Weather Forecasts

Chapter 4 tells the story of numerical weather forecasting from its inception to today’s supercomputing algorithms. In 1922, Lewis Fry Richardson proposed that, since the atmosphere is subject to the laws of physics, future weather can be predicted by means of algorithmic calculations. His attempt at forecasting a single day’s weather by means of manual calculations took several months. In the late 1940s, John von Neumann resurrected Richardson’s idea and launched a project to conduct the first weather forecast by computer. The world’s first operational electronic computer – ENIAC - completed a 24-hour forecast in just one day. It appeared that accurate forecasting simply required faster computers. In 1969, Edward Lorenz discovered that tiny errors in weather measurements can accumulate during numerical forecasting to produce large errors. The so-called Butterfly Effect was alleviated by the Monte Carlo simulation method invented by Stanislaw Ulam for particle physics.

Classifying Graphs

This chapter considers the richness of mathematics and mathematicians' responses to it, with a particular focus on various types of graphs. It begins with a discussion of theorems from many areas of mathematics that have been judged among the most beautiful, including the Euler Polyhedron Formula; the number of primes is infinite; there are five regular polyhedra; there is no rational number whose square is 2; and the Four Color Theorem. The chapter proceeds by describing regular graphs, irregular graphs, irregular multigraphs and weighted graphs, subgraphs, and isomorphic graphs. It also analyzes the degrees of the vertices of a graph, along with concepts and ideas concerning the structure of graphs. Finally, it revisits a rather mysterious problem in graph theory, introduced by Stanislaw Ulam and Paul J. Kelly, that no one has been able to solve: the Reconstruction Problem.

ENIAC Goes to Monte Carlo

Traces the series of Monte Carlo simulations run on ENIAC from their genesis in January 1947 exchanges between John von Neumann, Robert Richtmyer, and Stanislaw Ulam through the completion of detailed planning work for the initial batch of calculations in December 1947. Close attention to successive drafts illuminates the process by which John and Klara von Neumann worked with Adele Goldstine to transform the former’s outline plan of computation into a fully developed flow diagram documenting the flow of control and manipulation of data for a program written in the new style.

Stanisław Ulam 1909-1984

Prodigiously talented with a remarkable flair for anticipating correct results and initiating fruitful areas of research, Stanislaw Ulam (‘Stan’ to his friends) was an unusual mathematician. The considerable breadth of his preoccupations, even in the halcyon days of his youth in Poland, distinguished him from his peers. Sustained by a supreme self-confidence, Ulam preferred pioneering new fields of mathematics to elaborating the ideas of others. Indeed, impatient with detail he tended to leave technicalities to those with whom he collaborated. Uprooted from his native Poland in his twenties, Ulam was to spend two thirds of his working life associated with or employed by the Los Alamos National Laboratory.