Adventures of a Mathematician

Preface to the 1991 edition by William G. Mathews and Daniel Hirsch

It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.

This remark of Stanislaw Ulam’s is particularly appropriate to his own career. Our world is very different today because of Ulam’s contributions in mathematics, physics, computer science, and the design of nuclear weapons.

While still a schoolboy in Lwów, then a city in Poland, he signed his notebook “S. Ulam, astronomer, physicist and mathematician.” Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam’s early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day. The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam’s highly verbal and collaborative style early on. Ulam’s early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam’s lifelong, highly creative quest for new mathematical and scientific problems.

As conditions in prewar Poland deteriorated, Ulam welcomed opportunities to visit Princeton and Harvard, eventually accepting a faculty position at the University of Wisconsin. As United States involvement in World War II deepened, Ulam’s students and professional colleagues began to disappear into secret government laboratories. Following a failed attempt to contribute to the Allied war effort by enlisting in the U. S. military, Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. It was at Los Alamos that Ulam’s scientific interests underwent a metamorphosis and where he made some of his most far-reaching contributions.

On his very first day at Los Alamos he was asked to work with Edward Teller’s group on the “Super” bomb project, an early attempt to design a thermonuclear or hydrogen bomb. Except for Teller’s small group, the scientists at Los Alamos were working on the design and construction of an atomic bomb based on the energy released by the fission or breakup of uranium or plutonium nuclei. Although there was a general consensus at Los Alamos that the fission bomb would have to precede the Super for which it would serve as an ignition device, Teller was already preoccupied with the Super and re-fused to work on the fission bomb calculations. As a means of retaining Teller at Los Alamos, Robert Oppenheimer as lab director allowed Teller to work on the Super bomb with several scientists and assistants. Teller’s assignment for Ulam on his arrival at Los Alamos was to study the exchange of energy between free electrons and radiation in the extremely hot gas anticipated in thermonuclear bombs. Ironically, this first-day problem for Ulam in 1943 would later become a critical part of Ulam’s work with Cornelius Everett in 1950 in which he demonstrated that Teller’s design for the Super bomb was impractical.

This first problem in theoretical physics was the beginning of a major scientific transition for Ulam from the esoteric, abstract world of pure mathematics to a quite different kind of applied mathematics necessary to visualize and solve problems in physics. The mathematics relevant to the physical problems at Los Alamos involved differential and integral equations that describe the motion of gas, radiation, and particles. The transition from pure mathematics to physics is seldom attempted and very rarely accomplished at Ulam’s level. The creative process and the initial guesswork that lead to significant new ideas in physics involve an added dimension of taste and judgment extending beyond the rigorous logic of mathematics alone. Physical intuition which “very few mathematicians seem to possess to any great degree” is constrained by knowledge of natural phenomena determined from experiment. Ulam claims not to have experienced this “gap between the mode of thinking in pure mathematics and the thinking in physics.” Indeed, in these memoirs Ulam discusses his transition from pure mathematics to mathematical physics and hopes that his analysis ’’of thinking in science is one of the possible interests of this book.”

Ulam could hardly have been in better company to learn physics. During the war years the scientists assembled at Los Alamos represented a Who’s Who of modern physical science. The large number of eminent physicists — Hans Bethe, Niels Bohr, Enrico Fermi, Richard Feynman, Ernest Lawrence, J. Robert Oppenheimer, and so on — formed an intellectual powerhouse of physics that has not been surpassed before or since.

During the war years Ulam contributed to the development of the fission bomb with statistical studies on the branching and multiplication of neutrons responsible for initiating and sustaining the chain reaction and energy release in uranium or plutonium. A critical problem on which Ulam worked with von Neumann was the detailed calculation of the implosion or compression of a sphere of uranium effected by an external chemical detonation. When uranium is compressed the small number of naturally occuring neutrons created by random fissions of uranium nuclei collide more easily with other uranium nuclei. Some of these collisions result in further fissions, multiplying further the number of neutrons until a rapid chain reaction ensues, ultimately releasing an extraordinary amount of energy in a powerful explosion. In order to predict the amount of energy released, Los Alamos scientists needed to estimate the detailed behavior of the uranium as it was being compressed. Although this problem was conceptually straightforward, accurate solutions were not possible using standard mathematical analyses. This problem was quite literally at the secret core of atomic bomb research at Los Alamos — even the word “implosion” was classified during the war.

