Fractals, Chaos, Power Laws:
Minutes from an Infinite Paradise

Manfred Schroeder

Freeman 1991
A book review by Danny Yee © 2016
Taking self-similarity and scaling as its unifying themes, Fractals, Chaos, Power Laws covers an immense range of material, with sections only a few pages in length outlining aspects of mathematics or natural phenomena. Schroeder takes the latter mostly from physics, ranging from astronomy to acoustics (one of his special interests), but also draws on biology, gambling, finance, music, and other areas. He occasionally touches on history and philosophy in this, but mostly remains focused on the science and mathematics.

A sampling of the section titles gives a feel for the variety: "A New Dimension for Fractals" (on Hausdorff dimension); "To Scale or Not to Scale: A Bit of Biology and Astrophysics"; "More Self-Similarity in Music: The Tempered Scales of Bach"; "Fresh and Tired Mountains" (Brownian mountains with different dimensions); "Invariant Distributions: Gauss, Cauchy, and Beyond" (how "the cherished Gaussian distribution with D = 2 stands revealed as but an extreme, albeit ubiquitous, member of an entire clan of distributions"); "Symbolic Dynamics and Deterministic Chaos"; "The Multifractal Spectrum: Turbulence and Diffusion-Limited Aggregation"; "The Fractal Dimensions of Fracture Surfaces"; "Self-Similarity in the Logistic Parabola"; "More Forbidden Symmetries" ("having tasted a first forbidden fruit of fivefold symmetry, we might ask whether there are quasicrystals with other outlawed symmetry axes that can be distilled from self-similar iterated maps"); "The Golden-Mean Route to Chaos"; "Critical Conflagration on a Square Lattice" (a percolation model); and "The Game of Life".

Schroeder offers no proofs or derivations. And he makes no attempt to teach any mathematics, but simply assumes familiarity with a good deal of both mathematics — exponents and logarithms, trigonometry, limits and calculus, matrices, and so forth — and physics — with off-hand references to the thermodynamic equipartition of energy, characteristic times, correlation lengths, and so forth. And in some places the presentation is dense. So the primary target audience is physicists and engineers, other scientists with a mathematical background, and mathematicians interested in applications. In some ways Fractals, Chaos, Power Laws resembles Mandelbrot's 1982 classic The Fractal Geometry of Nature, but it is pitched at a broader audience and is almost a survey of the first decade's response to that manifesto.

Fractals, Chaos, Power Laws is more accessible than this would suggest, however, because it doesn't systematically develop any body of theory and consists of short sections that are largely independent. The more involved material can easily be skipped over and there's a wealth of examples and applications as motivation. So it could be read by undergraduates or anyone else with some general mathematics and physics, and it would make an excellent complement to textbooks that teach the mathematics of fractal geometry and chaotic dynamics without much on scientific applications and examples.

May 2016

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Related reviews:
- Peitgen et al. - Chaos and Fractals: New Frontiers of Science
- Benoit Mandelbrot - The Fractal Geometry of Nature
- books about mathematics
- books about physics
%T Fractals, Chaos, Power Laws
%S Minutes from an Infinite Paradise
%A Schroeder, Manfred
%I Freeman
%D 1991
%O paperback, references, index
%G ISBN 0716723573
%P 429pp