Generative eBooks

10 More pictures

Generative eBooks — 2023-08-25T13:17:35Z

We have collected a small selection of pictures illustrating the preceding material. Needless to say that we have omitted a great deal of pictures that we would have liked to include.

- Computer experiments and visualization in mathematics and physics

9 Iterating complex polynomials : Hubbard and Mandelbrot

Generative eBooks — 2023-08-25T13:00:17Z

We end this article with complex dynamics, a field which was pioneered by the French mathematicians Pierre Fatou and Gaston Julia at the beginning of the 20th century. The fact that it became dormant for about sixty years is not by accident : without the visualization of Julia sets provided by computers, it is obvious that nobody had only the haziest notion of what these sets might look like when plotted on the plane. In the preface of the book “The Mandelbrot set, theme and variations” , (…)

- Computer experiments and visualization in mathematics and physics

8 The period-doubling cascade of Feigenbaum, Coullet and Tresser

Generative eBooks — 2023-08-25T12:52:45Z

In 1978, a young researcher named Pierre Coullet (born in 1949) from the University of Nice and Charles Tresser (born in 1950), a graduate student, got interested in the mechanisms leading from laminar to turbulent fluids. Such questions are notoriously hard and they decided to start with very simple mathematical models based on iterating a map of the unit interval into itself. As for Hénon's model, numerical experiments were decisive. The novelty they bring, which seems obvious nowadays, is (…)

- Computer experiments and visualization in mathematics and physics

7 Hénon : from astrophysics to strange attractors

Generative eBooks — 2023-08-25T12:45:35Z

Michel Hénon (1931-2013) has put numerical experiments at the center of his scien- tific practice to which he accorded the same status as physics experiments. He was interested in astrophysics – an area where experiments with controlled conditions are obviously impossible and numerical computations the only way to experiment. In the 1950s, Hénon worked on the construction of analogic computers be- fore making his own one. This was just before the advent and democratization of digital (…)

- Computer experiments and visualization in mathematics and physics

6 Kruskal, Zabusky, solitons, and the visiometrics

Generative eBooks — 2023-08-25T12:34:05Z

At the beginning of the 1960s, in USA, the mathematician and physicist Martin Kruskal (1925-2006) and the physicist Norman Zabusky (1929-2018) started to investigate again the FPUT system (see Section 2). They modified the nonlinear term in the interaction between springs and, together with their programmer Gary Deem, they conducted detailed numerical experiments leading to the discovery of a new phe- nomenon : they observed “solitary waves” they called “solitons” . Remarkably, two solitons (…)

- Computer experiments and visualization in mathematics and physics

5 Lorenz : from meteorology to strange attractors

Generative eBooks — 2023-08-25T12:22:23Z

At MIT in 1953, Edward Lorenz was put in charge of a project devoted to statistical forecasting, exploring how the newly available digital computer could be put to use. A key issue was to know how well such numerical tools, based on linear statistical models, could predict complex weather patterns. Lorenz was skeptical of the appropriateness of such tools as most atmospheric phenomena are non-linear. He sought a set of simple nonlinear differential equations that would mimic meteorological (…)

- Computer experiments and visualization in mathematics and physics

3 Turing, morphogenesis, and computers

Generative eBooks — 2023-08-25T12:10:20Z

Alan Turing (1912-1954) is best known for the creation of the “Turing machine” which he used to solve, in a masterful way, Hilbert's decision problem, and also for helping crack the code on intercepted Nazi messages helping the Allies win many major engagements during World War 2. Turing can also be considered a forefather of artificial intelligence with his seminal paper “Computing Machinery and Intelligence” published in 1950. In this section we give a very brief account on his pioneering (…)

- Computer experiments and visualization in mathematics and physics

2 Fermi, Pasta, Ulam and Tsingou : The FPUT experiments on MANIAC

Generative eBooks — 2023-08-25T12:09:57Z

To begin this section, let us quote Ulam :
“Computers were brand-new ; in fact the Los Alamos Maniac was barely finished. The Princeton von Neumann machine had met with technical and engineering difficulties that had prolonged its perfection. [...] As soon as the machines were finished, Fermi, with his great common sense and intuition, recognized immediately their importance for the study of problems in theoretical physics, astrophysics, and classical physics. We discussed this at length (…)

- Computer experiments and visualization in mathematics and physics

1 Von Neumann and Ulam, and “synergesis”

Generative eBooks — 2023-08-25T12:09:54Z

John von Neumann (1903-1957) and Stanislaw Ulam (1909-1984) are likely the very first people to have understood the potential of computers in mathematics and physics :
“Almost immediately after the war Johny and I also began to discuss the possibilities of using computers heuristically to try to obtain insights into questions of pure mathematics. By producing examples and by observing the properties of special mathematical objects one could hope to obtain clues as to the behavior of (…)

- Computer experiments and visualization in mathematics and physics

Introduction

Generative eBooks — 2023-08-25T12:09:50Z

The purpose of this article is to illustrate the impact of computer experiments on mathematics and the physical sciences from a historical perspective. We are especially interested in showing the role of visualization and interactivity. Let us start by giving the floor to John Hubbard, who will meet again in the section on complex dynamics :
“I mention computer graphics because faster and cheaper computers alone would not have had the same impact ; without pictures, the information pouring (…)

- Computer experiments and visualization in mathematics and physics