But Ulam’s most remarkable achievement at Los Alamos was his contribution to the postwar develoment of the thermonuclear or hydrogen bomb in which nuclear energy is released when two hydrogen or deuterium nuclei fuse together. Ulam was a participant at a Los Alamos meeting in April, 1946, at which the wartime efforts on the Super bomb were discussed and evaluated. The conceptual idea of the “Classical Super” was to heat and ignite some part of a quantity of liquid deuterium by using an atomic bomb. The thermal energy deposited in this part would initiate deuterium reactions which would in turn heat adjacent regions, inducing further thermonuclear reactions, until the detonation would propagate through the entire amount of deuterium fuel. Deuterium, a heavier osotope of hydrogen having an extra neutron in its nucleus, was preferred since it reacts at significantly lower temperatures than ordinary hydrogen. Tritum, a third and even heavier form of hydrogen with two neutrons, reacts at even lower temperatures but, unlike deuterium, is virtually nonexistent in nature and was extremely expensive to make in nuclear reactors.

The evaluation of Teller’s Super project at the 1946 meeting was guardedly optimistic, but the participants were aware of major technical uncertainties and potential difficulties with the Super design. In discussing the conclusions reached at the meeting J. Carson Mark has written, “The estimates available of the behavior of the various steps and links in the sort of device considered were rather qualitative and open to question in detail. The main question of whether there was a specific design of that type which would work well was not answered.” Studies prior to 1946 had established that the net balance of energy gains over losses in the Super bomb was marginal; there was no large margin of design flexibility for which a successful detonation could be guaranteed. According to Mark,

As it was, the studies of this question had merely sufficed to show that the problem was very difficult indeed; that the mechanisms by which energy would be created in the system and uselessly lost from it were comparable; and that because of the great complexity and variety of processes which were important, it would require one of the most difficult and extensive mathematical analyses which had ever been contemplated to resolve the question — with no certainty that even such an attempt could succeed in being conclusive.

The uncertainties regarding the ignition and sustenance of fusion reactions in the Super bomb design as developed by Teller’s group during the war years were still present in late 1949 and early 1950. Nevertheless, this was the hydrogen bomb design that Teller lobbied for in Washington and that formed the basis of President Truman’s decision early in 1950 to accelerate work on the fusion bomb.

The two main questions about the Super design were (1) whether it would be possible to ignite some of the deuterium to get the thermonuclear reactions started, and (2) whether a thermonulcear reaction in the liquid deuterium, once started, would be self-sustaining or, alternatively, would slow down and fizzle away if the rate that energy is lost from the reacting regions exceeded that produced by the reactions. The ignition of the Super would require a gun-type atomic bomb trigger in which two subcritical masses of fissionable uranium would be rapidly united to form a supercritical explosive mass, as in the Hiroshima bomb. The ignition problem was difficult. The unusually high temperatures required for ignition would require a trigger A-bomb that would need to have a yield, reach temperatures, and use a quantity of fissionable material substantially in excess of the bombs in the arsenal in 1950. Even under the most favorable circumstances, the deuterium could not be ignited directly. It was thought that a small amount of tritium could be used to help initiate deuterium-burning in the region initially heated by the fission bomb.

The first a major difficulty for the Super, the problem of ignition, was attacked by Ulam on his own initiative but in collaboration with Cornelius Everett, a mathematical colleague of Ulam’s at the University of Wisconsin who had come to Los Alamos after the war at Ulam’s invitation. These calculations followed in detail the initial evolution of the nuclear reactions in tritium and deuterium and included an estimate of the heating of the unburnt nuclear fuel by the hot reacting regions with allowances made for the energy lost due to expansion and radiation. The Ulam-Everett calculation was tedius and exacting. While each step of the computation was understood, the complex interplay among the many components involved made the whole calculation extremely difficult. The exchange of energy between electrons and radiation. Ulam’s first problem at Los Alamos, was just one part of this monumental calculation. For several months Ulam and Everett worked in concentrated effort from four to six hours a day. Since each step in the calculation depended on the previous work, it was necessary to complete each stage virtually without error; fortunately, freedom from error was one of Everett’s specialties. It is hard to imagine today that these calculations were performed with slide rules and old-fashioned manually operated mechanical desk calculators. Ulam and Everett had to make many approximations and educated guesses in order for a solution to be possible at all. By this time Ulam had clearly mastered the physical intuition and judgment needed to make sensible estimates. When the calculation was finished, however, their conclusion was negative. The deuterium could not be ignited without spectacular amounts of tritum, amounts sufficient to make the entire Super project impractical and uneconomical. Within a few months the conclusions of the Ulam-Everett calculation were confirmed by von Neumann using an early electronic computer at Princeton.

