The Fermi-Pasta-Ulam-Tsingou Problem: A Foray Into The Beautifully Simple And The Simply Beautiful

In November 1918, a 17-year-student from Rome sat for the entrance examination of the Scuola Normale Superiore in Pisa, Italy’s most prestigious science institution. Students applying to the institute had to write an essay on a topic that the examiners picked. The topics were usually quite general, so the students had considerable leeway. Most students wrote about well-known subjects that they had already learnt about in high school. But this student was different. The title of the topic he had been given was “Characteristics of Sound”, and instead of stating basic facts about sound, he “set forth the partial differential equation of a vibrating rod and solved it using Fourier analysis, finding the eigenvalues and eigenfrequencies. The entire essay continued on this level which would have been creditable for a doctoral examination.” The man writing these words was the 17-year-old’s future student, friend and Nobel laureate, Emilio Segre. The student was Enrico Fermi. The examiner was so startled by the originality and sophistication of Fermi’s analysis that he broke precedent and invited the boy to meet him in his office, partly to make sure that the essay had not been plagiarized. After convincing himself that Enrico had done the work himself, the examiner congratulated him and predicted that he would become an important scientist.

Twenty five years later Fermi was indeed an important scientist, so important in fact that J. Robert Oppenheimer had created an entire division called F-Division under his name at Los Alamos, New Mexico to harness his unique talents for the Manhattan Project. By that time the Italian emigre was the world’s foremost nuclear physicist as well as perhaps the only universalist in physics – in the words of a recent admiring biographer, “the last man who knew everything”. He had led the creation of the world’s first nuclear reactor in a squash court at the University of Chicago in 1942 and had won a Nobel Prize in 1938 for his work on using neutrons to breed new elements, laying the foundations of the atomic age.

The purpose of F-division was to use Fermi’s unprecedented joint abilities in both experimental and theoretical physics to solve problems that stumped others. To Fermi other scientists would take their problems in all branches of physics, many of them current or future Nobel laureates. They would take advantage of Fermi’s startlingly simple approach to problem-solving, where he would first qualitatively estimate the parameters and solution and then plug in complicated mathematics only when necessary to drive relentlessly toward the solution. He had many nicknames including “The Roadroller”, but the one that stuck was “The Pope” because his judgement on any physics problem was often infallible and the last word.

Fermi’s love for semi-quantitative, order-of-magnitude estimates gave him an unusual oeuvre. He loved working out the most rigorous physics theories as much as doing back-of-the-envelope calculations designed to test ideas; the latter approach led to the famous set of problems called ‘Fermi problems‘. Simplicity and semi-quantitative approaches to problems are the hallmark of models, and Fermi inevitably became one of the first modelers. Simple models such as the quintessential “spherical cow in a vacuum” are the lifeblood of physics, and some of the most interesting insights have come from using such simplicity to build toward complexity. Interestingly, the problem that the 17-year-old Enrico had solved in 1918 would inspire him in a completely novel way many years later. It would be the perfect example of finding complexity in simplicity and would herald the beginnings of at least two new, groundbreaking fields.

Los Alamos was an unprecedented exercise in bringing a century’s worth of physics, chemistry and engineering to bear on problems of fearsome complexity. Scientists quickly realized that the standard tools of pen and paper that they had been using for centuries would be insufficient, and so for help they turned to some of the first computers in history. At that time the word “computer” meant two different things. One meaning was women who calculated. The other meaning was machines which calculated. Women who were then excluded from most of the highest echelons of science were employed in large numbers to perform repetitive calculations on complicated physics problems. Many of these problems at Los Alamos were related to the tortuous flow of neutrons and shock waves from an exploding nuclear weapon. Helping the female computers were some of the earliest punched card calculators manufactured by IBM. Although they didn’t know it yet, these dedicated women working on those primitive calculators became history’s first pioneering programmers. They were the forerunners of the women who worked at NASA two decades later on the space program.

Fermi had always been interested in these computers as a way to speed up calculations or to find new ways to do them. At Los Alamos a few other far-seeing physicists and mathematicians had realized their utility, among them the youthful Richard Feynman who was put in charge of a computing division. But perhaps the biggest computing pioneer at the secret lab was Fermi’s friend, the dazzling Johnny von Neumann, widely regarded as the world’s foremost mathematician and polymath and fastest thinker. Von Neumann who had been recruited by Oppenheimer as a consultant because of his deep knowledge of shock waves and hydrodynamics had become interested in computers after learning that a new calculating machine called ENIAC was being built at the University of Pennsylvania by engineers J. Presper Eckert, John Mauchly, Herman Goldstine and others. Von Neumann realized the great potential of what we today call the shared program concept, a system of programming that contains both the instructions for doing something and the process itself in the same location, both coded in the same syntax.

