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The subject of space-filling curves has fascinated mathematicians for over a century and has intrigued many generations of students of mathematics. Working in this area is like skating on the edge of reason. Unfortunately, no comprehensive treatment has ever been attempted other than the gallant effort by W. Sierpiriski in 1912. At that time, the subject was still in its infancy and the most interesting and perplexing results were still to come. Besides, Sierpiriski's paper was written in Polish and published in a journal that is not readily accessible (Sierpiriski [2]). Most of the early literature on the subject is in French, German, and Polish, providing an additional raison d'etre for a comprehensive treatment in English. While there was, understandably, some intensive research activity on this subject around the turn of the century, contributions have, nevertheless, continued up to the present and there is no end in sight, indicating that the subject is still very much alive. The recent interest in fractals has refocused interest on space­ filling curves, and the study of fractals has thrown some new light on this small but venerable part of mathematics. This monograph is neither a textbook nor an encyclopedic treatment of the subject nor a historical account, but it is a little of each. While it may lend structure to a seminar or pro-seminar, or be useful as a supplement in a course on topology or mathematical analysis, it is primarily intended for self-study by the aficionados of classical analysis.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
In 1878, George Cantor demonstrated that any two finite-dimensional smooth manifolds, no matter what their dimensions, have the same cardinality, and Mathematics has never been the same since. Cantor’s finding implies, in particular, that the interval [0,1] can be mapped bijectively onto the square [0,1]2. The question arose almost immediately whether or not such a mapping can possibly be continuous. In 1879, E. Netto put an end to such speculation by showing that such a bijective mapping is necessarily discontinuous. Suppose the condition of bijectivity were dropped, is it still possible to obtain a continuous surjective mapping from [0,1] onto [0,1]2? Since a continuous mapping from [0,1] (or any other interval, for that matter) into the plane (or space) was and, to a large extent, still is called a curve, the question may be rephrased as follows: Is there a curve that passes through every point of a two-dimensional region (such as, for example, [0,1]2) with positive Jordan content (area)? G. Peano settled this question once and for all in 1890 by constructing the first such curve. Curves with this property are now called space-filling curves or Peano curves. Further examples by D. Hilbert (in 1891), E.H. Moore (in 1900), H. Lebesgue (in 1904), W. Sierpiński (in 1912), G. Pólya (in 1913), and others followed. Around the turn of the century, when many of these curves were discovered, the term space was primarily used for the three-dimensional space, and these curves were called surface-filling curves, as is apparent from the titles of the early papers on the subject (“Abbildung einer Linie auf ein Flächenstück” = “mapping of a line onto a piece of a surface,” “Une courbe qui remplit une aire plane” = “A curve that fills a plane region,” or, more specifically, “O krzywych, wypolniajacych kwadrat” = “On curves that fill a square”).
Hans Sagan

Chapter 2. Hilbert’s Space-Filling Curve

Abstract
David Hilbert (1862–1943), who, more than anybody else, set the course for the mathematicians of the 20th century, was born in Königsberg, East Prussia (which was renamed Kaliningrad when it was incorporated into Russia in 1945) and died in Göttingen. He studied at the University of Königsberg, except for the second semester, which he spent at the University of Heidelberg, and received his doctor’s degree in 1884 C.L.F. Lindemann (who, in 1882, succeeded in proving that π is transcendental) and, especially, A. Hurwitz were his most influential mentors at that time. After some postdoctoral studies in Leipzig and Paris, he returned to the University of Königsberg in 1886. In 1892, he became the successor of A. Hurwitz (who had left for Zürich), and in 1893, he succeeded to the chair that was held up to that time by Lindemann. In 1895, he followed a call to the University of Göttingen, where he taught until his retirement in 1930.
Hans Sagan

Chapter 3. Peano’s Space-Filling Curve

Abstract
Giuseppe Peano (1858–1932) was born in Spinetta, Italy, and died in Turin. He completed his studies at the University of Turin in 1880 and became a professor there in 1890. He held that position until his death. He is known for his pioneering work in symbolic logic, the axiomatic method, and for his contributions to mathematical analysis.
Hans Sagan

Chapter 4. Sierpiński’s Space-Filling Curve

Abstract
Waclaw Sierpiński (1882–1969) was born and died in Warsaw, Poland. In 1899, he entered the (Russian) University of Warsaw. He graduated in 1904 with a degree in science. He went on to teach grammar school and participated in the great school strike that was connected with the 1905 revolution. He then went on to Cracow where, at the University of Cracow, he received his doctor’s degree in 1906. After teaching at a grammar school, he became docent at the University of L’vov in 1908 and professor in 1910. At that time, he also changed the focus of his studies from number theory to topology, not returning to number theory until around 1950. In his publications during the last twenty years of his life, number theory was the predominant topic. He was a most prolific mathematician, having published about 700 papers and books, about 600 of which were in topology. In 1914, he was interned by the Tsarist authorities.
Hans Sagan

