day. As her language progressed she titled them: Book About the Number with Three in It; Book About the Number of Three in It; Book About the Bump; Book About the Light in the Science Building; Book About the Shadow. The records of her preoccupations, her enjoyments, her anxieties, her desolations. I have been in no hurry to put them away, thinking she would be interested in these relics of past absorptions. In the years when talking — and human interaction — was an educational project, the best way to elicit speech was to revisit this library of former experience. Her books were made for her own satisfaction, not to communicate experience but to record it. Still, she would explain if we asked the right questions and we didn’t press too hard. Though intentional, eager communication («Come see!») lay far in the future, the books, with their successive layers of explanations, allowed us glimpses of the world within.
She wouldn’t — couldn’t — say much about her books in the early years, often not more than a few garbled words. But I’d write them down. A year or two later, returning to the same book, I could gauge how much her language had progressed, when she now had the words to clarify the explanations that had been a puzzle. I’d write those down too.
Jessy remembered then and she remembers now. Briefly the books about the numbers catch her interest; she wants to explain the «difference between ‘with’ and ‘of’». «Of» means divisible by 3; «with» means there is an actual 3 to be seen. But she looks no further. It’s I who am interested. This is the past; she has better things to do.
The little imitation people came first, inspired by the illustrations for Gulliver’s Travels and The Borrowers. Before she understood words, I was always drawing for her, showing her pictures. We looked at The Treasury of Art Masterpieces; we looked at the nice explicit illustrations in beginning readers. We looked at Harold and the Purple Crayon, where what Harold draws with his crayon becomes his own story. And the year she turned nine she too began to draw her own stories and enter them. The books began.
Jessy's «little imitation people»
The first were the series of what Jessy called «comic books». Harold and his crayon provided both inspiration and model; Jessy’s adventures followed his closely. TV was another source; Batman appeared, and renditions of TV logos — NBC, ABC and true to form, nonsense acronyms, VBC, ZBC, KBC. Jessy was now interested enough in letters to use them. Mysterious words appeared — not words she might be expected to know, like «cake», but GAKE, VAKE, GOKE, day after day. More nonsense, we thought — until, years later, she explained those as the noise of the heat coming on in the radiator and we understood why she had refused to enter rooms where she might hear the offending sound. Sometimes, however, the secret life remained secret. «Not for Mama. Oh oh, do not look at any more pictures, please!» and she crumpled it up and threw it away. Enough was enough.
Jessy was quite conscious of her sources. «This is all from Batman. I never understood the Batman plot, which involved two tigers, a fight, and a fall in the water. But the progression was orderly, the drawings clearly sequential. If the characters climbed a hill on one page, on the next they descended, or („like Harold!“) they fell from it, their streaming hair obeying gravity to register their fall. I had made her figures out of pipe cleaners, and Piper Cleaner Man, Piper Cleaner Lady, Piper Cleaner Girl and Boy, Piper Cleaner Fairy — and Paper Doll Jessy — formed the basic cast, occasionally joined by Big Girl Jessy, Mama, and Daddy». Piper Cleaner people came and went mysteriously, Jessy keeping careful count. Sometimes she doubled the group, and there would be two Paper Doll Jessies as well. The cast increased as she grew more interested in numbers: twelve, sixteen, eighteen, twenty-one.
The next year numbers took over. An isolated sheet shows the numbers from o to 8, drawn hollow so a corresponding number of Piper Cleaner people could cavort with Jessy inside each one. (The zero, of course, is empty.) Then Piper Cleaners became rare, replaced by row after row of standing figures whose bodies are actual numbers. There are odd consistencies; the hair on each 5 figure 5, 15, 50 always stands on end, and there is always a 200-person, often Jessy herself, to remind us that that busy mind knew exactly what it was doing, even if we didn’t.[25]
Jessy’s simple plots were uniformly upbeat. Paper Doll Jessy may open a door and fall into water (bubbles rising), but she finds another door, a rope, and returns to the Piper Cleaner family. The Piper Cleaners, naturally thin, procure sticks of gum and grow fat. If, as in Batman, they fight, the fight is «for fun». If they fall, they and Paper Doll Jessy still proceed home, to end safely in her bed, or occasionally with a party. That was natural enough; Harold’s adventures ended happily too. But the next year, when she’d done with Piper Cleaner book
s, Slovenly Peter hardly encouraged so benign a vision. Heinrich Hoffmann’s verses for bad children can raise the hair on an ordinary child. But Jessy made a book of two of the worst; in one a cry-baby literally cries her eyes out, in the other a hyperactive boy romps so hard he breaks off his foot. The German illustrator showed the detached foot and the two eyes on the floor, and so did Jessy. In her version, however, «ring a bell, girl and dog come» to restore the eyes and bring another foot. «Put the foot on, stand up!» In other books poor families are given meat, new houses, and finally a Christmas tree. Even a book about a terrifying night-long lightning storm ends with Jessy peacefully in bed, the storm over and a cerulean blue window heralding the dawn.
* * *
But the child’s world of happy endings was coming to an end. Numbers could turn bad. Flavor tubes and weather anxieties lay ahead. And Jessy, entering adolescence, was more and more engaged in the human world. We were glad of that. But the human world is not Nirvana. Her family, her companions, her teachers, had been endlessly patient, but «endlessly’ is all too easily written. No one’s patience is endless, and one expects more from a fourteen-year-old, even from a handicapped fourteen- year-old, than from a child. Not everyone — no one — can stand everything all the time. It is not always possible to call on emotional reserves when so much effort, so much affection, is rewarded only with a hostile „Go away!“ Though anger is always regrettable, it is more than rationalization to recognize that it may convey, in the only way it can be conveyed, the important social knowledge that people do have limits, that actions can have unpleasant consequences, and that there are good reasons to undertake the hard work of self-control.
Exiting Nirvana e-2 Page 9