Fermi was a good friend of von Neumann’s, but his best friend was Stanislaw Ulam, a mathematician of stunning versatility and simplicity who had been part of the famous Lwów School of mathematics in Poland. Ulam belonged to the romantic generation of Central European mathematics, a time during the early twentieth century when mathematicians had marathon sessions fueled by coffee in Lwów, Vienna and Warsaw’s famous cafes, where they scribbled on the marble tables and argued mathematics and philosophy late into the night. Ulam had come to the United States in the 1930s; by then von Neumann had already been firmly ensconced at Princeton’s Institute for Advanced Study with a select group of mathematicians and physicists including Einstein. Ulam had started his career in the most rarefied parts of mathematics including set theory; he later joked that during the war he had to stoop to the level of manipulating actual numbers instead of merely abstract symbols. After the war started Ulam had wanted to help with the war effort. One day he got a call from Johnny, asking him to a move to a secret location in New Mexico. At Los Alamos Ulam worked closely with von Neumann and Fermi and met the volatile Hungarian physicist Edward Teller with whom he began a fractious, consequential working relationship.

Fermi, Ulam and von Neumann all worked on the intricate calculations involving neutron and thermal diffusion in nuclear weapons and they witnessed the first successful test of an atomic weapon on July 16th, 1945. All three of them realized the importance of computers, although only von Neumann’s mind was creative and far-reaching enough to imagine arcane and highly significant applications of these as yet primitive machines – weather control and prediction, hydrogen bombs and self-replicating automata, entities which would come to play a prominent role in both biology and science fiction. After the war ended, computers became even more important in the early 1950s. Von Neumann and his engineers spearheaded the construction of a pioneering computer in Princeton. After the computer achieved success in doing hydrogen bomb calculations at night and artificial life calculations during the day, it was shut down because the project was considered too applied by the pure mathematicians. But copies started springing up at other places, including one at Los Alamos. Partly in deference to the destructive weapons whose workings would be modeled on it, the thousand ton Los Alamos machine was jokingly christened MANIAC, for Mathematical Analyzer Numerical Integrator and Computer. It was based on the basic plan proposed by von Neumann which is still the most common plan used for computers worldwide – the von Neumann architecture.

After the war, Enrico Fermi had moved to the University of Chicago which he had turned into the foremost center of physics research in the country. Among his colleagues and students there were T. D. Lee, Edward Teller and Subrahmanyan Chandrasekhar. But the Cold War imposed on his duties, and the patriotic Fermi started making periodic visits to Los Alamos after President Truman announced in 1951 that he was asking the United States Atomic Energy Commission to resume work on the hydrogen bomb as a top priority. Ulam joined him there. By that time Edward Teller had been single-mindedly pushing for the construction of a hydrogen bomb for several years. Teller’s initial design was highly flawed and would have turned into a dud. Working with pencil and paper, Fermi, Ulam and von Neumann all confirmed the pessimistic outlook for Teller’s design, but in 1951, Ulam had a revolutionary insight into how a feasible thermonuclear weapon could be made. Teller honed this insight into a practical design which was tested in November 1952, and the thermonuclear age was born. Since then, the vast majority of thermonuclear weapons in the world’s nuclear arsenals have been based on some variant of the Teller-Ulam design.

By this time Fermi had acutely recognized the importance of computers, to such an extent in fact that in the preceding years he had taught himself how to code. Work on the thermonuclear brought Fermi and Ulam together, and in 1955 Fermi proposed a novel project to Ulam. To help with the project Fermi recruited a visiting physicist named John Pasta who had worked as a beat cop in New York City during the Depression. With the MANIAC ready and standing by, Fermi was especially interested in problems where highly repetitive calculations on complex systems could take advantage of the power of computing. Such calculations would be almost impossible in terms of time to perform by hand. As Ulam recalled later,

“Fermi held many discussions with me on the kind of future problems which could be studied through the use of such machines. We decided to try a selection of problems for heuristic work where in the absence of closed analytic solutions experimental work on a computing machine might perhaps contribute to the understanding of properties of solutions. This could be particularly fruitful for problems involving the asymptotic, long time or “in the large” behavior of non-linear physical systems…Fermi expressed often a belief that future fundamental theories in physics may involve non-linear operators and equations, and that it would be useful to attempt practice in the mathematics needed for the understanding of non-linear systems. The plan was then to start with the possibly simplest such physical model and to study the results of the calculation of its long-term behavior.”

