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Space-filling curves. (English) Zbl 0806.01019
Universitext. New York: Springer-Verlag. xv, 193 p. (1994).
A first comprehensive treatment in English of the history and content of a “small but venerable part of mathematics” (cited from the author’s Preface). The topic is presented in its genesis from Cantor and Netto (1878-79) through Peano and Hilbert (1890-91), Moore, Osgood and Lebesgue (1900-04), Sierpiński, Mazurkiewicz, Hahn and Pólya (1912-13), Steinhaus and Schoenberg (1936-38), Salem and Zygmund (1945) until the recent development, in particular in the author’s investigations. Also a new life of the subject in the study of fractals is enlightened. All this is given in an elegant manner. Historical comments (with biographies and portraits) are alternated by the relevant materials, treated, some in detail, some summarized, from the aspects elaborated by the author. A special emphasis is laid on the visual aids, using a lot of good figures. Correspondingly a great importance is given to the method of approximating polygons, similarity transformations applied in infinitely many steps, attractor sets etc. Also the main analytical tools are explained up to computer programs. So the both old and new material is presented in a good unified manner, in a modern style. Each of the 9 chapters ends with a problem section.
Reviewer: Ü.Lumiste (Tartu)

MSC:
01A60 History of mathematics in the 20th century
28A75 Length, area, volume, other geometric measure theory
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
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