Space-Filling Curves (Universitext) | Hans Sagan

Space-Filling Curves (Universitext) | Hans Sagan | Pretty amusing stuff

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Space-Filling Curv...

	Space-Filling Curves (Universitext) Hans Sagan Springer, 1994 - 193 pages average customer review:based on 2 reviews view larger image
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The subject of space-filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by Peano. Cantor, Hilbert, Moore, Knopp, Lebesgue, and Polya are among the prominent mathematicians who have contributed to the field. However, there have been no comprehensive treatments of the subject since Siepinsky's in 1912. Cantor showed in 1878 that the number of points on an interval is the same as the number of points in a square (or cube, or whatever), and in 1890 Peano showed that there is indeed a continuous curve that continuously maps all points of a line onto all points of a square, though the curve exists only as a limit of very convoluted curves. This book discusses generalizations of Peano's solution and the properties that such curves must possess and discusses fractals in this context. The only prerequisite is a knowledge of advanced calculus.

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a neat book

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.

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Pretty amusing stuff

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.

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