Fermi and Ulam had caught the bull by its horns. Crudely speaking, linear systems are systems where the response is proportional to the input. Non-linear systems are ones where the response can vary disproportionately. Linear systems are the ones which many physicists study in textbooks and as students. Non-linear systems include almost everything encountered in the real world. In fact, the word “non-linear” is highly misleading, and Ulam nailed the incongruity best: “To say that a system is non-linear is to say that most animals are non-elephants.” Non-linear systems are the rule rather than the exception, and by 1955 physics wasn’t really well-equipped to handle this ubiquity. Fermi and Ulam astutely realized that the MANIAC was ideally placed to attempt a solution to non-linear problems. But what kind of problem would be complex enough to attempt by computer, yet simple enough to provide insights into the workings of a physical system? Enter Fermi’s youthful fascination with vibrating rods and strings.

The simple harmonic oscillator is an entity which physics students encounter in their first or second year of college. Its distinguishing characteristic is that the force applied to it is proportional to the displacement. But as students are taught, this is an approximation. Real oscillators – real pendulums, real vibrating rods and strings in the real world – are not simple. The force applied results in a complicated function of the displacement. Fermi and Ulam set up a system consisting of a string attached to one end. They considered four models; one where the force is proportional to the displacement, one where the force is proportional to the square of the displacement, one where it’s proportional to the cube, and one where the displacement varies in a discontinuous way with the force, going from broken to linear and back. In reality the string was modeled as a series of 64 points all connected through these different forces. The four graphs from the original paper are shown below, with force on the x-axis and displacement on the y-axis and the dotted line indicating the linear case.

Here’s what the physicists expected: the case for a linear oscillator, familiar to physics students, is simple. The string shows a single sinusoidal node that remains constant. The expectation was that when the force became non-linear, higher frequencies corresponding to two, three and more sinusoidal modes would be excited (these are called harmonics or overtones). The global expectation was that adding a non-linear force to the system would lead to an equal distribution or “thermalization” of the energy, leading to all modes being excited and the higher modes being heavily so.

What was seen was something that was completely unexpected and startling, even to the “last man who knew everything.” When the quadratic force was applied, the system did indeed transition to the two and three-mode system, but the system then suddenly did something very different.

“Starting in one problem with a quadratic force and a pure sine wave as the initial position of the string, we indeed observe initially a gradual increase of energy in the higher modes as predicted. Mode 2 starts increasing first, followed by mode 3 and so on. Later on, however, this gradual sharing of energy among successive modes ceases. Instead, it is one or the other mode that predominates. For example, mode 2 decides, as it were, to increase rather rapidly at the cost of all other modes and becomes predominant. At one time, it has more energy than all the others put together! Then mode 3 undertakes this role.”

Fermi and Ulam could not resist adding an exclamation point even in the staid language of scientific publication. Part of the discovery was in fact accidental; the computer had been left running overnight, giving it enough time to go through many more cycles. The word “decides” is also interesting; it’s as if the system seems to have a life of its own and starts dancing of its own volition between one or two lower modes; Ulam thought that the system was playing a game of musical chairs. Finally it comes back to mode 1, as if it were linear, and then continues this periodic behavior. An important way to describe this behavior is to say that instead of the initial expectation of equal distribution of energy among the different modes, the system seems to periodically concentrate most or all of its energy in one or a very small number of modes. The following graph for the quadratic case makes this feature clear: on the y-axis is energy while on the x-axis is the number of cycles ranging into the thousands (as an aside, this very large number of cycles is partly why it would be impossible to solve this problem using pen and paper in reasonable time). As is readily seen, the height of modes 2 and 3 is much larger than the higher modes.

The actual shapes of the string corresponding to this asymmetric energy exchange are even more striking, indicating how the lower modes are disproportionately excited. The large numbers here again correspond to the number of cycles.

