Fractals in Science

1. From Coin Flipping to Motion of Molecules.- 1.1 Introduction.- 1.2. Randomness: Making Predictions Even When No One Knows What Will Happen Next.- Activity 1.1 Lottery Game.- 1.3. How Often Something Happens (Probability Distributions).- Activity 1.2 Coin-Flipping.- Activity 1.3 Flipping Coins by Computer.- 1.4 Random Movement (Random Walks).- Activity 1.4 Ten-Step Random Walk by Hand.- Activity 1.5 Random Walk Program: 1-D Random Walk.- 1.5 Pascal's Triangle.- Activity 1.6 Random Walk Program: Pascal's Triangle.- Activity 1.7 Many Walkers Program: Distribution Width.- Activity 1.8 Average Position After N Steps.- 1.6 Average Displacement and Average Squared Displacement.- Activity 1.9 Measures of Average Squared Displacement.- 1.7 Diffusion.- Activity 1.10 Hands-on Random Walk in 2-Dimensions.- Activity 1.11 Computer Random Walk in 2-Dimensions.- Activity 1.12 Number of Distinct Sites Visited.- Activity 1.13 Biased Random Walk: Approximating Nature.- Activity 1.14 Population Dynamics: The Deer Program.- Activity 1.15 Diffusion Chamber Experiment.- 1.8 Periodic Precipitation: Liesegang Rings.- Activity 1.16 Liesegang Experiment.- 1.9 Postlude: The Meaning of Models.- Appendix A Demonstration that Average Squared Displacement Is Equal to Number of Steps Taken.- Appendix B. Preparation for Liesegang Ring Experiment.- 2. Fractals in Nature. Growing and Measuring Random Fractals.- 2.1 Introduction.- 2.2. Coastline.- Activity 2.1 "Walking" Along a Coastline.- Activity 2.2 Measuring the Dimension of a Coastline.- Activity 2.3 Creating Your Own Coastline By Hand.- Activity 2.4 Computer Program for "Walking" Along a Coastline.- Activity 2.5 Covering a Coastline with Boxes by Hand.- Activity 2.6. Computer Covering of a Coastline.- 2.3. The Meaning of Dimension.- Activity 2.7 Hands-on Circle Measurement of Dimension.- 2.4. Growing Random Patterns.- Activity 2.8 Growing a Pattern in the Laboratory.- 2.5. Computer Measurement of Fractal Dimension.- Activity 2.9 Fractal Dimension Computer Program.- 2.6. Modeling the Growth of a Fractal Pattern.- Activity 2.10 Building an Aggregate by Hand.- Activity 2.11 Building an Aggregate by Computer.- 2.7. The Hele-Shaw Experiment.- Activity 2.12 The Hele-Shaw Experiment.- 2.8 Branching Patterns Formed by Bacterial Colonies.- Activity 2.13 Modeling a Rough Surface.- Activity 2.14 Growth of Bacteria under Starvation Conditions.- 2.9 Termite Nesting and Foraging in Two Dimensions.- Activity 2.15 Termite Tunnel Patterns.- 2.10 Fractal Root Systems.- Activity 2.16 Growing Roots.- 2.11. Research!.- Activity 2.17 Research on Electrochemical Deposition.- Activity 2.18 Research on Viscous Fingering.- Activity 2.19 Research on Bacterial Colonits.- Activity 2.20 Research on Termite Colonies.- Activity 2.21 Research on Roots.- Appendix A. Dimensions and Logarithms.- Appendix B. Construction of ECD and Hele-Shaw Cells.- Appendix C. Using Video Frame Grabbing.- Appendix D. Using the Scanner for Imaging.- Appendix E. Preparing the Carrageenan Solutions.- Appendix F. Using the Vernier pH Meter.- Appendix G. Preparation of Bacterial Growth Experiment.- Appendix H. Constructing the Cell for a Termite Colony.- Appendix I. Constructing Rhizotron for Root Systems.- 3. Growth Patterns in Nature: Percolation.- 3.1 Growing a Forest.- Activity 3.1 Growing a Forest by Hand.- Activity 3.2 Growing a Forest by Computer.- Activity 3.3 Computer an Adequate Model? Forest Size.- 3.2 Burning a Forest.- Activity 3.4 Burning a Forest by Hand.- Activity 3.5 Burning a Forest by Computer.- Activity 3.6 Jello Experiment.- 4 DNA and Literature.- 4.1 DNA Sequences.- 4.1.1 The Basics.- 4.2 Storing Information in DNA.- 4.2.1 Coding and Non-Coding Sequences in DNA.- 4.2.2 Measures of Information.- 4.3 Word Occurences and Zipf's Law.- 4.3.1 Word Occurrences and Zipf's Law in Literature.- 4.3.2 Word Occurrences in DNA.- 4.3.3 n-Letter Words in Literature.- 4.3.4 n-Letter Words in DNA.- 4.3.5 Zipf's Law in DNA.- 4.4 Landscapes of Literature and DNA.- 4.4.1 Why a new approach?.- 4.4.2 Climb the "language mountain".- 4.4.3 DNA landscapes.- 4.4.4 The roughness exponent.- 4.4.5 Zero Information or Maximum Information?.- 4.5 Discover coding regions in DNA.- 4.5.1 Landscapes in Literature.- A Answers to problems..- B Log-Log-paper.- C This and That.- C.1 Translating codons.- C.1.1 Translation of the poem Nell.- 5 Spin Glasses and Neural Networks.- 5.1 Introduction.- 5.2 Magnetic Order.- 5.3 Ising Model.- 5.3.1 Activity 1: Energy of Ferromagnets.- 5.3.2 Activity 2: Energy of Antiferromagnets.- 5.4 Magnetic Disorder.- 5.5 Spin Glasses.- 5.5.1 Activity 3: Frustrated Magnetic Bonds.- 5.6 Hopfield Model.- 5.6.1 Activity 4: Ising Patterns.- 5.6.2 The Hopfield model as a model of neural networks.- 5.7 Bibliography.- 6 Lightning and Soap Films.- 6.1 Introduction.- 6.2 Electric Field.- 6.2.1 Activity 1: Hands-on-Your Room.- 6.3 Electrostatics.- 6.3.1 Activity 2: Hands-on - The Earth.- 6.4 Laplace Equation.- 6.4.1 Activity 3: Hands-On: An Old Recipe.- 6.5 Numerical Approach to our Main Task.- 6.5.1 Activity 4: Hands-On: Field Research I.- 6.5.2 Activity 5: Hands-On: Wood Work.- 6.6 Fractal Growth.- 6.6.1 Activity 6: Hands-On: Field Research II.- 6.7 Bibliography.- 7 Analyzing Rough Surfaces Digitally.- 7.1 Introduction.- 7.2 Defining the Self-Affine Surface.- 7.2.1 The Reading Glass.- 7.2.2 Constructing the Self-affine Object.- 7.3 Activity: The Paper Tear Experiment.- 7.3.1 The experiment.- 7.3.2 Measuring the roughness exponent with the computer.- 7.3.3 Scanning the paper tear.- 7.3.4 The Rough Surface Analyzer Program.- 7.4 The Paper Tear: Understanding Experiments with Models.- 7.4.1 The Random Walk Model.- 7.4.2 The Minimal Energy Route Model.- 7.4.3 How the Paper Finds the Minimal Energy Route.- 7.5 Testing the Minimal Energy Model.- 7.6 Conclusion.