Customer Reviews

Space-Filling Curves (Universitext)

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I really liked this book and especially enjoyed reading some parts. But I cannot give the book more than 4 stars. Several times in the text, his proofs could be much better written. Part of the difficulty, it seems to me, is his reluctance to rely on basic topology; I guess since he said you only need a background in analysis in his preface, he felt obliged to do everything with epsilons and deltas.

The beginning was very nice, introducing the Hilbert Curve the way Hilbert thought of it(geometric), and then generalizing to an arithmetic representation. A similar thing is done with the Peano Curve and the theme of the dual perspective of geometry and analysis continues throughout. For a beginner, I think this book goes a long way to explaining space-filling curves and self-similar fractals.

Space-filling curves, such as the Hilbert and Sierpinski curves, have found applications in many domains. (I happened to become interested in them through the literature on spatial clustering.) This book gives a nice historical treatment of the development of space-filling curves, relating many of the families together. It also tries to provide a reasonable discussion of the related algorithms, e.g., for curve generation. Even if you can't follow all of the math (and I certainly can't), I think it helps to have this book around if you're trying to understand why and when these techniques work.