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Fractals and Chaos : The Mandelbrot Set and Beyond - Benoit B. Mandelbrot

Fractals and Chaos

The Mandelbrot Set and Beyond

By: Benoit B. Mandelbrot, C.J.G. Evertsz (As told to), M.C. Gutzwiller (As told to), P.W. Jones (Foreword by)


Published: 9th January 2004
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It has only been a couple of decades since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot set. That picture, now seeming graphically primitive, has changed our view of the mathematical and physical universe. The properties and circumstances of the discovery of the Mandelbrot Set continue to generate much interest in the research community and beyond. This book contains the hard-to-obtain original papers, many unpublished illustrations dating back to 1979 and extensive documented historical context showing how Mandelbrot helped change our way of looking at the world.

From the reviews:

"It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot Set. The graphics were state of the art, though now they may seem primitive. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scientific conscience that it is difficult to remember the psychological shock of their arrival. 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, but contains a view on how to approach the mathematical and physical universe. This view is certain not to fade, but to be part of the working philosophy of the next mathematical revolution, wherever it may take us. So read the book, look at the beautiful pictures that continue to fascinate and amaze, and enjoy! "
--From the foreword by Peter W Jones, Yale University

Praise for Mandelbrot, Benoit B, Fractals and Chaos

American Scientist

"Manderlbrot brings together 25 papers from the past 25 years. Many of them are related in one way or another to the famous inkblot figure to which Mandelbrot's name is now firmly affixed. Of historical interest are some early images of this fractal object, produced with a crude dot-matrix printer. A few items in the collection have not been previously published, and all are accompanied by feisty commentary."

"This is the fourth volume of Mandelbrot's Selecta, comprising edited reprints of the author's papers. ... One chapter has been written specifically to help the non-expert appreciate the rest of the book. ... accessible to a wide readership. It provides a fascinating insight into the author's journey of seeing and discovering as the early pictures of the Mandelbrot set started to reveal a whole new world. It gives a feeling for his philosophy and approach of experimental mathematics ... ." (Kenneth Falconer, NATURE, Vol. 430, July1, 2004)

"The well-illustrated book combines hard-to-find early papers by the author with additional chapters that describe the historical background and context. Key topics are quadratic dynamics and its Julia and Mandelbrot sets, nonquadratic dynamics, Kleinian limit sets, and the Minkowski measure." (Zentralblatt fur Didaktik der Mathematik, Juli, 2004)

"Mandelbrot's book is a scientific, philosophic and pictorial treatise at the same time and it is one of the rare specimens of serious mathematics books that can be read and re-read at many different levels. ... The style is what one would call `truly Mandelbrotian', a mixture of hard science, often with a personal touch ... . a book that will be as important for the scientific community ... as it will be appealing to a general informed audience." (Rene L. Schilling, The Mathematical Gazette, March, 2005)

"Benoit Mandelbrot has produced a comprehensive, well-presented review of essential topics ... . All chapters are assembled in a way that the overall mix becomes a very well integrated source of know-how ... . The author made a serious and effective effort to realize a book that contains more than history, more than mathematics ... . it is an excellent book also for supporting courses at University, PhD and Post doc level. Moreover, it is indispensable for scientists ... ." (Biomedical Engineering OnLine, May, 2005)

"Chaotic and fractal work has become popular among the mathematicians. ... The introductions to reprinted classical texts ... . illustrate how a theory emerges, how answers create new questions, and how progress is made. The reader will be grateful not only for these historical explanatory remarks ... . Collectively they give a compelling account of how a new branch of mathematics was created by the author. This delightful book makes good reading." (Helmut Kirchner, Pure and Applied Geophysics, Vol. 162, 2005)

"Benoit Mandelbrot, an award winning, academic nomad ... is credited with changing the way scientists in many fields, look at the world. ... founded the influential fields of fractal and multifractal geometry. ... The book is well populated with black and white graphical computer outputs of fractals and conceptual illustrations. Never before published illustrations ... are included. ... Recommended for all university science libraries." (James A. Buczynski, E-Streams, Vol. 7 (10), 2004)

