Space-Filling Curves

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. Pólya'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.