Chapter 5. Lebesgue’s Space-Filling Curve

Abstract
In a paper on infinite linear point manifolds written in 1883, in which Cantor searched for a characterization of the continuum, he offers in the appendix the set of all points that can be represented by
$$frac{{2{t_1}}}{3} + \frac{{2{t_2}}}{{{3^2}}} + \frac{{2{t_3}}}{{{3^3}}} + \frac{{2{t_4}}}{{{3^4}}} + ...,$$
where t j = 0 or 1, as an example of a perfect set (a set that is equal to the set of all its accumulation points) that is not dense in any interval, no matter how small (Cantor [2]). This set, which had its humble beginnings as a counterexample in an appendix has since taken on a life of its own and has served ever since its inception, as an example, counterexample, and inspiration for inquiries into the most remote recesses of mathematical analysis. It is no coincidence that it appears in a fundamental role in the study of space-filling curves. We will see in Chapter 6 that this role is even more fundamental than it would appear to be from a study of the present chapter.
Hans Sagan

Chapter 6. Continuous Images of a Line Segment

Abstract
With squares and triangles and all their continuous images revealed as continuous images of the interval I, the question arose as to the general characterization of such sets. In 1908, A. Schoenflies found such a characterization (see Schoenflies [1], p. 237), which, by its very nature, only applies to sets in the two-dimensional plane. In 1913, Hans Hahn and Stefan Mazurkiewicz independently arrived at a complete characterization of such sets in E n (Hahn [2], [3], Mazurkiewicz [1], [2], [3]). Their results can be extended to apply to even more general spaces, as we will point out at the end of Section 6.8. We will follow Hahn’s approach, commenting on Mazurkiewicz’ work briefly in Section 6.7. The development of this chapter may be viewed as a generalization of Lebesgue’s construction of a space-filling curve. Two elements made this construction possible: First, the square Q emerged as a continuous image of the Cantor set and, secondly, it was possible to extend the definition of the mapping continuously from Γ into I by linear interpolation, i.e., by joining the image of the left endpoint of an interval that has been removed in the construction of the Cantor set to the image of the right endpoint by a straight line (which lies in the square Q). Hausdorff has shown that every compact set is a continuous image of the Cantor set. But compactness is not enough to ensure that the above mentioned images can be joined by a continuous arc that remains in the set in such a manner that the extended map is continuous.
Hans Sagan

Chapter 7. Schoenberg’s Space-Filling Curve

Abstract
Isaac J. Schoenberg (1903–1990) was born in Galatz, Romania, and died in Madison, Wisconsin. He studied at the Universities of Jassy, Göttingen, and Berlin, and received his doctor’s degree from the University of Jassy in 1926. After emigrating to the United States in 1930, he held fellowships at the University of Chicago, Harvard, and the Institute for Advanced Studies at Princeton. From 1935 on, he held positions on the faculties of Swarthmore College, Colby College, and The University of Pennsylvania, where he became Professor in 1948, and the University of Wisconsin, from which he retired in 1973. From 1945 to 1946, he was chief of the Punched Card Section of the Computing Branch of the Ballistics Research Laboratory of the Aberdeen Proving Grounds. From 1965 until his retirement he held a joint appointment at the University of Wisconsin and the Army Research Center in Madison (Schoenberg [2], back cover.)
Hans Sagan

Chapter 8. Jordan Curves of Positive Lebesgue Measure

Abstract
We know from Netto’s theorem (Theorem 6.4) that plane curves that do not cross (or touch) themselves cannot be space-filling. We will show in this chapter that it is still possible for such curves to have a positive two-dimensional Lebesgue measure.
Hans Sagan

Chapter 9. Fractals

Abstract
If we apply the similarity transformations
$$ \eqalign{ & \xi = \frac{1}{3}\xi \cr & \xi = \frac{1}{3}\left( {\xi + 2} \right) \cr} $$
(9.1.1)
to the interval I in Fig. 9.1.1 (a), we obtain the configuration in Fig. 9.1.1 (b). If we apply (9.1.1) to the configuration in Fig. 9.1.1 (b), we obtain the configuration in Fig. 9.1.1(c). Applying it to Fig. 9.1.1(c), we obtain the configuration in Fig. 9.1.1 (d), etc. If we carry this on ad infinitum, we arrive at the Cantor set of Section 5.1 (or Cantor dust, as B. Mandelbrot so aptly called it).
Hans Sagan

Backmatter

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