Curious curves

Preface

Acknowledgments

List of Figures

List of Tables

1. Examples of curious curves

1.1. Variations of the Koch curve

1.1.1. Koch curve

1.1.2. Modified Koch curve

1.1.3. Basics of complex numbers

1.2. More examples of curious curves

1.2.1. The unit square is a curve

1.2.2. Iterated function systems produce curves

1.3. Construction of a family of Cantor sets

1.3.1. Middle-thirds Cantor set

1.3.2. Construction of generalized Cantor sets

1.3.3. The length of the Cantor set C 0

1.3.4. The sets C h

1.4. What is not a curve?

2. The Koch curve and tangent lines

2.1. Construction of the Koch curve

2.1.1. Representation in base 4

2.1.2. Formulas for f k

2.1.3. Convergence of the sequence {f k }

2.1.4. An equation for f

2.1.5. Length of the Koch curve

2.2. Tangent lines to simple curves in C

2.2.1. Definition of a tangent line to a simple curve

2.2.2. Another construction of the Koch curve

2.2.3. Tangent lines to graphs of continuous maps from I to R

2.2.3.1. Modified Cantor functions

2.2.3.2. The graph G φ of the Cantor function

2.3. Problems

3. Curves and Cantor sets

3.1. A square is a curve!

3.2. Simple curves

3.2.1. A homeomorphism g with C x C ≈ g(I)

3.2.2. Simple curves with positive area

3.2.3. Proofs of Propositions 3.1 and 3.2

3.2.3.1. Proof of Proposition 3.1

3.2.3.2. Proof of Proposition 3.2

3.3. Continuous images of the Cantor set

3.4. Subsets of C that are not curves

3.5. Generalized curves

3.6. More examples of curves

3.7. Problems

4. Generalizations of the Koch curve

4.1. Construction of generalizations

4.1.1. The iteration process

4.1.2. A decomposition of K a,θ

4.2. Double points in K a,θ with θ = π/3 and a = 1/4

4.2.1. The pivotal value a (θ) = 1/4 for θ =π/3

4.3. Investigation of K a,π/4

4.3.1. Verifying

4.3.2. Double points form Cantor sets

4.4. Problems

5. Metric spaces and the Hausdorff metric

5.1. Metric spaces

5.1.1. Equivalent metrics

5.1.2. Topological properties of metric spaces

5.1.3. Complete metric spaces

5.2. The Hausdorff metric

5.3. Metrics and norms

5.4. Problems

6. Contraction maps and iterated function systems

6.1. Contraction maps

6.2. Iterated function systems

6.3. An iterated function system defines a curve

6.4. Implementation of iterated function systems

6.5. Problems

7. Dimension, curves and Cantor sets

7.1. Intervals, squares and cubes

7.2. Hausdorff dimension of a bounded subset of R 2

7.2.1. Basic facts about dimension

7.3. Tent maps and Cantor sets with prescribed dimension

7.3.1. Dimension of Cantor sets

7.3.2. Dimension of Cantor sets in the plane

7.4. Dimension and simple curves

7.4.1. Simple curves with prescribed dimension

7.4.2. Dimension of the Koch curve

7.4.3. Functions with prescribed dimension of points of non-tangency

7.5. Symmetric Cantor sets (optional section)

7.5.1. Construction of a symmetric Cantor set

7.5.2. Definition of dimension of symmetric Cantor sets

7.6. Saw tooth maps

7.7. Problems

8. Julia sets and the Mandelbrot set

8.1. Theory of Julia sets

8.1.1. Observations

8.1.2. Visual images

8.1.3. Two facts about Julia sets

8.2. The Mandelbrot set

8.2.1. Fixed points of f c

8.2.2. The central cardioid

8.2.3. The great circle

8.2.3.1. Period two points

8.2.3.2. Description of the great circle

8.2.4. Super-attracting fixed points

8.2.5. The two large bulbs adjoining the central cardioid

8.3. Generalized curves and Julia sets

8.4. Problems

Appendix A. Points on a line

A.1. Labeling points on a line

A.l.l. base b representations

A.2. Convergence

A.2.1. The geometric series

A.3. The special nested interval property

A.4. Bounds on subsets of a line

A.4.1. Bounded sequences have convergent subsequences

A.5. The real numbers R

A.6. Eventually periodic base b representations

A.7. Problems

Appendix B. Length and area

B.l. Intervals and length

B.2. Lengths of subsets of intervals

B.3. Intervals and rectangles in the plane

B.4. Length of a curve

B.5. Areas of subsets of the plane

B.5.1. Areas of rectangles

B.5.2. Areas of general subsets of the plane

B.6. Problems

Appendix C. Maps and sets in the plane

C.l. Definition of a map

C.2. Properties of points in the plane

C.3. Continuity and limits

C.4. Topological properties of subsets of R 2

C.4.1. Closed sets

C.4.2. Compact sets

C.4.3. Connected sets

C.4.4. Fixed points of maps

C.4.5. Uniform continuity of maps

C.5. Convergence of maps

C.6. Linear maps from R 2 to R 2

C.7. Homeomorphisms: Inverse maps on compact subsets of R 2

C.8. Problems

Appendix D. Infinite sets

D.l. Countable and uncountable sets

D.l.l. The positive rational numbers are countably infinite

D.l.2. The Cantor set is not a countable set

D.1.3. The continuum question

D.2. Problems

Bibliography

Solutions to selected problems

Index