Space-filling curves

the long version

The subject of space-filling curves has generated a great deal of interest since the first such curve was discovered by Peano over a century ago. Hilbert, Lebesque, and Sierpinski were among the prominent mathematicians who made significant contributions to the field in its early stages of development. Cantor showed in 1878 that there is a one-to-one correspondence between an interval and a square (or cube, or any finite-dimensional manifold) and Netto demonstrated that such a correspondence cannot be continuous. Dropping the requirement that the mapping be one-to-one, Peano found a continuous map from the interval onto the square (or cube) in 1890. In other words: He found a continuous curve that passes through every point of the square (or cube). This book discusses generalizations and modifications of Peano's constructions, the properties of such curves, and their relationship to fractals. Surprisingly, there has not been a comprehensive treatment of space-filling curves since Sierpinski's in 1912, when the subject was still in its infancy. The author, who has established his credentials through a series of publications on space-filling curves, provides a rigorous and comprehensive treatment, but also reflects on the subject's historical development and the personalities involved. The only prerequisite is a solid knowledge of Advanced Calculus.