"This book is a collection of early papers by Benoit B. Mandelbrot, combined with additional papers ... . The reader surely will appreciate the number of pictures and illustrations ... . The book will be valued by both mathematicians and physicists who are interested in the field. It is written in a clear and straightforward way whilst pointing out lots of applications. A non-mathematician will appreciate the minimal number of formulas ... . the mathematician's eye will be pleased by the exactness of the presentation." (EMS Newsletter, December, 2005)

"This book contains early papers by Benoit Mandelbrot, as well as additional chapters ... . the mathematical papers are extremely interesting, and a collection ... is really a treat, but what I found even more fascinating (and more entertaining to read, even for non-specialists) are the papers dealing with background, historical notes, bibliographical notes, commentaries etc. ... this is a wonderful book for a large group of readers ... . Reading this book was a pleasure." (Mihaela Poplicher, MathDL, January, 2005)

"Benoit Mandelbrot was 80 last year; and it is 25 years since he first saw the ubiquitious set that bears his name. ... This book is a selection of articles from the 1980s and early 1990s, together with previously unpublished material from the same period ... . The informal mix of mathematics and commentary in Mandelbrot's book provides a fascinating insight into his motivation and method." (Shaun Bullett, Times Higher Education Supplement, August, 2005)

"A scientist becomes rarely the historian of his own work, however here it is the case. ... Mandelbrot comments upon each of the selected articles ... . The reader will be grateful not only for these historical explanatory remarks, but also for pieces adapted or written by Mandelbrot especially for this book. Collectively they give a compelling account of how a new branch of mathematics was created by the author. This delightful book makes good reading." (Helmut Kirchner, Pure and Applied Geophysics, Vol. 162, 2005)

Forewordp. ix
Preface (2003)p. 1
Quadratic Julia and Mandelbrot Sets
Introduction to papers on quadratic dynamics: a progression from seeing to discovering (2003)p. 9
Acknowledgments related to quadratic dynamics (2003)p. 27
Fractal aspects of the iteration of z to [lambda] z (1-z) for complex [lambda] and z (M1980n)p. 37
Cantor and Fatou dusts; self-squared dragons (M 1982F)p. 52
The complex quadratic map and its M-set (M1983p)p. 73
Bifurcation points and the "n squared" approximation and conjecture (M1985g)p. 96
The "normalized radical" of the M-set (M1985g)p. 100
The boundary of the M-set is of dimension 2 (M1985g)p. 110
Certain Julia sets include smooth components (M1985g)p. 114
Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs (M1985g)p. 117
Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets (M1985n)p. 125
Nonquadratic Rational Dynamics
Introduction to chaos in nonquadratic dynamics: rational functions devised from doubling formulas (2003)p. 137
The map z to [lambda] (z + 1/ z) and roughening of chaos from linear to planar (computer-assisted homage to K. Hokusai) (M1984k)p. 146
Two nonquadratic rational maps, devised from Weierstrass doubling formulas (1979-2003)p. 157
Iterated Nonlinear Function Systems and the Fractal Limit Sets of Kleinian Groups
Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments (2003)p. 171
Self-inverse fractals, Apollonian nets, and soap (M 1982F)p. 178
Symmetry by dilation or reduction, fractals, roughness (M2002w)p. 193
Self-inverse fractals osculated by sigma-discs and limit sets of inversion ("Kleinian") groups (M1983m)p. 205
Multifractal Invariant Measures
Introduction to measures that vanish exponentially almost everywhere: DLA and Minkowski (2003)p. 221
Invariant multifractal measures in chaotic Hamiltonian systems and related structures (Gutzwiller & M 1988)p. 231
The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems (M1993s)p. 239
Harmonic measure on DLA and extended self-similarity (M & Evertsz 1991)p. 251
Background and History
The inexhaustible function z squared plus c (1982-2003)p. 259
The Fatou and Julia stories (2003)p. 268
Mathematical analysis while in the wilderness (2003)p. 276
Cumulative Bibliographyp. 281
Indexp. 299
Table of Contents provided by Rittenhouse. All Rights Reserved.

ISBN: 9780387201580
ISBN-10: 0387201580
Series: Selecta
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 308
Published: 9th January 2004
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.91
Weight (kg): 0.59