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.
Author(s): Hans Sagan
Edition: 1
Publisher: Springer
Year: 1994
Language: English
Pages: 208
Cover......Page 1
Titlepage......Page 2
Preface......Page 6
Contents......Page 12
1.1. A Brief History of Space-Filling Curves......Page 16
1.2. Notation......Page 17
1.3. Definitions and Netto's Theorem......Page 19
1.4. Problems......Page 21
2.1. Generation of Hilbert's Space-Filling Curve......Page 24
2.2. Nowhere Differentiability of the Hilbert Curve......Page 27
2.3. A Complex Representation of the Hilbert Curve......Page 28
2.4. Arithmetization of the Hilbert Curve......Page 33
2.5. An Analytic Proof of the Nowhere Differentiability of the Hilbert Curve......Page 34
2.6. Approximating Polygons for the Hilbert Curve......Page 36
2.7. Moore's Version of the Hilbert Curve......Page 39
2.8. A Three-Dimensional Hilbert Curve......Page 41
2.9. Problems......Page 44
3.1. Definition of Peano's Space-Filling Curve......Page 46
3.3. Geometric Generation of the Peano Curve......Page 49
3.4. Proof that the Peano Curve and the Geometric Peano Curve are the Same......Page 51
3.5. Cesaro's Representation of the Peano Curve......Page 55
3.6. Approximating Polygons for the Peano Curve......Page 57
3.7. Wunderlich's Versions of the Peano Curve......Page 58
3.8. A Three-Dimensional Peano Curve......Page 60
3.9. Problems......Page 61
4.1. Sierpinski's Original Definition......Page 64
4.2. Geometric Generation and Knopp's Representation of the Sierpinski Curve......Page 66
4.3. Representation of the Sierpinski-Knopp Curve in Terms of Quaternaries......Page 71
4.4. Nowhere Differentiability of the Sierpinski-Knopp Curve......Page 73
4.5. Approximating Polygons for the Sierpinski-Knopp Curve......Page 75
4.6. Po1ya's Generalization of the Sierpinski-Knopp Curve......Page 77
4.7. Problems......Page 82
5.1. The Cantor Set......Page 84
5.2. Properties of the Cantor Set......Page 86
5.3. The Cantor Function and the Devil's Staircase......Page 89
5.4. Lebesgue's Definition of a Space-Filling Curve......Page 91
5.5. Approximating Polygons for the Lebesgue Curve......Page 94
5.6. Problems......Page 97
6.1. Preliminary Remarks and a Global Characterization of Continuity......Page 100
6.2. Compact Sets......Page 106
6.3. Connected Sets......Page 109
6.4. Proof of Netto's Theorem......Page 112
6.5. Locally Connected Sets......Page 113
6.6. A Theorem by Hausdorff......Page 114
6.7. Pathwise Connectedness......Page 116
6.8. The Hahn-Mazurkiewicz Theorem......Page 121
6.9. Generation of Space-Filling Curves by Stochastically Independent Functions......Page 123
6.10. Representation of a Space-Filling Curve by an Analytic Function......Page 127
6.11. Problems......Page 130
7.1. Definition and Basic Properties......Page 134
7.2. The Nowhere Differentiability of the Schoenberg Curve......Page 136
7.3. Approximating Polygons......Page 138
7.4. A Three-Dimensional Schoenberg Curve......Page 142
7.5. An No-Dimensional Schoenberg Curve......Page 143
7.6. Problems......Page 144
8.1. Jordan Curves......Page 146
8.2. Osgood's Jordan Curves of Positive Measure......Page 147
8.3. The Osgood Curves of Sierpinski and Knopp......Page 151
8.4. Other Osgood Curves......Page 155
8.5. Problems......Page 157
9.1. Examples......Page 160
9.2. The Space where Fractals are Made......Page 164
9.3. The Invariant Attractor Set......Page 169
9.4. Similarity Dimension......Page 171
9.5. Cantor Curves......Page 174
9.6. The Heighway-Dragon......Page 177
9.7. Problems......Page 180
A.1.1. Computation of the Nodal Points of the Hilbert Curve......Page 184
A.1.2. Computation of the Nodal Points of the Peano Curve 1......Page 85
A.1.3. Computation of the Nodal Points of the Sierpinski-Knopp Curve......Page 186
A.1.4. Plotting Program for the Approximating Polygons of the Schoenberg Curve 1......Page 87
A.2.1. Binary and Other Representations......Page 188
A.2.5. Measure of the Intersection of a Decreasing Sequence of Sets......Page 189
A.2.7. Infinite Products......Page 190
References......Page 192
Index......Page 202
