Genius (18 page)
Authors: James Gleick
Embrace the field or abhor it—either way, by the nineteen-thirties the choice seemed more one of method than reality. The events of 1926 and 1927 had made that clear. No one could be so naïve now as to ask whether Heisenberg’s matrices or Schrödinger’s wave functions
existed
. They were alternative ways of viewing the same processes. Thus Feynman, looking for a new eyepiece himself, began drifting back to a classical notion of unfieldlike particle interaction. The wavelike transmission of energy and the hocus-pocus of action at a distance were issues that he would have to address. In the meantime, Wheeler, too, had reasons to be drawn toward this implausibly pure conception. Electrons might interact directly, without the mediation of the field.
Feynman tended to associate more with the mathematicians than the physicists at the Graduate College. Students from the two groups joined each afternoon for tea in a common lounge—more English tradition transplanted—and Feynman would listen to an increasingly alien jargon. Pure mathematics had swerved away from the fields of direct use to contemporary physicists and toward such seeming esoterica as topology, the study of shapes in two, three, or many dimensions without regard to rigid lengths or angles. An effective divorce had occurred between mathematics and physics. By the time practitioners reached the graduate level, they shared no courses and had nothing practical to say to one another. Feynman listened to the mathematicians standing in groups or sitting on the couch at tea, talking about their proofs. Rightly or wrongly he felt he had an intuition for what theorems could be derived from what lemmas, even without quite understanding the subject. He enjoyed the strange rhetoric. He enjoyed trying to guess the counterintuitive answers to their nearly unvisualizable questions, and he enjoyed applying the physicist’s favorite needle, the claim that mathematicians spent their time proving the obvious. Although he teased them, he thought they were an exciting group—happy and interested in a kind of science that was getting beyond him. One friend was Arthur Stone, a patient young man attending Princeton on a fellowship from England. Another was John Tukey, who later became one of the world’s leading statisticians. These men spent their leisure time in curious ways. Stone had brought with him English-standard loose-leaf notebooks. The American-standard paper he bought at Woolworth’s overhung the notebooks by an inch, so he presently found himself with a supply of inch-wide paper ribbons, suitable for folding and twisting in different configurations. He tried diagonal folds at the 60-degree angle that produced rows of equilateral triangles. Then, following these folds, he wrapped a strip into a perfect hexagon.
Flexing a hexaflexagon.
When he closed the loop by taping the ends together, he found that he had created an odd toy: by pinching opposite corners of the hexagon, he could perform a queer origami-like fold, producing a new hexagon with a different set of triangles exposed. Repeating the operation exposed a third face. One more “flex” brought back the original configuration. In effect, he had a flattened tube that he was steadily turning inside out.
He considered this overnight. In the morning he took a longer strip and confirmed a new hypothesis: that a more elaborate hexagon could be made to cycle through not three but six different faces. The cycling was not so straightforward this time. Three of the faces tended to come up again and again, while the other three seemed harder to find. This was a nontrivial challenge to his topological imagination. Centuries of origami had not produced such an elegantly convoluted object. Within days copies of these “flexagons”—or, as this subspecies came to be more precisely known, “hexahexaflexagons” (six sides, six internal faces)—were circulating across the dining hall at lunch and dinner. The steering committee of the flexagon investigation soon comprised Stone, Tukey, a mathematician named Bryant Tuckerman, and their physicist friend Feynman. Honing their dexterity with paper and tape, they made hexaflexagons with twelve faces buried amid the folds, then twenty-four, then forty-eight. The number of varieties within each species rose rapidly according to a law that was far from evident. The theory of flexigation flowered, acquiring the flavor, if not quite the substance, of a hybrid of topology and network theory. Feynman’s best contribution was the invention of a diagram, called in retrospect the Feynman diagram, that showed all the possible paths through a hexaflexagon.
Seventeen years later, in 1956, the flexagons reached
Scientific American
in an article under the byline of Martin Gardner. “Flexagons” launched Gardner’s career as a minister to the nation’s recreational-mathematics underground, through twenty-five years of “Mathematical Games” columns and more than forty books. His debut article both captured and fed a minor craze. Flexagons were printed as advertising flyers and greeting cards. They inspired dozens of scholarly or semischolarly articles and several books. Among the hundreds of letters the article provoked was one from the Allen B. Du Mont Laboratories in New Jersey that began:
Sirs: I was quite taken with the article entitled “Flexagons” in your December issue. It took us only six or seven hours to paste the hexahexaflexagon together in the proper configuration. Since then it has been a source of continuing wonder.
But we have a problem. This morning one of our fellows was sitting flexing the hexahexaflexagon idly when the tip of his necktie became caught in one of the folds. With each successive flex, more of his tie vanished into the flexagon. With the sixth flexing he disappeared entirely.
We have been flexing the thing madly, and can find no trace of him, but we have located a sixteenth configuration of the hexahexaflexagon… .
