by James Gleick
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.
Escalation of the computation effort came in the fall of 1943 with an order to IBM for business machines to be delivered to an unknown location: three 601 multipliers, one 402 tabulator, one reproducer-summary punch, one verifier, one keypunch, one sorter, and one collator. Astronomers at Columbia had been experimenting with punch-card computing before the war. A multiplier, an appliance the size of a restaurant stove, could process calculations in large batches. Electrical probes found the holes in the cards, and operations could be configured by plugging groups of wires into a patchboard. Among the computation-minded at Los Alamos, the prospect of such machines caused excitement. Even before they arrived, one of the theorists, Stanley Frankel, set about devising improvements: for example, tripling the output by rearranging the plugs so that three sets of three- or four-digit numbers could be multiplied in a single pass. Having requisitioned the machines, the scientists now also requisitioned a maintenance man—an IBM employee who had been drafted into the army. They were gaining adroitness at military procurement. The crates arrived two days before the repairman; in those two days Feynman and his colleagues managed to get the machines unpacked and assembled, after a fashion, with the help of nothing but a set of wiring blueprints. So much more powerful were they that Feynman—sensitive to rhythms as always—rapidly discovered that he could program them to clatter out the cadence of well-known songs. The theorists began to organize something new in the annals of computation: a combination of the calculating machine and the factory assembly line. Even before the IBM machines arrived Feynman and Metropolis set up an array of people—mostly wives of scientists, working at three-eighths salary—who individually handled pieces of complex equations, one cubing a number and passing it on, another performing a subtraction, and so on. It was mass production married to numerical calculation. The banks of women wielding Marchants simulated the internal workings of a computer. As a later generation would discover, there was something mentally seductive in the act of breaking calculus into the algorithmic cogs needed for machine computation. It forced the mind back down into the essence of arithmetic. It also began a long transformation in the understanding of what kinds of equations were solvable. Stacks of punch cards could solve equations for a ball of fire rising through a suddenly turbulent atmosphere, by stepping through successive approximations for time 0:01, time 0:02, time 0:03 … though by the lights of traditional analysis those sharply nonlinear equations were unsolvable.
Of the many problems put to the Los Alamos computers, none better anticipated the coming age of massive scientific simulation than implosion itself: how to calculate the motion of an inward-flowing shock wave. An explosive charge wrapped around the bomb was to set the shock wave in motion, and the pressure would crush a nugget of plutonium into criticality. How should the bomb assembly be configured to assure a stable detonation? What kind of fireball would ensue? Such questions required a workable formula for the propagation of a spherical detonation wave in a compressible fluid, the “
compressible fluid” in this case being the shotput-size piece of plutonium liquefied in the microseconds before it became a nuclear blast. The pressure would be more intense than at the earth’s center. The temperature would reach 50 million degrees Centigrade. The theorists were on their own here; experimentalists could offer little more than good wishes. All during 1944 the computation effort grew. John von Neumann served as a traveling consultant with an eye on the postwar future. Von Neumann—mathematician, logician, game theorist (he was more and more a fixture in the extraordinary Los Alamos poker game), and one of the fathers of modern computing—talked with Feynman while they worked on the IBM machines or walked though the canyons. He left Feynman with two enduring memories. One was the notion that a scientist need not be responsible for the entire world, that social irresponsibility might be a reasonable stance. The other was a faint, early recognition of the mathematical phenomena that would later be called chaos: a persistent, repeatable irregularity in certain equations as they prepared to run them through their primitive computers. As a shock wave, for example, passed though a material, it left oscillations in its wake. Feynman thought at first that the irregular wiggles must be numerical errors. Von Neumann told him that the wiggles were actually features of interest.
