| Preface | p. v |
| Acknowledgments | p. ix |
| List of Figures | p. xvii |
| List of Tables | p. xxi |
| Examples of curious curves | p. 1 |
| Variations of the Koch curve | p. 2 |
| Koch curve | p. 2 |
| Modified Koch curve | p. 2 |
| Basics of complex numbers | p. 4 |
| More examples of curious curves | p. 6 |
| The unit square is a curve | p. 6 |
| Iterated function systems produce curves | p. 7 |
| Construction of a family of Cantor sets | p. 10 |
| Middle-thirds Cantor set | p. 11 |
| Construction of generalized Cantor sets | p. 12 |
| The length of the Cantor set C0 | p. 13 |
| The sets Ch | p. 15 |
| What is not a curve? | p. 15 |
| The Koch curve and tangent lines | p. 19 |
| Construction of the Koch curve | p. 19 |
| Representation in base 4 | p. 20 |
| Formulas for fk | p. 21 |
| Convergence of the sequence {fk} | p. 24 |
| An equation for f | p. 25 |
| Length of the Koch curve | p. 26 |
| Tangent lines to simple curves in C | p. 26 |
| Definition of a tangent line to a simple curve | p. 27 |
| Another construction of the Koch curve | p. 28 |
| Tangent lines to graphs of continuous maps from I to R | p. 30 |
| Modified Cantor functions | p. 34 |
| The graph G¿ of the Cantor function | p. 34 |
| Problems | p. 35 |
| Curves and Cantor sets | p. 37 |
| A square is a curve! | p. 37 |
| Simple curves | p. 40 |
| A homeomorphism g with C x C ≈ g(I) | p. 42 |
| Simple curves with positive area | p. 44 |
| Proofs of Propositions 3.1 and 3.2 | p. 48 |
| Proof of Proposition 3.1 | p. 48 |
| Proof of Proposition 3.2 | p. 49 |
| Continuous images of the Cantor set | p. 52 |
| Subsets of C that are not curves | p. 55 |
| Generalized curves | p. 56 |
| More examples of curves | p. 57 |
| Problems | p. 63 |
| Generalizations of the Koch curve | p. 67 |
| Construction of generalizations | p. 67 |
| The iteration process | p. 70 |
| A decomposition of Ka,¿ | p. 70 |
| Double points in Ka,¿ with ¿ = ¿/3 and a = 1/4 | p. 71 |
| The pivotal value a (¿) = 1/4 for ¿ =¿/3 | p. 75 |
| Investigation of Ka,¿/4 | p. 76 |
| Verifying <$$> | p. 78 |
| Double points form Cantor sets | p. 80 |
| Problems | p. 81 |
| Metric spaces and the Hausdorff metric | p. 83 |
| Metric spaces | p. 83 |
| Equivalent metrics | p. 84 |
| Topological properties of metric spaces | p. 85 |
| Complete metric spaces | p. 89 |
| The Hausdorff metric | p. 90 |
| Metrics and norms | p. 92 |
| Problems | p. 94 |
| Contraction maps and iterated function systems | p. 101 |
| Contraction maps | p. 101 |
| Iterated function systems | p. 104 |
| An iterated function system defines a curve | p. 106 |
| Implementation of iterated function systems | p. 108 |
| Problems | p. 112 |
| Dimension, curves and Cantor sets | p. 115 |
| Intervals, squares and cubes | p. 115 |
| Hausdorff dimension of a bounded subset of R2 | p. 117 |
| Basic facts about dimension | p. 117 |
| Tent maps and Cantor sets with prescribed dimension | p. 118 |
| Dimension of Cantor sets | p. 122 |
| Dimension of Cantor sets in the plane | p. 123 |
| Dimension and simple curves | p. 123 |
| Simple curves with prescribed dimension | p. 123 |
| Dimension of the Koch curve | p. 123 |
| Functions with prescribed dimension of points of non-tangency | p. 124 |
| Symmetric Cantor sets (optional section) | p. 125 |
| Construction of a symmetric Cantor set | p. 125 |
| Definition of dimension of symmetric Cantor sets | p. 126 |
| Saw tooth maps | p. 127 |
| Problems | p. 127 |
| Julia sets and the Mandelbrot set | p. 129 |
| Theory of Julia sets | p. 129 |
| Observations | p. 130 |
| Visual images | p. 131 |
| Two facts about Julia sets | p. 132 |
| The Mandelbrot set | p. 132 |
| Fixed points of fc | p. 133 |
| The central cardioid | p. 136 |
| The great circle | p. 136 |
| Period two points | p. 136 |
| Description of the great circle | p. 137 |
| Super-attracting fixed points | p. 138 |
| The two large bulbs adjoining the central cardioid | p. 139 |
| Generalized curves and Julia sets | p. 140 |
| Problems | p. 140 |
| Points on a line | p. 143 |
| Labeling points on a line | p. 143 |
| base b representations | p. 143 |
| Convergence | p. 144 |
| The geometric series | p. 145 |
| The special nested interval property | p. 147 |
| Bounds on subsets of a line | p. 147 |
| Bounded sequences have convergent subsequences | p. 149 |
| The real numbers R | p. 150 |
| Eventually periodic base b representations | p. 150 |
| Problems | p. 151 |
| Length and area | p. 155 |
| Intervals and length | p. 155 |
| Lengths of subsets of intervals | p. 157 |
| Intervals and rectangles in the plane | p. 158 |
| Length of a curve | p. 159 |
| Areas of subsets of the plane | p. 159 |
| Areas of rectangles | p. 160 |
| Areas of general subsets of the plane | p. 162 |
| Problems | p. 163 |
| Maps and sets in the plane | p. 165 |
| Definition of a map | p. 165 |
| Properties of points in the plane | p. 166 |
| Continuity and limits | p. 168 |
| Topological properties of subsets of R2 | p. 169 |
| Closed sets | p. 170 |
| Compact sets | p. 171 |
| Connected sets | p. 174 |
| Fixed points of maps | p. 176 |
| Uniform continuity of maps | p. 177 |
| Convergence of maps | p. 178 |
| Linear maps from R2 to R2 | p. 182 |
| Homeomorphisms: Inverse maps on compact subsets of R2 | p. 184 |
| Problems | p. 185 |
| Infinite sets | p. 191 |
| Countable and uncountable sets | p. 191 |
| The positive rational numbers are countably infinite | p. 192 |
| The Cantor set is not a countable set | p. 192 |
| The continuum question | p. 193 |
| Problems | p. 193 |
| Bibliography | p. 195 |
| Solutions to selected problems | p. 197 |
| Index | p. 207 |
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