Department of Mathematics MAT 580 — SPECIAL TOPICS IN MATHEMATICS: FRACTALS COURSE DESCRIPTION Through hands-on activities, computer experiments, visuals, readings, discussions, and lectures, this course will examine the concepts of iteration, fractals, and chaos, as well as the history and people involved in this fascinating field. The course will begin by looking at the basic concept of iteration: repeatedly applying some specified operation. Iteration leads naturally to, among other things, the geometric representation of a fractal, self-similarity, fractal dimension, and the notion of mathematical chaos. Students will learn concepts from classical mathematics and ways in which these concepts can be applied to model and explain real-world scenarios. Additionally, they will be shown that mathematics is a current, active, useful, and dynamic field of study. 3 c.h., 3 s.h. RATIONALE The concept of fractional dimension has been around for about 80 years, but the term “fractal” itself and the interest in them, both popular and scientific, date from the proliferation of microcomputers in the early 1980s. For example, using the self-similarity feature of fractals allows for efficient data compression in such computer applications as Microsoft’s Encarta encyclopedia and in video games. As a result of all this interest, fractals have begun appearing in junior and high school textbooks as a way of demonstrating to today’s younger students that mathematics is an active and dynamic field of study, one for which all the answers are not known, indeed one for which all the questions have not even been asked. Our current mathematics curriculum at KU, both undergraduate and graduate, is decidedly more classical in nature. Indeed, we only peripherally touch on the subject of fractals in a few of our undergraduate courses. Because the study of fractals is a very popular, cutting edge, experimental branch of mathematics, it is one to which our students, in particular those in our Master of Education in Secondary Education with Specialization in Mathematics program, should be exposed. This course satisfies one of the course requirements in AREA IV: Subject Specialization of the Master of Education in Secondary Education degree. COURSE OBJECTIVES Students will be able to: Define the concepts of fractals, self-similarity, and fractal dimension, and and apply them to various other parts of mathematics. Generate classic fractals such as the Cantor Set, the Sierpinski Gasket, and the Koch Snowflake. Compute the limiting perimeter and area enclosed by fractal sets. Encode images by simple transformations. Use technology to perform the iterations needed to generate such fractal sets as Julia and Mandelbrot sets. Use their proof skills in the area of analysis as it applies to fractals and dynamical systems. Model real-world systems of interest so they can be investigated by mathematical tools. Explain how and when dynamical systems can demonstrate chaotic behavior. 1 COURSE OUTLINE I. Feedback and the Iterator A. The Principle of Feedback B. The Multiple Reduction Copy Machine (MRCM) C. Basic Types of Feedback Processes II. Classical Fractals and Self-Similarity A. The Cantor Set B. The Sierpinski Gasket and Carpet C. The Pascal Triangle D. The Koch Curve E. Space-Filling Curves F. Fractals and the Problem of Dimension G. The Universality of the Sierpinski Carpet H. Julia Sets I. Pythagorean Trees III. Limits and Self-Similarity A. Similarity and Scaling B. Geometric Series and the Koch Curve C. Comer the New from Several Sides: Pi and the Square Root of Two D. Fractals as Solution of Equations E. Box Self-Similarity: Grasping the Limit IV. Length, Area and Dimension: Measuring Complexity and Scaling Properties A. Finite and Infinite Length of Spirals B. Measuring Fractal Curves and Power Laws C. Fractal Dimension D. The Box-Counting Dimension E. Borderline Fractals: Devil's Staircase and Peano Curve V. Encoding Images by Simple Transformations A. The Multiple Reduction Copy Machine Metaphor B. Composing Simple Transformations C. Classical Fractals by Iterated Function Systems (IFSs) D. Image Encoding by IFSs . E. Foundation of IFS: The Contraction Mapping Principle F. Choosing the Right Metric G. Composing Self-Similar Images . H. Breaking Self-Similarity and Self-Affinity, or, Networking with MRCMs 2 VI. The Chaos Game: How Randomness Creates Deterministic Shapes A. The Fortune Wheel Reduction Copy Machine B. Addresses: Analysis of the Chaos Game C. Tuning the Fortune Wheel D. Random Number Generator Pitfall E. Adaptive Cut Methods VII. Irregular Shapes: Randomness in Fractal Constructions A. Randomizing Deterministic Fractals B. Percolation: Fractals and Fires in Random Forests C. Random Fractals in a Laboratory Experiment D. Simulation