It takes the form of an extended essay, "a casebook and a manifesto", making the argument that, rather than being pathological constructs of only mathematical interest, fractals are essential tools for understanding the natural world. Mandelbrot goes through a range of case studies, drawn from physics, geography, astronomy and assorted other disciplines and looking at coastlines, rivers and watersheds, galactic clusters, turbulence, lattices, market prices and a broad range of other phenomena. Much of this centres on dimensions, both on the observed dimensions of natural phenomena and on the construction of fractals with particular dimensions, but Mandelbrot also introduces attempts to measure "texture" such as succolarity and lacunarity.
"This chapter begins with pleas for a more geometric approach to turbulence and for the use of fractals. These pleas are numerous but each is brief, because they involve suggestions with few hard results as yet.
After that, we focus on the problem of intermittency, which I have investigated actively. My most important conclusion is that the region of dissipation, namely the spatial set in which turbulent dissipation is concentrated, can be modeled by a fractal. Measurements done for different purposes suggest that this region's D lies around 2.5 to 2.6, but probably below 2.66."
The Fractal Geometry of Nature is light on formal mathematics, omitting any proofs and largely referencing results rather than recapitulating them. (And most of the more formal material is relegated to a forty-page "Mathematical Backup and Addenda".) It is not a popular work, however, as it assumes a solid familiarity and comfort not just with mathematical ideas and ways of thinking but with concepts from physics such as percolation, dissipation, and power laws. It is not inaccessible to students or lay readers, but the intended audience is mathematicians and mathematically aware scientists.
Some historical background is woven into everything, but thirty five pages at the end add some details. Some "Biographical Sketches" look at a few of the less well-known "mavericks" who contributed to the history of the ideas presented: Bachelier, Fournier D'Albe, Hurst, Lévy, Richardson, and Zipf. Some "Historical Sketches" explore earlier premonitions and dead-ends and half-starts. And Mandelbrot includes a brief account of his own "path to fractals".
A sixteen page colour insert constitutes a self-contained essay on the history of fractals in art and visualisation. The graphics of The Fractal Geometry of Nature, here and in the black and white illustrations accompanying the main text, were superb for their time but now seem relatively ordinary. Some of Mandelbrot's terminological suggestions — such as trema for the "holes" or "gaps" removed in the construction of fractals — have clearly failed to get traction. And a huge amount of work has been done both on the mathematics of fractal geometry and on its application. But otherwise The Fractal Geometry of Nature has survived thirty years surprisingly well.
In many ways Mandelbrot's goal of introducing fractal geometry to scientists has been achieved: it is hard to imagine anyone becoming a physicist now without having had at least some exposure to fractal ideas, and fractals have entered popular awareness to the extent that they can be mentioned in a Disney song. On the other hand, some of the early excitement about fractals seems to have died down — a look at Google's English book corpus suggests that the relative frequency of "fractal" has declined by a third since a peak around 1993 — and fractal geometry remains a relatively esoteric topic, despite the availability of some excellent textbooks.
Many of the specific ideas and above all the overall message of The Fractal Geometry of Nature remain important. It is also unique in the way it situates fractal ideas in their broader scientific and historical contexts, and in its vision and its passion.
January 2016
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