Overview

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.

Full Product Details

Author:   Hans Sagan
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1994
Dimensions:   Width: 15.50cm , Height: 1.10cm , Length: 23.50cm
Weight:   0.670kg
ISBN:  

9780387942650

ISBN 10:   0387942653
Pages:   194
Publication Date:   02 September 1994
Audience:   College/higher education ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1. Introduction.- 1.1. A Brief History of Space-Filling Curves.- 1.2. Notation.- 1.3. Definitions and Netto's Theorem.- 1.4. Problems.- 2. Hilbert's Space-Filling Curve.- 2.1. Generation of Hilbert's Space-Filling Curve.- 2.2. Nowhere Differentiability of the Hilbert Curve.- 2.3. A Complex Representation of the Hilbert Curve.- 2.4. Arithmetization of the Hilbert Curve.- 2.5. An Analytic Proof of the Nowhere Differentiability of the Hilbert Curve.- 2.6. Approximating Polygons for the Hilbert Curve.- 2.7. Moore's Version of the Hilbert Curve.- 2.8. A Three-Dimensional Hilbert Curve.- 2.9. Problems.- 3. Peano's Space-Filling Curve.- 3.1. Definition of Peano's Space-Filling Curve.- 3.2. Nowhere Differentiability of the Peano Curve.- 3.3. Geometric Generation of the Peano Curve.- 3.4. Proof that the Peano Curve and the Geometric Peano Curve are the Same.- 3.5. Cesaro's Representation of the Peano Curve.- 3.6. Approximating Polygons for the Peano Curve.- 3.7. Wunderlich's Versions of the Peano Curve.- 3.8. A Three-Dimensional Peano Curve.- 3.9. Problems.- 4. Sierpi?ski's Space-Filling Curve.- 4.1. Sierpi?ski's Original Definition.- 4.2. Geometric Generation and Knopp's Representation of the Sierpi?ski Curve.- 4.3. Representation of the Sierphiski-Knopp Curve in Terms of Quaternaries.- 4.4. Nowhere Differentiability of the Sierpi?ski-Knopp Curve.- 4.5. Approximating Polygons for the Sierpi?ski-Knopp Curve.- 4.6. Polya's Generalization of the Sierpi?ski-Knopp Curve.- 4.7. Problems.- 5. Lebesgue's Space-Filling Curve.- 5.1. The Cantor Set.- 5.2. Properties of the Cantor Set.- 5.3. The Cantor Function and the Devil's Staircase.- 5.4. Lebesgue's Definition of a Space-Filling Curve.- 5.5. Approximating Polygons for the Lebesgue Curve.- 5.6. Problems.- 6. Continuous Images of a Line Segment.- 6.1. Preliminary Remarks and a Global Characterization of Continuity.- 6.2. Compact Sets.- 6.3. Connected Sets.- 6.4. Proof of Netto's Theorem.- 6.5. Locally Connected Sets.- 6.6. A Theorem by Hausdorff.- 6.7. Pathwise Connectedness.- 6.8. The Hahn-Mazurkiewicz Theorem.- 6.9. Generation of Space-Filling Curves by Stochastically Independent Functions.- 6.10. Representation of a Space-Filling Curve by an Analytic Function.- 6.11. Problems.- 7. Schoenberg's Space-Filling Curve.- 7.1. Definition and Basic Properties.- 7.2. The Nowhere Differentiability of the Schoenberg Curve.- 7.3. Approximating Polygons.- 7.4. A Three-Dimensional Schoenberg Curve.- 7.5. An No-Dimensional Schoenberg Curve.- 7.6. Problems.- 8. Jordan Curves of Positive Lebesgue Measure.- 8.1. Jordan Curves.- 8.2. Osgood's Jordan Curves of Positive Measure.- 8.3. The Osgood Curves of Sierpi?ski and Knopp.- 8.4. Other Osgood Curves.- 8.5. Problems.- 9. Fractals.- 9.1. Examples.- 9.2. The Space where Fractals are Made.- 9.3. The Invariant Attractor Set.- 9.4. Similarity Dimension.- 9.5. Cantor Curves.- 9.6. The Heighway-Dragon.- 9.7. Problems.- A.1. Computer Programs 169 A.1.1. Computation of the Nodal Points of the Hilbert Curve.- A.1.2. Computation of the Nodal Points of the Peano Curve.- A.1.3. Computation of the Nodal Points of the Sierpi?ski-Knopp Curve.- A.1.4. Plotting Program for the Approximating Polygons of the Schoenberg Curve.- A.2. Theorems from Analysis.- A.2.1. Binary and Other Representations.- A.2.2. Condition for Non-Differentiability.- A.2.3. Completeness of the Euclidean Space.- A.2.4. Uniform Convergence.- A.2.5. Measure of the Intersection of a Decreasing Sequence of Sets.- A.2.6. Cantor's Intersection Theorem.- A.2.7. Infinite Products.- References.

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