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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 chal enge 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 hal 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, cal ed in retrospect the Feynman diagram, that showed al 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 Al en 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 fel ows 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 intel ectual 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 Jel -O with a spoon and shouting

“Don’t bother me!” He was trying to see how the Jel -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 stil 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 smal 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 Sitwel , 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 gleeful y.

“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 spil ing out into his fingertips, restlessly tapping on desks and notebooks. “While a universe grows in my head,—” Sitwel wrote,

I have dreams, though I have not a bed—

The thought of a world and a day

When al may be possible, stil come my way.

Forward or Backward?

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, pul ing 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 stil had the capacity to surprise. One did not have to probe far into physicists’ understanding of Newton’s laws before reaching a shal ow 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