Джеймс Глик - Genius - The Life and Science of Richard Feynman

Piano lessons dismayed him, too, not just because he played so poorly, but because he kept playing an exercise cal ed “Dance of the Daisies.” For a while this verged on

obsession. Anxiety would strike when his mother sent him to the store for “peppermint patties.”

As a natural corol ary he was shy about girls. He worried about getting in fights with stronger boys. He tried to ingratiate himself with them by solving their school problems or showing how much he knew. He endured the canonical humiliations: for example, watching helplessly while some neighborhood children turned his first chemistry set into a brown, useless, sodden mass on the sidewalk in front of his house. He tried to be a good boy and then worried, as good boys do, about being too good—“goody-good.” He could hardly retreat from intel ect to athleticism, but he could hold off the taint of sissiness by staying with the more practical side of the mental world, or so he thought. The practical man—that was how he saw himself.

At Far Rockaway High School he came upon a series of mathematics primers with that magical phrase in the title

— Arithmetic for the Practical Man ; Algebra for the Practical Man —and he devoured them. He did not want to let himself be too “delicate,” and poetry, literature, drawing, and music were too delicate. Carpentry and machining were activities for real men.

For students whose competitive instincts could not be satisfied on the basebal field, New York’s high schools had the Interscholastic Algebra League: in other words, math team. In physics club Feynman and his friends studied the wave motions of light and the odd vortex phenomenon of smoke rings, and they re-created the already classic experiment of the California physicist Robert Mil ikan, using

suspended oil drops to measure the charge of a single electron. But nothing gave Ritty the thril of math team.

Squads of five students from each school met in a classroom, the two teams sitting in a line, and a teacher would present a series of problems. These were designed with special cleverness. By agreement they could require no calculus—nothing more than standard algebra—yet the routines of algebra as taught in class would never suffice within the specified time. There was always some trick, or shortcut, without which the problem would just take too long.

Or else there was no built-in shortcut; a student had to invent one that the designer had not foreseen.

According to the fashion of educators, students were often taught that using the proper methods mattered more than getting the correct answer. Here only the answer mattered. Students could fil the scratch pads with gibberish as long as they reached a solution and drew a circle around it. The mind had to learn indirection and flexibility. Head-on attacks were second best. Feynman lived for these competitions. Other boys were president and vice president, but Ritty was team captain, and the team always won. The team’s number-two student, sitting directly behind Feynman, would calculate furiously with his pencil, often beating the clock, and meanwhile he had a sensation that Feynman, in his peripheral vision, was not writing—never wrote, until the answer came to him. You are rowing a boat upstream. The river flows at three miles per hour; your speed against the current is four and one-quarter. You lose your hat on the water. Forty-five minutes later you realize it

is missing and execute the instantaneous, acceleration-free about-face that such puzzles depend on. How long does it take to row back to your floating hat?

A simpler problem than most. Given a few minutes, the algebra is routine. But a student whose head starts fil ing with 3s and 4¼s, adding them or subtracting them, has already lost. This is a problem about reference frames. The river’s motion is irrelevant—as irrelevant as the earth’s motion through the solar system or the solar system’s motion through the galaxy. In fact al the velocities are just so much foliage. Ignore them, place your point of reference at the floating hat—think of yourself floating like the hat, the water motionless about you, the banks an irrelevant blur—

now watch the boat, and you see at once, as Feynman did, that it wil return in the same forty-five minutes it spent rowing away. For al the best competitors, the goal was a mental flash, achieved somewhere below consciousness.

In these ideal instants one did not strain toward the answer so much as relax toward it. Often enough Feynman would get this unstudied insight while the problem was stil being read out, and his opponents, before they could begin to compute, would see him ostentatiously write a single number and draw a circle around it. Then he would let out a loud sigh. In his senior year, when al the city’s public and private schools competed in the annual championship at New York University, Feynman placed first.

For most people it was clear enough what mathematics was—a cool body of facts and rote algorithms, under the established headings of arithmetic, algebra, geometry,

trigonometry, and calculus. A few, though, always managed to find an entry into a freer and more colorful world, later cal ed “recreational” mathematics. It was a world where rowboats had to ferry foxes and rabbits across imaginary streams in nonlethal combinations; where certain tribespeople always lied and others always told the truth; where gold coins had to be sorted from false-gold in just three weighings on a balance scale; where painters had to squeeze twelve-foot ladders around inconveniently sized corners. Some problems never went away. When an eight-quart jug of wine needed to be divided evenly, the only measures available were five quarts and three. When a monkey climbed a rope, the end was always tied to a balancing weight on the other side of a pul ey (a physics problem in disguise). Numbers were prime or square or perfect. Probability theory suffused games and paradoxes, where coins were flipped and cards dealt until the head spun. Infinities multiplied: the infinity of counting numbers turned out to be demonstrably smal er than the infinity of points on a line. A boy plumbed geometry exactly as Euclid had, with compass and straightedge, making triangles and pentagons, inscribing polyhedra in circles, folding paper into the five Platonic solids. In Feynman’s case, the boy dreamed of glory. He and his friend Leonard Mautner thought they had found a solution to the problem of trisecting an angle with the Euclidean tools—a classic impossibility. Actual y they had misunderstood the problem: they could trisect one side of an equilateral triangle, producing three equal segments, and they mistakenly

assumed that the lines joining those segments to the far corner mark off equal angles. Riding around the neighborhood on their bicycles, Ritty and Len excitedly imagined the newspaper headlines: “Two Children in High School First Learning Geometry Solve the Age-Old Problem of the Trisection of the Angle.”

This cornucopian world was a place for play, not work.

Yet unlike its stolid high-school counterpart it actual y connected here and there to real, adult mathematics.

Il usory though the feeling was at first, Feynman had the sense of conducting research, solving unsolved problems, actively exploring a live frontier instead of passively receiving the wisdom of a dead era. In school every problem had an answer. In recreational mathematics one could quickly understand and investigate problems that were open. Mathematical game playing also brought a release from authority. Recognizing some il ogic in the customary notation for trigonometric functions, Feynman invented a new notation of his own: √x for sin √x for cos (x),

√x for tan (x). He was free, but he was also extremely methodical. He memorized tables of logarithms and practiced mental y deriving values in between. He began to fil notebooks with formulas, continued fractions whose sums produced the constants π and e.