Fractals in Science : An Introductory Course

Nature is full of spidery patterns: lightning bolts, coastlines, nerve cells, termite tunnels, bacteria cultures, root systems, forest fires, soil cracking, river deltas, galactic distributions, mountain ranges, tidal patterns, cloud shapes, sequencing of nucleotides in DNA, cauliflower, broccoli, lungs, kidneys, the scraggly nerve cells that carry signals to and from your brain, the branching arteries and veins that make up your circulatory system. These and other similar patterns in nature are called natural fractals or random fractals. This chapter contains activities that describe random fractals. There are two kinds of fractals: mathematical fractals and natural (or random) fractals. A mathematical fractal can be described by a mathematical formula. Given this formula, the resulting structure is always identically the same (though it may be colored in different ways). In contrast, natural fractals never repeat themselves; each one is unique, different from all others. This is because these processes are frequently equivalent to coin-flipping, plus a few simple rules. Nature is full of random fractals. In this book you will explore a few of the many random fractals in Nature. Branching, scraggly nerve cells are important to life (one of the patterns on the preceding pages). We cannot live without them. How do we describe a nerve cell? How do we classify different nerve cells? Each individual nerve cell is special, unique, different from every other nerve cell. And yet our eye sees that nerve cells are similar to one another.

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.