The spirits of play and intellectual inquiry ran together. Feynman spent slow afternoons sitting in the bay window of his room, using slips of paper to ferry ants back and forth to a box of sugar he had suspended with string, to see what he could learn about how ants communicate and how much geometry they can internalize. One neighbor barged in on Feynman sitting by the window, open, on a wintry day, madly stirring a pot of Jell-O with a spoon and shouting “Don’t bother me!” He was trying to see how the Jell-O would coagulate while in motion. Another neighbor provoked an argument about the motile techniques of human spermatozoa; Feynman disappeared and soon returned with a sample. With John Tukey, Feynman carried out a long, introspective investigation into the human ability to keep track of time by counting. He ran up and down stairs to quicken his heartbeat and practiced counting socks and seconds simultaneously. They discovered that Feynman could read to himself silently and still keep track of time but that if he spoke he would lose his place. Tukey, on the other hand, could keep track of the time while reciting poetry aloud but not while reading. They decided that their brains were applying different functions to the task of counting: Feynman was using an aural rhythm, hearing the numbers, while Tukey visualized a sort of tape with numbers passing behind his eyes. Tukey said years later: “We were interested and happy to be empirical, to try things out, to organize and reduce to simple things what had been observed.”
Once in a while a small piece of knowledge from the world outside science would float Feynman’s way and stick like a bur from a chestnut. One of the graduate students had developed a passion for the poetry of Edith Sitwell, then considered modern and eccentric because of her flamboyant diction and cacophonous, jazzy rhythms. He read some poems aloud, and suddenly Feynman seemed to catch on; he took the book and started reciting gleefully. “Rhythm is one of the principal translators between dream and reality,” the poet said of her own work. “Rhythm might be described as, to the world of sound, what light is to the world of sight.” To Feynman rhythm was a drug and a lubricant. His thoughts sometimes seemed to slip and flow with a variegated drumbeat that his friends noticed spilling out into his fingertips, restlessly tapping on desks and notebooks. “While a universe grows in my head,—” Sitwell wrote,
I have dreams, though I have not a bed—
The thought of a world and a day
When all may be possible, still come my way.
For a while the tea-time conversation among the physicists both at Princeton and at the Institute for Advanced Study was dominated by the image of a rotating lawn sprinkler, an S-shaped apparatus spun by the recoil of the water it sprays forth. Nuclear physicists, quantum theorists, and even pure mathematicians were consumed by the problem: What would happen if this familiar device were placed under water and made to suck water in instead of spewing it out? Would it spin in the reverse direction, because the direction of the flow was now reversed, pulling rather than pushing? Or would it spin in the same direction, because the same twisting force was exerted by the water, whichever way it flowed, as it was bent around the curve of the S? (“It’s clear to me at first sight,” a friend of Feynman’s said to him some years later. Feynman shot back: “It’s clear to
everybody
at first sight. The trouble was, some guy would think it was perfectly clear one way, and another guy would think it was perfectly clear the other way.”) In an increasingly sophisticated time the simple problems still had the capacity to surprise. One did not have to probe far into physicists’ understanding of Newton’s laws before reaching a shallow bottom. Every action produces an equal and opposite reaction—that was the principle at work in the lawn sprinkler, as in a rocket. The inverse problem forced people to test their understanding of where, exactly, the reaction wielded its effects. At the point of the nozzle? Somewhere in the curve of the S, where the twisted metal forces the water to change course? Wheeler was asked for his own verdict one day. He said that Feynman had absolutely convinced him the day before that it went around backward; that Feynman had absolutely convinced him today that it went around forward; and that he did not yet know which way Feynman would convince him the next day.
If the mind was the most convenient of laboratories, it was not proving the most trustworthy. Because the
Gedankenexperiment
was failing, Feynman decided to bring the lawn-sprinkler problem back into the world of matter—stiff metal and wet water. He bent a piece of tubing into an S. He ran a piece of soft rubber hose into it. Now he needed a convenient source of compressed air.
The Palmer Physical Laboratory at Princeton housed a magnificent array of facilities, though not quite up to the standards of MIT. There were four large laboraories and several smaller ones, with a total floor space of more than two acres. Machine shops supplied electrical charging devices, storage batteries, switchboards, chemical equipment, and diffraction gratings. The third floor was devoted to a high-voltage laboratory capable of direct currents at 400,000 volts. A low-temperature laboratory had machinery for liquefying hydrogen. Palmer’s pride, however, was its new cyclotron, built in 1936. Feynman had made a point of wandering over the day after he arrived at Princeton and had tea with the Dean. By comparison, MIT’s even newer cyclotron was an elegant futuristic masterpiece of shiny metal and geometrically arrayed dials; when MIT had finally decided to invest in high-energy physics, it had not stinted. Princeton’s gave Feynman a shock. He made his way down into the basement of Palmer, opened the door, and saw wires hanging like cobwebs from the ceiling. Safety valves for the cooling system were exposed, and water dripped from them. Tools were scattered on tables. It could not have looked less like Princeton. He thought of his wooden-crate laboratory at home in Far Rockaway.
The mystery of the lawn sprinkler. When it sprays water, it spins counterclockwise.But what happens when it is made to suck water in?
Amid the chaos, it seemed reasonable enough for Feynman to borrow the use of an outlet for compressed air. He attached the rubber tube and pushed the end through a large cork. He lowered his miniature lawn sprinkler through the neck of a giant glass water bottle and sealed the bottle with the cork. Rather than try to suck water from the tube, he was going to pump air into the top of the bottle. That would increase the pressure of the water, which would then flow backward into the S-shaped pipe, up the rubber hose, and out the bottle.