The graphs for the cubic and broken displacement case are similar but even more complex, leading to higher modes being excited more often but the energy still concentrated into the lower modes. Needless to say, these results were profoundly unexpected and fascinating. The physicists did not quite know what to make of them, and Ulam found them “truly amazing”. Fermi told him that he thought they had made a “little discovery”.

The 1955 paper contains an odd footnote: “We thank Ms. Mary Tsingou for efficient coding of the problems and for running the computations on the Los Alamos MANIAC machine.” Mary Tsingou was the underappreciated character in the story. She was a Greek immigrant whose family barely escaped Italy before Mussolini took over. With bachelor’s and master’s degrees in mathematics from Wisconsin and Michigan, in 1955 she was a “computer” at Los Alamos, just like many other women. Her programming of the computer was crucial and non-trivial, but she was acknowledged in the work and not in the writing. She worked later with von Neumann on diffusion problems, was the first FORTRAN programmer, and even did some calculations for Ronald Reagan’s infamous “Star Wars” program. As of 2020, Mary Tsingou is still alive and 92 and living in Los Alamos. The Fermi-Pasta-Ulam problem should be called the Fermi-Pasta-Ulam-Tsingou problem.

Fermi’s sense of having made a “little discovery” has to be one of the great understatements of 20th century physics. The results that he, Ulam, Pasta and Tsingou obtained went beyond harmonic systems and the MANIAC. Until then there had been two revolutions in 20th century physics that changed our view of the universe – the theory of relativity and quantum mechanics. The third revolution was quieter and started with French mathematician Henri Poincare who studied non-linear problems at the beginning of the century. It kicked into high gear in the 1960s and 70s but still evolved under the radar, partly because it spanned several different fields and did not have the flashy reputation that the then-popular fields of cosmology and particle physics had. The field went by several names, including “non-linear dynamics”, but the one we are most familiar with is chaos theory.

As James Gleick who gets the credit for popularizing the field in his 1987 book says, “Where chaos begins, classical science stops.” Classical science was the science of pen and pencil and linear systems. Chaos was the science of computers and non-linear systems. Fermi, Ulam, Pasta and Tsingou’s 1955 paper left little reverberations, but in hindsight it is seminal and signals the beginning of studies of chaotic systems in their most essential form. Not only did it bring non-linear physics which also happens to be the physics of real world problems to the forefront, but it signaled a new way of doing science by computer, a paradigm that is the forerunner of modeling and simulation in fields as varied as climatology, ecology, chemistry and nuclear studies. Gleick does not mention the report in his book, and he begins the story of chaos with Edward Lorenz’s famous meteorology experiment in 1963 where Lorenz discovered the basic characteristic of chaotic systems – acute sensitivity to initial conditions. His work led to the iconic figure of the Lorenz attractor where a system seems to hover in a complicated and yet simple way around one or two basins of attraction. But the 1955 Los Alamos work got there first. Fermi and his colleagues certainly demonstrated the pull of physical systems toward certain favored behavior, but the graphs also showed how dramatically the behavior would change if the coefficients for the quadratic and other non-linear terms were changed. The paper is beautiful. It is beautiful because it is simple.

It is also beautiful because it points to another, potentially profound ramification of the universe that could extend from the non-living to the living. The behavior that the system demonstrated was non-ergodic or quasiergodic. In simple terms, an ergodic system is one which visits all its states given enough time. A non-ergodic system is one which will gravitate toward certain states at the expense of others. This was certainly something Fermi and the others observed. Another system that as far as we know is non-ergodic is biological evolution. It is non-ergodic because of historical contingency which plays a crucial role in natural selection. At least on earth, we know that the human species evolved only once, and so did many other species. In fact the world of butterflies, bats, humans and whales bears some eerie resemblances to the chaotic world of pendulums and vibrating strings. Just like these seemingly simple systems, biological systems demonstrate a bewitching mix of the simple and the complex. Evolution seems to descend on the same body plans for instance, fashioning bilateral symmetry and aerodynamic shapes from the same abstract designs, but it does not produce the final product twice. Given enough time, would evolution be ergodic and visit the same state multiple times? We don’t know the answer to this question, and finding life elsewhere in the universe would certainly shed light on the problem, but the Fermi-Pasta-Ulam-Tsingou problem points to the non-ergodic behavior exhibited by complex systems that arise from simple rules. Biological evolution with its own simple rules of random variation, natural selection and neutral drift may well be a Fermi-Pasta-Ulam-Tsingou problem waiting to be unraveled.

