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
Preface to the 1983 edition
In writing a preface to another edition of this book I cannot resist the temptation to compare the present with the guesses and timid predictions I made about the future of science as it looked to me ten years ago. If anything, the present looks even more exciting than I had hoped. It is wonderful to observe how many unforeseen or unforeseeable facts and ideas have emerged. While I shall mention just a few of the many developments in recent science, it is important to realize that the rate at which we comprehend the universe is as vital as what we finally understand.
Progress in science and technology has proceeded at an ever-increasing pace, making the short period since I wrote this book as significant as any in the history of science. To see this one has only to think of the landings on the moon, the now commonplace launching of satellites, and the enormous discoveries made in astronomy and in the study of the earth itself.
Most notable has been the exponential growth in the technology of electronic computers, whose use pervades many aspects of daily life. Now elements of a "meta-theory" of computing are being outlined and problems of computability in the general sense, especially with respect to its limits, are being studied successfully.
I wonder what John von Neumann's reaction would have been had he lived to see it all. He prophesied the growing importance of the computer's role, but even he would probably have been amazed at the scope of the computer age and the rapidity of its appearance.
One could say that after the atomic age there came the computer age, which, in turn, made the space age possible. All space vehicles — rockets, satellites, projectiles, shuttles, and so on — depend on the feasibility of very fast calculations that must be instantly transmitted to them in outer space to correct their orbits. Before the advent of the fastest electronic computers this kind of remote control was not possible.
Recently a great wealth of observations in physics and astronomy has increased the perplexity of the description of the universe. The enigma of quasars is still unresolved. These quasi-stellar objects seem to be billions of light-years away with an intrinsic luminosity hundreds of times greater than that of the galaxies in their foreground. In the few years since I wrote this book, vast "empty regions" hundreds of millions of light-years wide have been found. These areas make us question the sameness and isotropy of the universe suggested by the apparent uniformity of the cosmic radiation remaining from the Big Bang. It is now widely believed that black holes do exist. They may explain the behavior of several observed astronomical objects. In addition, growing evidence supports the theory that violent processes cause gigantic explosions in starlike objects and galaxies.
To a mathematician like myself, the question, "Is the universe in space finite and bounded, or does it extend indefinitely?" remains the number-one problem of cosmogony and cosmology.
In physics, the number of new, fundamental, or primary particles is constantly increasing. Quarks seem more and more to represent real, not merely mathematical, constituents of matter, but their number and nature remain unverifiable, and scientists are considering the existence of subparticles, such as gluons.