Von Neumann also kept these new computer specialists up to date with the other sites he visited. He brought news of an electromechanical Mark I under construction at Harvard, a relay calculator at Bell Laboratories, human neuronal research at the University of Illinois, and at the Aberdeen Proving Ground in Maryland, where problems of ballistic trajectories motivated the calculators, a more radical device with a new kind of acronym: ENIAC, for Electronic Numerical Integrator and Computer, a machine composed of eighteen thousand vacuum tubes. The tubes controlled binary on-off flip-flops; in a bow to the past, the flip-flops were arranged in rings of ten, to simulate the mechanical wheels used in decimal calculating machines. The ENIAC had too many tubes to survive. Von Neumann estimated: “Each time it is turned on, it blows two tubes.” The army stationed soldiers carrying spare tubes in grocery baskets. The operators borrowed mean free path terminology from the ricocheting particles of diffusion theory; the computer’s mean free path was its average time between failures.
Meanwhile, under the influence of this primal dissection of mathematics, Feynman retreated from pragmatic engineering long enough to put together a public lecture on “Some Interesting Properties of Numbers.” It was a stunning exercise in arithmetic, logic, and—though he would never have used the word—philosophy. He invited his distinguished audience (“all the mighty minds,” he wrote his mother a few days later) to discard all knowledge of mathematics and begin from first principles—specifically, from a child’s knowledge of counting in units. He defined addition, a + b, as the operation of counting b units from a starting point, a. He defined multiplication (counting b times). He defined exponentiation (multiplying b times). He derived the simple laws of the kind a + b = b + a and (a + b) + c = a + (b + c), laws that were usually assumed unconsciously, though quantum mechanics itself had shown how crucially some mathematical operations did depend on their ordering. Still taking nothing for granted, Feynman showed how pure logic made it necessary to conceive of inverse operations: subtraction, division, and the taking of logarithms. He could always ask a new question that perforce required a new arithmetical invention. Thus he broadened the class of objects represented by his letters a, b, and c and the class of rules by which he was manipulating them. By his original definition, negative numbers meant nothing. Fractions, fractional exponents, imaginary roots of negative numbers—these had no immediate connection to counting, but Feynman continued pulling them from his silvery logical engine. He turned to irrational numbers and complex numbers and complex powers of complex numbers—these came inexorably as soon as one from facing up to the question: What number, i, when multiplied by itself, equals negative one? He reminded his audience how to compute a logarithm from scratch and showed how the numbers converged as he took successive square roots often and thus, as an inevitable by-product, derived the “natural base” e, that ubiquitous fundamental constant. He was recapitulating centuries of mathematical history—yet not quite recapitulating, because only a modern shift of perspective made it possible to see the fabric whole. Having conceived of complex powers, he began to compute complex powers. He made a table of his results and showed how they oscillated, swinging from one to zero to negative one and back again in a wave that he drew for his audience, though they knew perfectly well what a sine wave looked like. He had arrived at trigonometric functions. Now he posed one more question, as fundamental as all the others, yet encompassing them all in the round recursive net he had been spinning for a mere hour: To what power must e be raised to reach i? (They already knew the answer, that e and i and ? were conjoined as if by an invisible membrane, but as he told his mother, “I went pretty fast & didn’t give them a hell of a lot of time to work out the reason for one fact before I was showing them another still more amazing.”) He now repeated the assertion he had written elatedly in his notebook at the age of fourteen, that the oddly polyglot statement eπi + 1 = 0 was the most remarkable formula in mathematics. Algebra and geometry, their distinct languages notwithstanding, were one and the same, a bit of child’s arithmetic abstracted and generalized by a few minutes of the purest logic. “Well,” he wrote, “all the mighty minds were mightily impressed by my little feats of arithmetic.”
Indeed, if Feynman was, as his friend Welton thought, consciously trying to establish himself among these influential physicists, he was succeeding even more than he knew. As early as November 1943, seven months after the Los Alamos project began, Oppenheimer began trying to persuade his department at Berkeley to hire Feynman for after the war. He wrote to the department chairman, Birge:
He is by all odds the most brilliant young physicist here, and everyone knows this. He is a man of thoroughly engaging character and personality, extremely clear, extremely normal in all respects, and an excellent teacher with a warm feeling for physics in all its aspects.