The Los Alamos report was written in 1955, but Enrico Fermi was not one of the actual co-authors because he had tragically died in November 1954, the untimely consequence of stomach cancer. He was still at the height of his powers and would have likely made many other important discoveries compounding his reputation as one of history’s greatest physicists. When he was in the hospital Stan Ulam paid him a visit and came out shaken and in tears, partly because his friend seemed so composed. He later remembered the words Crito said in Plato’s account of the death of Socrates: “That now was the death one of the wisest men known.” Just three years later Ulam’s best friend Johnny von Neumann also passed into history. Von Neumann had already started thinking about applying computers to weather control, but in spite of the great work done by his friends in 1955, he did not realize that chaos might play havoc with the prediction of a system as sensitive to initial conditions as the global climate. It took only seven years before Lorenz found that out. Ulam himself died in 1984 after a long and productive career in physics and mathematics. Just like their vibrating strings, Fermi, Ulam and von Neumann had ascended to the non-ergodic, higher modes of the metaphysical universe.

First posted on 3 Quarks Daily.

How chemistry exemplifies the Fermi method

In my review of the new biography of Enrico Fermi I alluded to one of Fermi's most notable qualities - his uncanny ability to reach rapid conclusions to tough problems based on order of magnitude, back of the envelope calculations. This method of approximation has since come to be known as the Fermi method, and problems which can especially benefit from applying it are called Fermi problems.

It struck me that chemistry is an especially fertile ground for applying the Fermi method, and in fact many chemists probably use the technique unconsciously in their daily work without explicitly realizing it. For understanding why this so, it's worth taking a look at some of the details of the method and the kinds of problems to which it can be fruitfully applied.

At the heart of the Fermi method is a way to make educated guesses about different factors and quantities that could affect the answer to a problem. Usually when you are looking at complicated problems not just in physics or chemistry but in psychology or economics for that matter, much of complex problem solving involves examining different factors that could influence the magnitude and nature of the solution. For instance, say you were calculating the trajectory of a bomb dropped from an airplane. In that case you would consider parameters like the velocity of the plane, the velocity of the bomb, air resistance, the weight of the bomb, the angle at which it was dropped etc. If you were trying to gauge the impact of a certain economic proposal on the economy, you would consider the market and demographic to which the proposal was applied, the presence or absence of existing elements which could interact positively or negatively with the proposed policy, rates of inflation, potential changes in the prices of certain goods relevant to the policy etc. The first part of the Fermi method simply involves writing down such factors and making sure you have a more or less comprehensive list.

The second part of the Fermi method consists of making educated guesses for each of these factors. The crucial aspect of this part is that you don't need to make highly accurate predictions for each factor to the fourth or fifth decimal place. In fact it was precisely this approach that made Fermi such a novelty in his time; it was because physicists could calculate quantities to four decimal places that they were often tempted to do this. Fermi showed that they didn't have to, and in some sense he weaned them away from this temptation. The fact of the matter is you don't always need a high degree of accuracy to reach actionable, semi-qualitative conclusions; you just need to know some rough numbers and get the answer right to an order of magnitude. That was the key insight from Fermi's technique.

Now, before I proceed and discuss how these two aspects of the Fermi method may apply to chemistry, it's worth noting that there are of course several examples in which an order of magnitude answer is simply not good enough. A famous example concerns the very Manhattan Project of which Fermi was such a valued member. In the early phases of the project when General Leslie Groves was picked as head of the project, he quizzed the scientists in Chicago about how much fissile material they would need. When they said that at that point all they give him was an answer correct to an order of magnitude, he was indignant and pointed out that that would be tantamount to ordering a wedding cake and not knowing whether to order enough cake for ten people or one person or a hundred people.

Notwithstanding such specific cases though, it's clear that there are in fact several example of general problems which can benefit from Fermi's technique. Chemistry in fact is a poster child for both the key aspects of the method illustrated above. Many problems in chemistry involve estimating the various kind of forces - electrostatic, hydrophobic, hydrogen bonds, Van der Waals - influencing the interaction of one molecule with another. For instance when a drug molecule is interacting with a protein, all these factors play an important role. Sometimes they synergize with each other and sometimes they oppose each other. Using the Fermi method then, you would first simply make sure you are listing all of them as comprehensively as possible. The goal is to come up with a total number resulting from all these contributions that would crucially provide you with the strength or free energy of interaction between the drug and the protein; a quantity measured in units of kcal/mol.

