Genius (31 page)

For Feynman, thinking in his spare time about the pure theory of particles and light, diffusion dovetailed peculiarly with quantum mechanics. The traditional diffusion equation bore a family resemblance to the standard Schrödinger equation; the crucial difference lay in a single exponent, where the quantum mechanical version was an imaginary factor,
i
. Lacking that
i
, diffusion was motion without inertia, motion without momentum. Individual molecules of perfume carry inertia, but their aggregate wafting through air, the sum of innumerable random collisions, does not. With the
i
, quantum mechanics could incorporate inertia, a particle’s memory of its past velocity. The imaginary factor in the exponent mingled velocity and time in the necessary way. In a sense, quantum mechanics was diffusion in imaginary time.

The difficulties of calculating practical diffusion problems forced the Los Alamos theorists into an untraditional approach. Instead of solving neat differential equations, they had to break the physics into steps and solve the problem numerically, in small increments of time. The focus of attention was pushed back down to the microscopic level of individual neutrons following individual paths. Feynman’s quantum mechanics was evolving along strikingly similar lines. His private work, like the diffusion work, embodied an abandonment of a too simple, too special differential approach; the emphasis on step-by-step computation; and above all the summing of paths and probabilities.

Computing by Brain

Walking around the hastily built wooden barracks that housed the soul of the atomic bomb project in 1943 and 1944, a scientist would see dozens of men laboring over computation. Everyone calculated. The theoretical department was home to some of the world’s masters of mental arithmetic, a martial art shortly to go the way of jiujitsu. Any morning might find men such as Bethe, Fermi, and John von Neumann together in a single small room where they would spit out numbers in a rapid-fire calculation of pressure waves. Bethe’s deputy, Weisskopf, specialized in a particularly oracular sort of guesswork; his office became known as the Cave of the Hot Winds, producing, on demand, unjustifiably accurate cross sections (shorthand for the characteristic probabilities of particle collisions in various substances and circumstances). The scientists computed everything from the shapes of explosions to the potency of Oppenheimer’s cocktails, first with rough guesses and then, when necessary, with a precision that might take weeks. They estimated by seat of the pants, as a cook who wants one-third cup of wine might fill half a juice glass and correct with an extra splash. Anyone who calculated logarithms by mentally interpolating between the entries in a standard table—a technique that began to vanish thirty years later, when inexpensive electronic calculators made it obsolete—learned to estimate this way, using some unconscious feeling for the right curve. Feynman had a toolbox of such curves in his head, precalibrated. His Los Alamos colleagues were sometimes amused to hear him, when thinking out loud, howl a sort of whooping glissando when he meant,
this rises exponentially
; a different sound signified
arithmetically
. When he started managing groups of people who handled laborious computation, he developed a reputation for glancing over people’s shoulders and stabbing his finger at each error: “That’s wrong.” His staff would ask why he was putting them to such labor if he already knew the answers. He told them he could spot wrong results even when he had no idea what was right—something about the smoothness of the numbers or the relationships between them. Yet unconscious estimating was not really his style. He liked to know what he was doing. He would rummage through his toolbox for an analytical gimmick, the right key or lock pick to slip open a complicated integral. Or he would try various simplifying assumptions:
Suppose we treat some quantity as infinitesimal.
He would allow an error and then measure the bounds of the error precisely.

It seemed to colleagues that some of his computation was a matter of conscious reputation building. One day Feynman, who had made a point of considering watches to be affectations, received a pocket watch from his father. He wore it proudly, and his friends began to needle him; they asked the time at every opportunity, until he began responding, with a glance at the watch: “Well, four hours and twenty minutes ago it was twelve before noon,” or “In three hours and forty-nine minutes it will be two seventeen.” Few caught on. He was doing no arithmetic at all. Rather, he had designed a simple parlor trick in the spirit of gauge theories to come. Each morning he would turn his watch to a fixed offset from the true time—three hours and forty-nine minutes fast one day; the next day four hours and twenty minutes slow. He had only to remember one number and read the other directly from the watch. (This was the same Feynman who, years later, trying to describe to a layman the intricate shiftings of time and orientation on which theoretical physics depended, said, “You know how it is with daylight saving time? Well, physics has a dozen kinds of daylight saving.”)

