What the rain knows

I’m going to divide the introduction into sections. The titles are mine, rather than the authors’, unless stated otherwise.

Warmup

• Comparison of predictable tides to unpredictable weather (forecasts good for much shorter intervals).
• Chaos is deterministic (future completely determined by past) - but small uncertainties can snowball.
• Mention of methodologies which have been designed for a precise scientific evaluation of the presence of chaotic behavior in mathematical models as well as in real phenomena and says that they can be used to estimate the “predictability horizon” of a system - mathematical/physical/time limit within which predictability is ideally possible.
• Predictability horizon for weather is not more than 2-3 weeks.

History of chaos theory

• Calculus developed by Newton and Leibniz.
• Laplace’s demon
• Uncertainty principle
• Lewis Fry Richardson’s book Weather Prediction by Numerical Process
• John von Neumann began building ENIAC (1st electronic computer) to try to put Richardson’s ideas about weather prediction into practice.
• It was found (by whom?) that Richardson hadn’t succeeded because the space and time increments used in his work had not met a computational stability criterion (Courant-Friedrichs-Lewy Criterion).
• Edward Norton Lorenz and the butterfly effect (aka “sensitive dependence on initial conditions”) mentioned.
• Near the end: determinism and predictability are not equivalent.

Also:

In other words, one of the lessons coming out of chaos theory is that the validity of the causality principle is narrowed by the uncertainty principle from one end as well as by the intrinsic instability properties of the underlying natural laws from the other end.

Notes

A Google search for “Courant-Friedrichs-Lewy Criterion” turned up only for-pay journal articles, all focused narrowly on particular applications/instances of its use. A Google Book search turned up a book that I’ve added to my Amazon wishlist, The Emergence of Numerical Weather Prediction: Richardson’s Dream (ISBN-13 978-0521857291), Richardson’s book, and the second volume of the authors other work, Fractals in the Classroom, which seems to have the same introduction, word for word, as CaF.

The edition of Richardson’s book cited in a footnote in the introduction is the 1965 Dover edition.

The Foreword to Chaos and Fractals: New Frontiers of Science was written by none other than Mitchell J. Feigenbaum.

Feigenbaum warms up with a discussion of fluid turbulence and then goes on to divide physical phenomena into three categories: linear (for which the rule that determines what a piece of a system is going to do next is not influenced by what it is doing now.), nonlinear (some of which can be predicted by using distorted models of linear phenomena), and strongly nonlinear (which includes “chaos”).

Finished. Feigenbaum ended up used the foreward to push the Feigenbaum constants (Wolfram MathWorld: Feigenbaum Constant).

Chaos and Fractals: New Frontiers of Science was first published by Springer in 1992. I’ve got the 2nd edition (first printing, it seems), published in 2004.

The authors, in order of appearance on the cover and in the signature of the preface: Heinz-Otto Peitgen, Hartmut Jürgens, and Deitmar Saupe.

According to the above-linked Wikipedia page for Saupe, the book won the American Publishers’ Association award for Best Mathematics Book of the Year in 1992. I can’t find an award site to confirm this fact and it isn’t noted in either of the prefaces.

Preface to the 2nd edition

The first preface (to the 2nd ed.) notes the differences between the first and second editions:

• Two appendices have been removed - one by Yuval Fisher on the subject of fractal image compression because Fisher’s since published a book on the subject. They don’t detail the subject of the other appendix and I can’t find a table of contents for the 1st edition.
• There was a BASIC program at the end of each chapter of the first book and those have been replaced with a set of 10 Java applets posted online at http://www.cevis.uni-bremen.de/fractals/. CEVIS is the Center for Complex Systems and Visualization at the University of Bremen, which has been home, at one time or another, to all of the authors.

I noticed at least two typos in the preface. Apostrophes were missing when the authors talked about Yuval Fisher’s appendix or Yuval’s book (i.e. Yuval Fishers and Yuvals, respectively).

Preface to the 1st edition

The preface begins with a quote from James Gleick’s Chaos: Making a New Science. I’ve read and didn’t care for Gleick’s book.

Points of interest from this preface:

• This book is written for everyone, even without much knowledge of technical mathematics, wants to know the details of chaos theory and fractal geometry. They go on to say that it’s not a textbook, but not written in popsci style either.
• Fractals and chaos have, for students, brought math out of the realm of ancient history into the twenty-first century.
• Elements of Euclidian geometry are basc visible forms like lines, circles, and spheres, the elements of fractal geometry are algorithms.
• Fractals and modern chaos theory are linked by the fact that many of the contemporary pace-setting discoveries in their fields were only possible using computers.
• This book is a close relative of the two-volume set Fractals for the Classroom, which was published by Springer-Verlag and the National Council of Teachers of Mathematics in 1991 and 1992. [Fractals for the Classroom was written by the same three mathematicians who wrote Chaos and Fractals].
• The other appendix removed from the 2nd edition is revealed to have been a treatment of multifractal measures by Carl J.G. Evertsz and Benoit B. Mandelbrot.

• The entire book has been produced using the TEX and LaTex typesetting systems where all figures (except for the half-tone and color images) were integrated in the computer files. Even though it took countless hours of sometimes painful experimentation setting up the necessary maxcros, it must be acknowledged that this approach immensely helped to streamline the writing, editing, and printing.