This part is where the method is especially useful. When you are trying to come up with numbers for each of these forces, it's valuable simply to know some ranges; you don't need to know the answers to three decimal places. For instance, you know that hydrogen bonds can contribute 2-5 kcal/mol, electrostatic interactions usually add 1-2 kcal/mol, and all the hydrophobic interactions will add a few kcal/mol to the mix. There are some trickier estimates such as those for the entropy of interaction, but there are also approximations for these. Sum up these interactions and you can come up with a reasonable estimate for the free energy of binding. The job becomes easier when what you are interested in are differences and not absolute values. For instance you may be given a list of small molecules and asked to rank these in order of their free energies. In those cases you just have to look at differences: for instance, if one molecule is forming an extra hydrogen bond and the other isn't, you can say that the first one is better by about 2-3 kcal/mol. You can also use your knowledge of experimental measurements for calibrating your estimates, another trait which Fermi supremely exemplified.

This then is the Fermi method of approximate guesses in action. One of the reasons it's far more prevalent in chemistry than physics is because unlike physics, in chemistry it's usually not even possible to calculate numbers to very high accuracy. Therefore unlike some physicists, chemists would not even be tempted to attempt to do this and would have already resigned themselves (if you will) to making do with approximate solutions. Today the Fermi method is incorporated in both the minds of seasoned working chemists as well as in computer programs which try to automate the process. Both the seasoned chemist and the computer program try to first list all the interactions between molecules and then try to estimate the strengths of each interaction based on rough numbers, adding up to a final value. 

The method does not work all the time since every interaction is modeled, so it may potentially miss some important real life component. But it works well enough for chemists and computers to employ it in a variety of useful tasks, from narrowing down the set of drug molecules that have to be made to prioritizing molecules for new materials and energy applications. Enrico Fermi's ghost lives on in test tubes, computers, fume hoods and spectrometers, more than even his wide-ranging mind could have imagined.

The man with an inside track to God: A new biography of Enrico Fermi

Scientists come in at least as many flavors as fruit. Some are inspired philosophers, others are get-your-hands-dirty mechanical craftsmen, yet others are like birds which can survey multiple parts of the scientific landscape from a very high altitude. But whatever other classification you may use, there are two distinctions which scientists have always exemplified. They can be either theoreticians or experimentalists, and especially these days, they are all specialists. In an age where it can take a lifetime to understand the complexities of even a narrow part of your science, excelling at every subfield of a scientific discipline, let alone both theory and experiment, would seem like an impossible feat.

Enter Enrico Fermi, the likes of whom we are unlikely to see for a very long time. Bucking almost every neat scientific distinction, Fermi was the only scientist of the twentieth century who was supremely accomplished in both theoretical and experimental physics. Almost any of his discoveries would have been enough to net a Nobel Prize, and yet he made at least half a dozen of them. In addition he was one of the three or four physicists of the century who were universalists, making contributions to and displaying a sound grasp of pretty much every branch of physics, from the microscopic to the cosmic. In my opinion, among his contemporaries only Hans Bethe, John von Neumann, Richard Feynman and Luis Alvarez came close to demonstrating the same breadth, and none of them excelled in both theory and experiment. You could ask Fermi any problem, and as long as he could calculate it he could give you an answer: no wonder that his colleagues called him the "Pope of Physics". It also helped that he lived through a century in which physics made momentous contributions to the human intellect and condition, and he was both fortunate and supremely qualified to be a major part of these contributions. As just one aspect of his extraordinary imprint on physics, no scientist has as many measurements, rules, laws, particles, statistics, units, and energy levels named after him as Fermi. He was also one of America's greatest immigrants.

This is a fine biography of Fermi written by Gino Segre and Bettina Hoerlin - a practicing physicist and a historian of science - who both had connections to Fermi through their families. Hoerlin's father worked on the Manhattan Project. Segre is the nephew of Emilio Segre, Nobel Prize-winning physicist and one of Fermi's closet friends and collaborators. The authors document Fermi's upbringing in Italy at the turn of the century. The Fermis came from a verdant, hilly region of Italy known for its industrious farming community, and throughout his life Fermi maintained his love for manual labor and the mountains, qualities endemic to many people from this region. His father was a railway inspector. Enrico was a child prodigy who combined great intellect with hard self-reliance and perseverance, qualities which were inculcated by his hardworking parents. A life-changing tragedy at age fifteen - the sudden death of his brother with whom he was best friends - turned him toward physics and mathematics. His performance as a seventeen year old in the entrance examination for a well-known university in Pisa displayed knowledge that would have been substantial for a graduate student. From then on his scientific development proceeded smoothly, and before he was 30 he was both Italy's leading physicist as well as one of the world's greatest scientists.

