★★★★★ Chaos mathematics coffee table book

The Beauty of Fractals: Images of Complex Dynamical Systems

Heinz-Otto Peitgen, Peter H. Richter

I first became aware of chaos mathematics in 1976, when Robert M. May published his famous Nature review "Simple mathematical models with very complicated dynamics", a preliminary exploration of what we now call the Logistic Map. Chaos research proceeded apace through the late 1970s and early 1980s, to the point where some of the questions May had raised were answered. It also became evident about then that there were connections between chaos mathematics and fractals, which had been explored earlier by Benoît B. Mandelbrot, who published The Fractal Geometry of Nature in 1977. In particular, there are deep connections between the logistic map explored by May and a particular fractal discovered by Mandelbrot that became known as the Mandelbrot Set.

Now, in 2024, many people have heard of the Mandelbrot Set and almost everyone has seen pictures of it. Before 1984 such pictures barely existed. Heinz-Otto Peitgen and Peter H. Richter, respectively professors of Mathematics and Physics at the University of Bremen, set about to produce color graphic representations of the Mandelbrot Set and other fractals. They had access to what at the time passed for powerful computers. Some of their pictures appeared at an exhibition entitled "Frontiers of Chaos". This book is a collection of pictures from the exhibition. In addition, the text of the book explains much of the math behind the pictures.

There are few things more subjective than beauty, so it will be understood that I express only a personal opinion when I say that the pictures are stunningly beautiful. I had seen images of the Mandelbrot Set and other fractals before I received this book as a gift, but none that I had seen were like this. The cover image, in particular, was one of the most beautiful pictures I had ever seen, and I still hold it so almost thirty years later.

In the thirty years since Peitgen and Richter produced these pictures powerful computers have become far more widely available. I include here a fractal image I produced on my desktop computer in 2015 as my answer to one of the questions in a problem set in Functional Analysis, a course I took as a first-year grad student in Applied Mathematics. This image reproduces Figure 42 of The Beauty of Fractals: Images of Complex Dynamical Systems, except that that image is in drab black and white, whereas mine, since I did the problem set over Halloween, I did in orange, black, and yellow. On my desktop it took mere seconds to produce this image.

Color images in The Beauty of Fractals are much prettier than this homegrown example.

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