The second uncertainty in the design of the Super was the question of the propagation of the deuterium-burning region throughout the entire amount of liquid deuterium. Would the fusion reaction be self-sustaining assuming that the ignition difficulty could somehow be overcome? This fundamental problem was solved by Ulam in collaboration with the brilliant physicist Enrico Fermi. Again using slide rules and desk calculators — and great care in making the appropriate physical approximations — they reached another negative conclusion; the heat lost form the deuterium burning region was too great to sustain the reaction. In discussing the conclusions of the Ulam-Fermi calculations, Fermi noted cautiously that “if the cross-sections for the nuclear reactions could somehow be two or three times larger than what was measured and assumed, the reaction could behave more successfully.” In fact the cross-sections (which characterize the rate that the reactions can occur) used by both Teller’s group and by Ulam and Fermi in 1950 were larger and therefore more otpimistic than the more accurate cross-sections obtained experimentally by James Tuck in the following year. In recent years the Ulam-Everett calcualtion has been redone in a much more refined manner using modern computers that have confirmed the marginal character of the self-sustaining propagation.

Within months of president Truman’s directive to expedite the development of a thermonuclear bomb, the two basic assumptions of Teller’s Super model were shown by Ulam and his colleagues to be incorrect. A crash program had begun on a project that was fundamentally flawed and which had never been seriously tested prior to Ulam’s work. According to Hans Bethe, Teller “was blamed at Los Alamos for leading the Laboratory, and indeed the whole country, into an adventurous program on the basis of calculations which he himself must have known to have been very incomplete.” The energy released by the deuterium reaction would be lost before adjacent regions could be ignited since, in Ulam’s explanation, “the hydrodynamical disassembly proceeded faster than the buildup and maintenance of the reaction.” Teller, who had worked on the Super during the war years and who later became a one-man political action committee urging a crash program for its construction, was distraught and practically undone by the Ulam-Everett-Fermi conclusions. Teller has written that “Ulam’s work indicated that we were on the wrong track, that the hydrogen bomb design we thought would work best would not work at all.”

The crisis of disappointment following these developments was quite stunningly resolved by Ulam in February, 1951, when he suggested a means of compressing the deuterium sufficiently to allow both ignition and self-sustaining propagation. According to Hans Bethe, director of the theoretical division at Los Alamos during the war, Ulam’s idea was to use “the propagation of [a] mechanical shock” (compression) wave from a fission explosion to induce a strong compression in the thermonuclear fuel, which would subsequently explode with great violence. The advantages of compression in helping to make thermonuclear reactions more efficient had been discussed even as early as the April 1946 meeting, but were never taken seriously since the compression required was far greater than could be achieved with chemical explosions. When Ulam told Teller of his idea of using a fission bomb to compress the deuterium just prior to its ignition, Teller immediately perceived the value of the idea. However, Teller suggested that the implosion could be achieved more conveniently by the action of radiation, with a so-called “radiation implosion,” rather than with the mechanical shock proposed by Ulam (which would also have worked). The new idea for the hydrogen bomb, known euphemistically as the “Teller-Ulam device,” was rapidly accepted by Los Alamos scientists and government officials. Since first proposed by Ulam, the coupling of a primary fission explosion with a secondary fusion explosion by means of implosion has been a standard feature of thermonuclear bombs.

All of these details concerning the origins of the hydrogen bomb, to the extent that we can put them together from declassified information, underscore Ulam as far more influential than has previously been known. Not only was he the first to dismantle the earlier Super concept which had been so inflexibly proposed for many years, he provided the key idea that resolved the difficulties of both ignition and propagation. In this instance, more than any other in Ulam’s scientific career, he demonstrated “how a few scribbles on a blackboard or on a sheet of paper” have quite radically and irreversibly changed the course of human affairs.”