The book lays out many of Fermi's major discoveries. Two in particular bracket his unsurpassed talents as both a theoretician and an experimentalist. In 1933 Fermi came up with a mathematical theory of radioactive decay and the weak nuclear force. And in 1942 he and his team assembled the world's first nuclear reactor. It is almost impossible to imagine any other scientist accomplishing these two very different and very important feats; the famed historian C. P. Snow paid Fermi the ultimate tribute in this regard when he said that, had Fermi been born twenty years before, he could have discovered both Niels Bohr's quantum theory of the atom (theory) and Ernest Rutherford's atomic nucleus (experiment). In the 1930s Fermi and his team became the world expert on neutrons; life in the physics institute on Via Panisperna in Rome was bucolic in spite of being intense. He almost single-handedly discovered the power of slow neutrons which are used to harness nuclear energy in reactors. He and other leading physicists also narrowly missed discovering nuclear fission, mistaking fission products for elements beyond uranium. Rome under his scientific tutelage became a magnet for scientists like Hans Bethe and Edward Teller who learnt the art of problem-solving in physics from the master. Fermi's marriage to a very intelligent and resourceful woman, Laura, cemented his family life. But the pall of fascism was dropping on Italy through the person of Benito Mussolini. Laura was Jewish, and by 1938 Fermi realized that he had to emigrate to another country. Fortunately the receipt of the 1938 Nobel Prize gave him the perfect opportunity to flee to the United States. Along with other brilliant scientists like Bethe, Albert Einstein, Leo Szilard and John von Neumann, Fermi became one of fascism's greatest gifts to this country.

In the United States Fermi was already known as the leading nuclear physicist of his generation. When nuclear fission was discovered in Germany at the end of 1938, there were legitimate fears that the Nazis would harness it to build an atomic bomb. Efforts to investigate fission in the US kicked into high gear, especially after Pearl Harbor. It was not surprising that the scientific community turned toward Fermi to assemble the world's first nuclear reactor. The book's account of this tremendous feat involving black graphite bricks and faces, the squash stand at the university and the sometimes amusing consequences of secrecy is worth reading. First at Columbia and then memorably at Chicago, Fermi and his team achieved the first self-sustaining nuclear reaction on December 6, 1942: a coded telegram went out to the leaders of the Manhattan Project saying that the "Italian navigator had landed in the New World". Even if he had accomplished nothing else this would have been sufficient to enshrine Fermi's name in history. But he kept on making major contributions, first at Chicago and then at Los Alamos. At Los Alamos Fermi's universal expertise was so valued that Oppenheimer created an entire division named after him (the F division). He became a kind of all-round troubleshooter who could solve any problem in theoretical or applied physics, or in engineering for that matter. He had an uncanny feel for numbers, and became known for posing and solving 'Fermi problems' which benefited from quick, back of the envelope, order-of-magnitude estimates. The iconic realization of the Fermi method was during the world's first atomic test in New Mexico on July 16, 1945, when, as the shockwave reached him, Fermi threw pieces of paper into the air and calculated the yield of the test based on the distance at which they fell. This calculation compared favorably with more sophisticated measurements that took several days to acquire.

After the war Fermi became a professor at Chicago where he again served as a magnet for the new generation of physicists exploring the frontiers of particle physics and cosmology. He was an incredibly clear and succinct teacher, and gave his students a true feel for the entire landscape of physics. Teaching was not just limited to classrooms but spilled over into the lunch cafeteria and on hikes. Physicists like Freeman Dyson and Richard Feynman made pilgrimages to see him from around the country, and six of his students received Nobel Prizes. Even after winning enough accolades for a lifetime, he worked harder and more diligently than anyone else. His colleagues joked that he was the man with an inside track to God, so all-encompassing were his scientific and computing abilities. His notes on thermodynamics, quantum mechanics and nuclear physics are still available and they attest to his clarity. At Chicago he not only made important contributions to experimental particle physics but he also made the first forays into computing. The so-called Monte-Carlo method which allows one to explore features of a system by making random jumps bears his imprint.

