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Fractals and Chaos: The Mandelbrot Set and Beyond

Benoit B. Mandelbrot
Springer Verlag
Publication Date: 
Number of Pages: 
[Reviewed by
Mihaela Poplicher
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This book contains early papers by Benoit Mandelbrot, as well as additional chapters describing the historical background and context. The material is grouped under five topics:

  • I Quadratic Julia and Mandelbrot Sets
  • II Nonquadratic Rational Dynamics
  • III Iterated Nonlinear Function Systems and the Fractal Limit Sets of Kleinian Groups
  • IV Multifractal Invariant Measures
  • V Background and History

Most of the papers included have been published before, beginning with the early 1980s until 2003, but there a few new ones. The work included in this book, "Selecta Volume C" was done by Mandelbrot while he was working at the IBM T. J. Watson Research Center and at Yale University. The book is dedicated to the memory of the author's uncle, Szolem Mandelbrojt, himself a mathematician who greatly influenced his nephew Benoit. The book also includes many illustrations, some of them very easily recognizable.

In his Foreword, Professor Peter W. Jones of Yale University notes: "It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. The graphics available at that time seem primitive today, and Mandelbrot's working drafts were even harder to interpret. But how that picture has changed our views of the mathematical and physical universe!" And later: "What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems; it also contains a view on how to approach the mathematical and physical universe."

In his Preface to the book, Mandelbrot emphasizes the fact that, although the book's main goal is to show the interconnections between fractals and chaotic dynamical systems, "this is neither a monograph on those interconnections, nor a textbook."

Of course the mathematical papers are extremely interesting, and a collection of all of them put together by their author is really a treat, but what I have found even more fascinating (and more entertaining to read, even for non-specialists) are the papers dealing with background, historical notes, biographical notes, commentaries, etc. Most of these have not been published before, so there is no hope finding them in another place. I will mention just a few examples, leaving the readers to discover the others for themselves.

  • Chapter C1, "Introduction to papers on quadratic dynamics: a progression from seeing to discovering", has tantalizing sections such as "Computing at Harvard in 1980" and "The culture of mathematics during the 1960s and 1970s".

  • Chapter C2, "Acknowledgments related to quadratic dynamics", contains wonderful references and tributes to big names in mathematics: Nicolas Bourbaki, Andre Weil, Jean Dieudonné, Laurent Schwartz, Marshall Stone, Gaston Julia, and others (including Szolem Mandelbrojt, the uncle who was Mandelbrot's "earliest and Foremost mentor").

  • Chapter C15, "Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments", contains sections such as "The early history of Poincaré's great innovation, one he chose to call 'Kleinian" groups'" and "Was the progress from pictures for their own sake, to open new mathematical vistas pre-ordained?" and "The notion of IFS (iterated function system or schemes) or decomposable dynamical systems".

  • The last part of the book, "Part V: Background and History", is my favorite. Here is how the author introduces it: "Some chapters in this part are introductions whose aim is to assist even the non-expert in gaining something from this book". And this is exactly what C23, "The inexhaustible function z squared plus c", C24 , "The Fatou and Julia stories", and C25, "Mathematical analysis while in the wilderness", are doing.

In summary, this is a wonderful book for a large group of readers: non-experts interested in some introduction to Mandelbrot's work and biography, with historical notes and commentaries; as well as for specialists learning and researching in quadratic and nonquadratic dynamics, Julia and Mandelbrot sets, Kleinian limit sets, Minkowski measure. Reading this book was a pleasure.

Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is

Foreword by Peter W Jones*Preface*Quadratic Julia and Mandelbrot Sets*NonQuadratic Rational Dynamics*Iterated Nonlinear Function Systems and the Fractal Limit Sets of Kleinian Groups*Multifractal Invariant Measures*Background and History*Bibliography* Index.