In view of the impact that the arsenal of nuclear weapons has had on world affairs, it is intriguing that Ulam returns in his autobiography several times to discuss the mindset and social role of weapons scientists who sequester themselves in top secret laboratories to invent and construct instruments of potential mass destruction. Most of the scientists who worked at Los Alamos during World War II were shocked by the annihilation of Japanese cities and elected to return to academic life after the war. It is likely that many of those who stayed on at Los Alamos or returned later were inherently apolitical and, like Ulam, were “mainly interested in the scientific aspects of the work,” having “no qualms about returning to the laboratory to contribute to further studies of the development of atomic bombs.” Although Ulam later felt that the stockpile of nuclear weapons had grown larger than necessary in his view there was nothing intrinsically “bad” about the mathematics or the laws of nature used in creating new weapons. Knowledge itself is without moral content. In particular, Ulam ’’never had any questions about doing purely theoretical work” on nuclear weapons, leaving to others their construction and application to political and military ends.

Ulam makes a curious distinction between the acquisition of knowledge concerning new instruments of mass destruction by scientists and its wider dissemination: “I sincerely felt it was safer to keep these matters in the hands of scientists and people who are accustomed to objective judgments rather than in those of demagogues or jingoists, or even well-meaning but technically uninformed politicians.” However, in a government-funded laboratory such as Los Alamos, the symbiosis that exists between weapons technology and political decisions is inescapable. While Ulam insists that “one should not initiate projects leading to possibly horrible ends,” it would nevertheless be “unwise for the scientists to turn away from problems of technology” since ’’this could leave it in the hands of dangerous and fanatical reactionaries.” In spite of these apparent contradictions, Ulam’s justifications of his role in weapons development provide us with one of the few insights into the personal attitudes of a Los Alamos scientist toward the end products of his work.

By virtue of his defense work at the Los Alamos Laboratory, Ulam enjoyed many advantages not available to academic scientists. Chief among these was his early access to the most powerful and fastest computers in existence. For several decades after the war, the computing facilities at the national weapons labs far exceeded those available to university scientists working on non-classified research. This was an advantage that Ulam exploited in a variety of remarkable ways.

The growth of powerful computers was initially driven by the war effort. At the beginning of World War II there were no electronic computers in the modern sense, only a few electromechanical relay machines. During the war, scientists at the University of Pennsylvania and at the Aberdeen Proving Ground in Maryland developed the ENIAC, the Electronic Numerical Integrator and Computer, which had circuitry specifically designed for computing artillery firing tables for the Army. By modern standards, this early computer was extremely slow and elephantine: the ENIAC operating at the University of Pennsylvania in 1945 weighed thirty tons and contained about eighteen thousand vacuum tubes with 500,000 soldered connections. While on a visit to the University of Pennsylvania in 1944, John von Neumann was inspired to design an electronic computer that could be programmed in the modern sense, one which could be instructed to perform any calculation and would not be restricted to computing artillery tables. The new computer would have circuits that could perform sequences of fundamental arithmetic operations such as addition and multiplication. Von Neumann desired a more flexible computer to solve the mathematically difficult A-bomb implosion problem being discussed at Los Alamos. The first electronic computer at Los Alamos, however, known as the MANIAC (Mathematical Analyzer, Numerical Integrator and Computer), was not available until 1952.

One of Ulam’s early insights was to use the fast computers at Los Alamos to solve a wide variety of problems in a statistical manner using random numbers, a method which has become appropriately known as the Monte Carlo method. It occurred to Ulam during a game of solitaire that the probability of various outcomes of the card game could be determined by programming a computer to simulate a large number of games. Newly selected cards could be chosen from the remaining deck at random, but weighted by the probability that such a card would be the next selected. The computer would use random numbers whenever an unbiased choice was necessary. When the computer had played thousands of games, the probabilty of winning could be accurately determined. In principle the probability of solitaire success could be rigorously calculated using probabilty theory rather than computers. However, this approach is impossible in practice since it would involve too many mathematical steps and exceedingly large numbers. The advantage of the Monte Carlo method is that the computer can be efficiently programmed to execute each step in a particular game according to known probabilities and the final outcome can be determined to any desired precision depending on the number of sample games computed. The game of solitaire is an example of how the Monte Carlo method can be used to solve otherwise intractable problems with brute computational power.

