Review of "Fractals and Chaos: The Mandelbrot Set and Beyond", by B. Mandelbrot
BioMedical Engineering OnLine
Book review Review of "Fractals and Chaos: The Mandelbrot Set and Beyond", by B. Mandelbrot Alberto Diaspro*
0 Address: Laboratory for Advanced Microscopy , Bioimaging and Spectroscopy (LAMBS) , MicroScoBio Research Center, IFOM, Department of Physics, University of Genoa , Via Dodecaneso, 33 16146 Genoa Italy
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Benoit Mandelbrot has produced a comprehensive,
wellpresented review of essential topics related to Mandelbrot
set theory and applications. The last part of the title "The
Mandelbrot set and beyond" fully describes its potential
allowing the reader to navigate through pictures,
hard-tofind early papers and important and effective chapters on
the historical background. All chapters are assembled in a
way that the overall mix becomes a very well integrated
source of know-how and knowledge bringing the readers
into the Mandelbrot set world. The spirit of the book is
well summarized in a sentence on page 34: "When seeking
new insights, I look, look, look, and play with many
pictures. (One picture is never enough)." It is certainly true
that in the last twenty years, mathematics has changed so
deeply that to younger persons some chapter's stories
might be simply incredible (p.36), as well, one should
admit that after Mandelbrot's sets, initially describing
trees, coastlines' shapes or allowing measuring the length
of the Britain coast, and after the seminal book on "The
Fractal Geometry of Nature" our way of looking at the
world changed. Mandelbrot wrote: "Why is geometry
often described as 'cold' and 'dry'? One reason lies in its
inability to describe the shape of a cloud, a mountain, a
coastline or a tree. Clouds are not spheres, mountains are
not cones, coastlines are not circles, and bark is not
smooth, nor does lightning travel in a straight line". I
think our vision of the world, from the atom to the higher
length scales, is still changing using those concepts clearly
illustrated in the current Mandelbrot's book. Selected
notes and papers make this book unique within the
several books published on this topic. It is clear the touch of
the author under all aspects: a touch of pure genius.
There are five main topics dominating the book, namely:
Quadratic iteration and its Mandelbrot set Quadratic
Julia and Mandelbrot sets; Nonquadratic iterations
Nonquadratic rational dynamics; Kleinian groups' limit
set Iterated nonlinear function systems and the fractal
limit sets of Kleinian groups; Multifractal invariant
measures Exponentially vanishing multifractal measures;
Background and History. Cumulative bibliography is
impressive and well done. It is clearly pointed out,
following the pathway through the book, how fractal geometry
played an important role in offering a quantitative tool in
several areas. Circumstances and facts are put together
also to bring important lessons for young scientists. The
author made a serious and effective effort to realize a book
that contains more than history, more than
mathematics... it is a sort of ideal book for stimulating new ideas,
new concepts, and new discoveries. So far, it is an
excellent book also for supporting courses at University, PhD
and Post doc level. Moreover, it is indispensable for
scientists not only as a lesson of a pathway in science but also
as an important source for science of tomorrow. This is a
valuable reference source to researchers from these and
related areas including bioengineering, biophysics,
nanobiosciences and, of course, applied mathematics.
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