Oppenheimer warned that Feynman was sure to have other job offers, because “a not inconsiderable number of ‘big shots’” had already noticed him. He quoted two of the big shots. Bethe, according to Oppenheimer, had said bluntly that he would sooner lose any two scientists than lose Feynman. And Wigner of Princeton had made what was, for a physicist’s physicist in the 1940s, perhaps the ultimate tribute.
“He is a second Dirac,” Wigner said, “only this time human.”
Fenced In
Feynman celebrated his wedding anniversary by grilling steak outdoors at the Presbyterian Sanatorium in a small charcoal broiler that Arline had ordered from a catalog. She also got him a chef’s hat, apron, and gloves. He wore them self-consciously, along with his new mustache, while she reveled in the domesticity of it all, until he could no longer stand the idea of people watching him from passing cars. She laughed, asking, as she so often did, why he cared what other people thought. Steak was an extravagance—eighty-four cents for two pounds. With it they ate watermelon, plums, and potato chips. The hospital lawn sloped down to Route 66, the cross-country highway, where the traffic roared by. Albuquerque was sweltering, and they were happy. Arline talked to her parents by long-distance telephone for seven minutes, another extravagance. After Richard left to hitchhike back north, a late-afternoon thundershower blackened the desert. Arline worried about him in the downpour. She still had not gotten used to the raw force of storms in the open West.
His near-weekly trips through the valley that lay between the Jemez and Sangre de Cristo mountains made him a rarity on the mesa. Few residents of that hermetic community had occasion to leave at all. Once, in a fanciful conversation about likely candidates to be a Nazi spy, one friend, Klaus Fuchs, a German turned Briton, suggested that it could only be Dick Feynman—who else had insinuated himself into so many different parts of the laboratory’s work? Who else had a regular rendezvous in Albuquerque
? In its unreal isolation, with its unusual populace, Los Alamos was growing into a parody of a municipality. It took its place in the mental geography of its residents as it was officially: not a village in the lee of the Jemez Mountains, not only a fenced-in circle of houses on dirt paths by a pond, with ducks, but also a fictitious abstraction, P. O. Box 1663, Santa Fe, New Mexico. To some it carried an ersatz resonance of a certain European stereotype of America, as one resident noted—“a pioneer people starting a new town, a self-contained town with no outside contacts, isolated in vast stretches of desert, and surrounded by Indians.” Victor Weisskopf was elected mayor. Feynman was elected to a town council. The fence that marked the city line heightened a magic-mountain atmosphere—it kept the world apart. An elite society had assembled on this hill. Elite and yet polyglot—in this cauldron, as in the other wartime laboratories, a final valedictory was being written to the Protestant, gentlemanly, leisurely class structure of American science. Los Alamos did gather an aristocracy—“the most exclusive club in the world,” one Oxonian said—yet the princely, exquisitely sensitive Oppenheimer made it into a democracy, where no invisible lines of rank or status were to impede the scientific discourse. The elected councils and committees furthered that impression. Graduate students were supposed to forget that they were talking to famous professors. Academic titles were mainly left behind with the business suits and neckties. It was a democracy by night, too, when inflamed parties brought together cuisines and cocktails of four continents, dramatic readings and political debates, waltzes and square dances (the same Oxonian, bemused amid the clash of cultures, asked, “What exactly is square about it—the people, the room, or the music?”), a Swede singing torch songs, an Englishman playing jazz piano, and Eastern Europeans playing Viennese string trios. Feynman played brassy drum duets with Nicholas Metropolis and organized conga lines. He had never been exposed to culture as such a flamboyant stew (certainly not when he was a student learning to disdain the packaged morsels that MIT handed to its would-be engineers). One party featured an original ballet, to modernistic-sounding music by Gershwin, titled Sacre du Mesa. At the end a clattering, flashing mechanical brain noisily revealed the sacred mystery of the mesa: 2 + 2 = 5.