While not a very sentimental man, Fermi's friendliness, integrity, modesty and impartial, non-emotional attitude endeared him to almost everyone he came in contact with. He was friendly and had an impish sense of humor, but while not cold was also not a warm person who engaged intimately with those around him; this quality led to a family life which while not unhappy was also not particularly joyous, and his relative lack of affection was reflected in the brisk relationship that Fermi had with his daughter and son. He despised politics but still served on important government committees because of his feelings of duty toward his adopted country. Remarkably, his neutrality through some very politically fraught times was not detested, and he was one of the very few scientists who was admired by people who were each other's sworn enemies. While he opposed the hydrogen bomb on moral terms and testified on behalf of Oppenheimer during the latter's infamous hearing, he also served as a consultant to Los Alamos once he realized that the Russians might also get the bomb; characteristically enough, he correctly predicted how long it would take them to build their first thermonuclear weapon. People looked to him for impartial guidance in almost every matter which could benefit from rational introspection.

Art and music baffled Fermi, but his rational analysis of these things only endeared him more to his friends and colleagues. At an art exhibit on the immigrant experience for instance, he calculated the ratio of the lengths of legs and heights of the immigrants in the photos and concluded that his own dimensions fit the statistical distribution. At Los Alamos he quickly memorized the rules of square dancing and danced with unerring accuracy but almost zero passion. His modesty and tendency to shun the limelight was also a great draw. He could as easily chat with janitors as with other Nobel Laureates. No task was beneath him, and his great ability to perform routine work without complaints or fatigue was instrumental in his success: whatever it took to solve a problem, Fermi would do it. When flabbergasted scientists asked him how he did it, Fermi would often reply with a smile, "C.i.f, con intuito formidable" ("with formidable intuition"). Often his distinguishing quality was pure stamina; whether it was a tennis match or a physics problem, he would beat the problem (and his opponents) into submission by sheer perseverance and doggedness. His manner of playing sports mirrored his manner of doing science: shun the style and elegance, and go straight and relentlessly for the solution using every technique at your disposal. The method of approximate guesses which came to be named after him has been used to estimate a wide variety of disparate numbers, from the number of extraterrestrial civilizations in the galaxy to the number of piano tuners in Chicago (his favorite example).

This giant of science was struck down by cancer in 1954 when he was still in his prime. The book talks about visits made by various famous scientists and friends to the hospital where he was installed after exploratory surgery indicated no hope. They could not believe that the indefatigable Enrico would soon be no more. All came away shaken, not because they saw an emotionally fraught man in pain but because they saw a perfectly calm and rational man who had reconciled himself with reality. He knew exactly what was happening to him and was making plans for publishing his last set of notes. Characteristically, he was measuring the rate of saline intake and calculating how many calories he was getting from it. When he came home and his wife rented a hospital bed for him, he predicted that he would only need it until the end of the month. True to his amazing calculating prowess, he passed away two days before the predicted date, on November 28, 1954.

This book in general lays out a warm and engrossing picture of Enrico Fermi. As I see it, it is up against two challenges. Firstly, it's relatively sparse on the science and does not always provide adequate background. In this context it is a light read and comes across unfavorably compared to Richard Rhodes' seminal book "The Making of the Atomic Bomb" which goes into great depth regarding Fermi's work, especially on the Chicago nuclear reactor. Rhodes' volume is also better on giving us a detailed picture of Fermi's contemporaries. Secondly, it cannot resist comparison with two old Fermi biographies. His wife Laura's endearing biography of him named "Atoms in the Family", published only a few months before his death, provides as intimate a picture of the personally reticent Fermi as we can expect. This book's view understandably is not as intimate. The same goes for "Enrico Fermi: Physicist", a biography of Fermi written by his friend, fellow Nobel laureate and uncle of one of the present book's authors, Emilio Segre. Segre was a top-notch physicist who worked with Fermi from the beginning and who does much recreating the early days of Fermi's childhood and his experiments in Rome. That description provides another personal touch which is again not as vivid in this volume.

Notwithstanding these comparisons, I am glad that Segre and Hoerlin wrote this book to introduce one of history's greatest and most unique scientists to a new generation. No scientist has contributed more practically and in a more versatile manner to modern physics. And few scientists have combined extraordinary and universal scientific talents with the kind of personal humility and decency that Fermi exemplified. For all this his life story needs to be known anew.