When Bethe and Feynman went up against each other in games of calculating, they competed with special pleasure. Onlookers were often surprised, and not because the upstart Feynman bested his famous elder. On the contrary, more often the slow-speaking Bethe tended to outcompute Feynman. Early in the project they were working together on a formula that required the square of 48. Feynman reached across his desk for the Marchant mechanical calculator.

Bethe said, “It’s twenty-three hundred.”

Feynman started to punch the keys anyway. “You want to know exactly?” Bethe said. “It’s twenty-three hundred and four. Don’t you know how to take squares of numbers near fifty?” He explained the trick. Fifty squared is 2,500 (no thinking needed). For numbers a few more or less than 50, the approximate square is that many hundreds more or less than 2,500. Because 48 is 2 less than 50, 48 squared is 200 less than 2,500—thus 2,300. To make a final tiny correction to the precise answer, just take that difference again—2—and square it. Thus 2,304.

Feynman had internalized an apparatus for handling far more difficult calculations. But Bethe impressed him with a mastery of mental arithmetic that showed he had built up a huge repertoire of these easy tricks, enough to cover the whole landscape of small numbers. An intricate web of knowledge underlay the techniques. Bethe knew instinctively, as did Feynman, that the difference between two successive squares is always an odd number, the sum of the numbers being squared. That fact, and the fact that 50 is half of 100, gave rise to the squares-near-fifty trick. A few minutes later they needed the cube root of 2½. The mechanical calculators could not handle cube roots directly, but there was a look-up chart to help. Feynman barely had time to open the drawer and reach for the chart before he heard Bethe say, “That’s 1.35.” Like an alcoholic who plants bottles within arm’s reach of every chair in the house, Bethe had stored away a device for anywhere he landed in the realm of numbers. He knew tables of logarithms and he could interpolate with unerring accuracy. Feynman’s own mastery of calculating had taken a different path. He knew how to compute series and derive trigonometric functions, and how to visualize the relationships between them. He had mastered mental tricks covering the deeper landscape of algebraic analysis—differentiating and integrating equations of the kind that lurk dragonlike in the last chapters of calculus texts. He was continually put to the test. The theoretical division sometimes seemed like the information desk at a slightly exotic library. The phone would ring and a voice would ask, “What is the sum of the series 1 + (½)
⁴
+ (⅓)
⁴
+ (¼)
⁴
+ … ?”

“How accurate do you want it?” Feynman replied.

“One percent will be fine.”

“Okay,” Feynman said. “One point oh eight.” He had simply added the first four terms in his head—that was enough for two decimal places.

Now the voice asked for an exact answer. “You don’t need the exact answer,” Feynman said.

“Yeah, but I know it can be done.”

So Feynman told him. “All right. It’s pi to the fourth over ninety.”

He and Bethe both saw their talents as labor-saving devices. It was also a form of jousting. At lunch one day, feeling even more ebullient than usual, he challenged the table to a competition. He bet that he could solve any problem within sixty seconds, to within ten percent accuracy, that could be stated in ten seconds. Ten percent was a broad margin, and choosing a suitable problem was hard. Under pressure, his friends found themselves unable to stump him. The most challenging problem anyone could produce was: Find the tenth binomial coefficient in the expansion of (1 +
x
)
²⁰
. Feynman solved that just before the clock ran out. Then Paul Olum spoke up. He had jousted with Feynman before, and this time he was ready. He demanded the tangent of ten to the hundredth. The competition was over. Feynman would essentially have had to divide one by ? and throw out the first one hundred digits of the result—which would mean knowing the one-hundredth decimal digit of ?. Even Feynman could not produce that on short notice.