An early application of the Monte Carlo method using high speed computers was to study the propagation of neutrons in fission bombs. This was accomplished by randomly picking the position of a radioactive nucleus that would release a neutron, then randomly selecting the neutron’s energy, its direction of motion, and the distance the neutron would travel before either escaping or colliding with the nucleus of another atom. In the latter event, the neutron would either be scattered, absorbed, or could induce nuclear fission according to probabilities again selected with random numbers. In this manner, after many neutron life experiences had been calculated, it was possible to determine the number of neutrons at any energy moving in a particular direction at any position in the apparatus. The Monte Carlo method is also well-suited to computing the equilibrium properties of materials, in estimating the efficiency of radiation or particle detectors having complicated geometries, and in simulating experimental data for a wide variety of physical problems.

Another early use of computer technology in which Ulam made contributions is the problem of determining the motion of compressible material. Indeed, it was the calculation of imploding compression waves in the fissionable core of atomic bombs that initially attracted Los Alamos scientists to the advantages of fast computers. One of Ulam’s contributions was his idea to represent the compressible material with an ensemble of representative points whose motion could be determined by the computer. Along similar lines, Ulam performed the first studies of the subtly complex collective motion of stars in a cluster, each mutually attracted to all the others by gravitational forces. The applications of computers to both compressible material and stellar systems along lines first explored by Ulam are major areas of research interest today.

Of particular interest is Ulam’s farsighted computer experiment in the mid-fifties with John Pasta and Enrico Fermi on the oscillations of a chain of small masses connected with slightly nonlinear springs. A nonlinear spring is one that does not quite stretch in exact proportion to the amount of force applied. When the group of masses simulated by the computer was started out in a particular rather simple motion, Ulam and his colleagues discovered to their amazement that the masses eventually returned nearly to the original motion but only after having gone through a bizarre and totally unanticipated intermediate evolution. Today computer studies of such nonlinear systems have become a major area of interdisciplinary scientific investigations. Many strange properties of dynamical systems have been discovered which have led to a deeper understanding of the long-term properties of nonlinear systems obeying deceptively simple physical laws.

A related computer experiment inspired by Ulam was the study of iterative nonlinear mappings. The computer is provided with a (nonlinear) rule for transforming one point in a mathematically defined region of space into another, then the same rule is applied to the new point and the process is continued for many iterations. When examined after only a few iterations, the pattern is generally uninteresting, but when a computer is used to generate thousands of iterations, Ulam and his colleague Paul Stein observed that a variety of strange patterns can result. In some cases after many iterations the points converge to a single point or are ordered along a curve within the given region of space. In other cases the successive images of iterated points appear to have disordered, chaotic properties. The final pattern of iterated images can be sensitive to the initial point chosen in generating them as well as the rules for (nonlinear) iteration. In recent years this early work of Ulam and Stein has been greatly extended at Los Alamos, now a major center of nonlinear studies.

Ulam also had an interest in the application of mathematics to biology. One example that may have biological relevance is the subfield of cellular automata founded by von Neumann and Ulam. As an example of this class of problems imagine dividing a plane into many small squares like a checkerboard with several objects placed in nearby squares. Then specify rules for the appearance of new objects (or the disappearance of old objects) in each square depending on whether adjacent squares are occupied or not. With each application of the rules to all the squares, the pattern of occupied squares evolves with time. Depending on the initial configuration and the rules of growth, some computer generated cellular automata evolve into patterns resembling crystals or snowflakes, others seem to have an ever-changing motion as if they were alive. In some cases colonies of self-replicating patterns expand to fill the available space like the growth of coral or bacteria in a petri dish.

Stanislaw Ulam was a man of many ideas and a fertile imagination. His creative and visionary talent planted intellectual seeds from Lwów to Los Alamos which have flourished into new disciplines of study throughout the world. Ulam’s scientific work was characterized by a singularly verbal style of inquiry begun during his early experiences in the coffeehouses of Lwów. The use of written material was also less essential for Ulam due to his formidable memory — he was able to recite many decades later the names of his classmates and to quote Greek and Latin poetry learned as a schoolboy. Ulam’s verbal and socially interactive approach was in fact well suited to the research environment at Los Alamos. Talented colleagues there were available to collaborate with Ulam, to provide the missing details of the ideas he sketched out, and to prepare the scientific papers and reports which changed the course of human affairs.

WILLIAM G. MATHEWS

DANIEL O. HIRSCH