of Brownian Motion E. Scaling Laws and Fractional Brownian Motion F. Fractal Landscapes INSTRUCTIONAL RESOURCES Alligood, K. T. (1997). Chaos: An introduction to dynamical systems. New York: Springer Verlag. Bannon, T. J. (1991). Fractals and transformations. Mathematics Teacher , 84 (3), 178-85. Barcellos, A. (1984). The fractal geometry of Mandelbrot. College Mathematics Journal, 15 (2), 98-114. Barnsley, M. F. (1988). Fractals everywhere. Boston: Academic Press. Barton, R. (1990). Chaos and fractals. Mathematics Teacher, 83 (7), 524-29. Briggs, J. (1992). Fractals: the patterns of chaos: A new aesthetic of art, science, and nature. New York: Simon & Schuster. Briggs, J., & Peat, F.D. (1989). Turbulent mirror. NY: Harper & Row. Bunde, A., & Havlin, S. (Eds.). (1994). Fractals in science: With a MS-DOS program diskette. Berlin ; New York: Springer-Verlag. Camp, D. R. (1995). Fractal geometry in the high school classroom. International Reviews on Mathematical Education, 27 (5), 143-52. Camp, D., Chiaverina, C., & Senior, T. (1999) Festive fractals. Physics Teacher, 37 (9), 532. Choate, J. (1999). Fractals: A tool kit of dynamics activities. Emeryville, CA: Key Curriculum Press. Cibes, M. (1990). The Sierpinski triangle: Deterministic versus random models. Mathematics Teacher, 83 (8), 617-21. Crilly, A.J., Earnshaw, R.A., & Jones, H. (Eds.). (1993). Applications of fractals and chaos: The shape of things. New York: Springer-Verlag. 3 Coes, L. (1993). Building fractal models with manipulatives. Mathematics Teacher, 86 (8), 646-51. Davis, D. M. (1993). The nature and power of mathematics. Princeton, N.J.: Princeton University Press. Devaney, R. L. (1992). A first course in chaotic dynamical systems: Theory and experiment. Reading, MA: Addison-Wesley. Devaney, R. L. (2000). The Mandelbrot and Julia sets: A tool kit of dynamics activities. Emeryville, CA: Key Curriculum Press. **Dewdney, A. K. (1991). Exploring fractals. Science Probe, 1, 49-57. Edgar, G. A. (Ed.). (1993). Classics on fractals. Reading, MA: Addison-Wesley. **Eglash, R. (1999). African fractals. New Brunswick, NJ: Rutgers University Press. **Falconer, K. J. (2003). Fractal geometry : Mathematical foundations and applications. Chichester, NY: Wiley. Falconer, K. J. (1997). Techniques in fractal geometry. Chichester, NY: Wiley. **Fauvel, J., Flood, R., Wilson, R., & Wilson, R.J. (2003). Music and mathematics: From Pythagoras to fractals. New York: Oxford University Press. Feder, J. (1988). Fractals. New York: Plenum Press. Fisher, Y. (Ed.). (1995). Fractal image compression: Theory and application. New York: Springer-Verlag. Forster, P. (1997). Using fractals to teach complex numbers with a constructivist approach. Australian Senior Mathematics Journal, 11 (2) 14-22. **Gary, W.F. (2000). The computational beauty of nature: Computer explorations of fractals, chaos, complex systems, and adaptation. Cambridge, MA: MIT Press. Gazalé, M. J. (1999). Gnomon: From pharaohs to fractals. Princeton, N.J.: Princeton University Press. **Gleick, J. (1988). Chaos: Making a new science. East Rutherford, NJ: Penguin. Gullberg, J. (1997). Mathematics:From the birth of numbers. New York: W.W. Norton. **Havil, J. (2003). Gamma: Exploring Euler's constant. Princeton, NJ: Princeton University Press. Iannaccone, P. M., & Khokha, M. (1996). Fractal geometry in biological systems: An analytical approach. Boca Raton, FL: CRC Press. Jurgens, H. (1990). The language of fractals. Scientific American, 263, 60-67. **Lapidus, M. L., & van Frankenhuysen, M. (2000). Fractal geometry and number theory. Boston: Birkhauser. **Lesmoir-Gordon, N., Rood, W., Edney, R., & Appignanesi, R. (2000). Introducing fractal geometry. Blue Ridge Summit, PA: Totem Books. 4 Liebovitch, L. S. (1998). Fractals and chaos simplified for the life sciences. New York: Oxford University Press. Mandelbrot, B. B. (1977). Fractals: Form, chance and dimension. San Francisco: W. H. Freeman. Mandelbrot, B. B. (1983). The fractal geometry of nature. New York: W.H. Freeman. **Mandelbrot, B. B. (2004). Fractals and chaos. New York: Springer-Verlag. McCauley, J. L. (1993). Chaos, dynamics, and fractals: An algorithmic approach to deterministic chaos. New York, NY: Cambridge University Press. McGuire, M. D. (1991). An eye for fractals: A graphic & photographic essay. Redwood City, CA.: Addison-Wesley. **Le Mehaute, A. (1991). Fractal geometries: Theory and applications. Boca Raton, FL: CRC Press. Milner, J., Hodgson, C., Moore, K., & Wheatley, V. (2002). Fractals with Word. Mathematics Teaching , 178, 23-26. Moon, F. C. (1992). Chaotic and fractal dynamics: An introduction for applied scientists and engineers. New York: Wiley. Nievergelt, Y. (1991). Fractals illustrate the mathematical way of thinking. College Mathematics