He integrated. He solved equations taking the spirit of infinite summation into more difficult realms. Some of these perilous, nontextbook, nonlinear equations could be integrated through just the right combination of mental gimmicks. Others could not be integrated exactly. One could plug in numbers, make estimates, calculate a little, make new estimates, extrapolate a little. One might visualize a polynomial expression to approximate the desired curve. Then one might try to see whether the leftover error could be managed. One day, making his rounds, Feynman found a man struggling with an especially complicated varietal, a nonlinear three-and-a-half-order equation. There was a business of integrating three times and figuring out a one-half derivative—and in the end Feynman invented a shortcut, a numerical method for taking three integrals at once and a half integral besides, all more accurately than had been thought possible. Similarly, working with Bethe, he invented a new and general method of solving third-order differential equations. Second order had been manageable for several centuries. Feynman’s invention was precise and practical. It was also doomed to a quick obsolescence in an age of machine computation, as was, for that matter, the skill of mental arithmetic that did so much to establish Feynman’s legend.

Computing by Machine

Not only the atomic era but also the computer era had its start in those years. Scattered about the nation’s military and civilian laboratories, a few researchers focused exclusively on the means of calculating instead of the calculations themselves. At Los Alamos, in particular, the demand for numerical computation grew more intense than anywhere else on earth. The means were mechanical and now partly electronic, though the crucial technological key, the transistor, remained to be invented at the decade’s end. Calculating technology became a hybrid with machine parts and human parts: people carrying cards from place to place served as the memories and logical-branching units of near computers that stretched across rows and columns of desks.

The bomb project could draw on the best technology available anywhere, but the best technology offered little to the working scientist. The manufacturers of such equipment—the International Business Machines Corporation already preeminent among them—considered the scientific market to be negligible. It could not imagine the vast clientele that would soon consume as much calculating capacity as could be created: for forecasting weather, designing engines, analyzing proteins, scheduling airplanes, and simulating everything from ecosystems to heart valves. Business was thought to be the sole potential consumer for business machines, and business meant accounting, which meant addition and subtraction. Multiplication seemed a luxury, although it might be necessary to multiply monthly sales by twelve. Division by machine was esoteric. Computation of mortgage payments and bond yields could be managed by humans with standard tables.

The workhorse of scientific calculating was the Marchant calculator, a clattering machine nearly as large as a typewriter, capable of adding, subtracting, multiplying, and with some difficulty dividing numbers of up to ten digits. (At first, to save money, the project ordered slower, eight-digit versions as well. They were rarely used.) In these machines a carriage spun around, propelled at first by a hand crank and later by an electric motor. Keys and levers pushed the carriage left or right. Counter and register dials displayed painted digits. There were rows and columns of keys for entering numbers, a plus bar and a minus bar, a multiplier key and a negative multiplier key, shift keys, and a key for stopping the machine when division went out of control, as it often did. Mechanical arithmetic was no simple affair. With all its buttons and linkages the Marchant was not quite as powerful as the giant Difference Engine and Analytical Engine, invented in England a century before by Charles Babbage in hopes of generating the printed tables of numbers on which navigators, astronomers, and mathematicians had to rely. Not only did Babbage solve the problem of carrying digits from one decimal place to the next; his machines actually used punched cards, borrowed from mechanical looms, to convey data and instructions. In the era of steam power, few of his contemporaries appreciated the point.

The Marchants took a hard pounding at Los Alamos. Metal parts wore thin and came out of alignment. The officially nonexistent laboratory was poorly suited to field-service visits by the manufacturer’s repair crews, so standard procedure required the shipping of broken machines back to California. Eventually three or four machines were in the pipeline at any one time. Feynman, frustrated, turned to Nicholas Metropolis, a mustached Greek mathematician who later became an authority on computation and numerical methods, and said, “Let’s learn about these damned things and not have to send them to Burbank.” (Feynman grew a temporary mustache, too.) They spent hours taking apart new and old machines for comparative diagnosis; learned where the jams and slippages began; and hung out a shingle advertising, “Computers Repaired.” Bethe was not amused at this waste of his theoreticians’ time. He finally ordered a halt to the tinkering. Feynman complied, knowing that within weeks the shortage of machines would change Bethe’s mind.