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.
Folds and Rhythms
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.
Forward or Backward?
For a while the tea-time conv
ersation 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.
He turned on the air valve. The apparatus gave a slight tremble, and water started to dribble from the cork. More air—the flow of water increased and the rubber tube seemed to shake but not to twist, at least not with any confidence. Feynman opened the valve farther, and the bottle exploded, showering water and glass across the room. The head of the cyclotron banished Feynman from the laboratory henceforth.
Sobering though Feynman’s experimental failure was, for years to come he and Wheeler both delighted in telling the story, and they were both scrupulous about never revealing the answer to the original question. Feynman had worked it out correctly, however. His physical intuition had never been sharper, nor his ability to translate fluently between a palpable sense of the physics and the formal mathematical equations. His experiment had actually worked, until it exploded. Which way does the lawn sprinkler turn? It does not turn at all. As the nozzles suck water in, they do not pull themselves along, like a rope climber pulling himself up hand over hand. They have no purchase on the water ahead. And the idea of force exerted as a torque within the curve of the S is beside the point. In the normal version, water sprays forth in organized jets. The action and reaction are straightforward and measurable. The momentum of the water spraying in one direction equals the momentum that spins the nozzle in the opposite direction. But in the inverse case, when water is sucked in, there are no jets. The water is not organized. It enters the nozzle from all directions and therefore applies no force at all.
A development in twentieth-century entertainment technology—the motion picture—incidentally provided an advance in the technology of thought experiments. It was now natural for a scientist, in his mind’s laboratory, to play the film backward. In the case of the lawn sprinkler, reversibility proved to be an illusion. If the flow of the water were visible, a motion picture of an ordinary lawn sprinkler played backward would look distinctly different from the sucking lawn sprinkler played forward. Filmmakers themselves had been seduced by the new, often comical insights that could be gained by taking a strip of celluloid and running it backward through the projector. Divers sprang feet first from lakes as a spray of water collapsed into the space left behind. Fires drew smoke from the air and created a trail of new-made paper. Fragmented eggshells assembled themselves around shuddering chicks.
For Feynman and Wheeler reversibility was becoming a central issue at the level of atomic processes, where spins and forces interacted more abstractly than in a lawn sprinkler. It was well known that the equations describing the motions and collisions of objects ran equally well forward and backward. They were symmetrical with respect to time, at least where just a few objects were concerned. How embarrassing, therefore, that time seemed so one-way in the real world, where a small amount of energy could scramble an egg or shatter a dish and where unscrambling and unshattering were beyond the power of science. “Time’s arrow” was already the catchphrase for this directionality, so evident to common experience, yet so invisible in the equations of physicists. There, in the equations, the road from past to future looked identical to the road from future to past. “There is no signboard to indicate that it is a one-way street,” complained Arthur Eddington. The paradox had been there all along, since Newton at least, but relativity had highlighted it. The mathematician Hermann Minkowski, by visualizing time as a fourth dimension, had begun to reduce past-future to the status of any pair of directions: left-right, up-down, back-front. The physicist drawing his diagrams obtains a God’s-eye view. In the space-time picture a line representing the path of a particle through time simply exists, past and future visible together. The four-dimensional space-time manifold displays all eternity at once.
The laws of nature are not rules controlling the metamorphosis of what is into what will be. They are descriptions of patterns that exist, all at once, in the whole tapestry. The picture is hard to reconcile with our everyday sense that time is special. Even the physicist has his memories of the past and his aspirations for the future, and no space-time diagram quite obliterates the difference between them.
Philosophers, in whose province such speculations had usually belonged, were left with
a muddy and senescent set of concepts. The distress of the philosophers of time spilled into their adverbs: sempiternally, hypostatically, tenselessly, retrodictably. Centuries of speculation and debate had left them unprepared for the physicists’ sudden demolition of the notion of simultaneity (in the relativistic universe it meant nothing to say that two events took place at the same time). With simultaneity gone, sequentiality was foundering, causality was under pressure, and scientists generally felt themselves free to consider temporal possibilities that would have seemed farfetched a generation before.