Taking the Koch snowflake as a key example, Falconer begins with a general explanation of what fractals are and where they are useful. There follow chapters on self-similarity and the iterative construction of fractals, fractal dimension, Julia and Mandelbrot sets, and random walks and Brownian motion.
The bulk of the material is descriptive, but some mathematics is introduced and used, albeit with concrete numbers and worked examples: the basic concepts of coordinate geometry, iteration, functions, logarithms, box-counting dimension, and complex numbers. There's the usual problem here: anyone who already understands coordinate geometry and logarithms will find the explanations of those unnecessary, while those with no acquaintance at all with these topics will likely find the explanations too terse. It's hard for me to judge, but overall I think Falconer finds a reasonable balance here.
He also does a good job of setting fractals in their broader context, for example in a discussion of when dimension measures are useful and their limitations. And he finishes with a survey of some "real world" examples/applications, a brief history, and a nice two page "Further Reading". Falconer makes good use of diagrams to support his explanations but, presumably constrained by the Very Short Introduction format, makes no real attempt to convey the visual appeal of fractals. There is also nothing on their computer exploration.
Hans Lauwerier's Fractals: Endlessly Repeated Geometric Figures — also published by Penguin as Fractals: Images of Chaos — covers these. It includes some nice colour images, of natural objects with fractal structure as well of computer-generated fractals. And it makes computer exploration central: parts of the presentation are almost built around it, and there's a final chapter "Making Your Own Fractals" and an appendix with twenty pages of BASIC code.
Otherwise Endlessly Repeated Geometrical Figures covers similar material to the Very Short Introduction, with some differences (its relatively large print and illustrations mean it is not much, if at all, longer). Lauwerier also looks at iteration, simple fractal dimensions, chance in fractals, and Julia and Mandelbrot sets, but in addition touches on different base systems and dynamics on fractals, countable and uncountable infinities, and limits. He leaves complex numbers to an appendix, using real number equations in generating Julia and Mandelbrot sets, but he assumes the reader already understands the basic ideas of functions and logarithms and coordinates.
The original Dutch Fractals: Meetkundige figuren in eindeloze herhaling dates to 1987 and though the mathematics itself hasn't changed there are many places where Endlessly Repeated Geometrical Figures shows its age. The colour illustrations are attractive, but clearly from an earlier generation. And the computer explorations include comments such as "It is important to have a screen with a high resolution (640x400 pixels)" and explanations of how to cope with very limited memory resources. There's a wealth of online resources available now for both visual and computational exploration of fractals, which is probably why there doesn't appear to be a newer equivalent of Images of Chaos. I can't help thinking there is still an opening for a book here, however.
Note: For those who want something a bit more substantial and are interested in learning more mathematics, it would be worth looking at Feldman's Chaos and Fractals: An Elementary Introduction. This is a textbook, complete with exercises, but one which assumes only elementary mathematics — no more than either of these books — even though it attempts to teach vastly more.
- External links:
Fractals: A Very Short Introduction
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Fractals: Endlessly Repeated Geometrical Figures
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- Related reviews:
- Kenneth Falconer - Fractal Geometry: Mathematical Foundations and Applications
- more Dutch literature
- books about popular mathematics
- books published by Oxford University Press
- books published by Princeton University Press