Advanced Mathematics from an Elementary Viewpoint: Chaos, Fractal Geometry, and Nonlinear Systems

Abstract

Most naturally occurring processes are inherently nonlinear and can give rise to very complex behaviors. Even very simple mathematical models can exhibit behavior gives rise to extremely convoluted (and often very beautiful) fractal shapes. The discovery of this fundamentally new area of mathematics has been crucially dependent on computationally intensive graphic methods and has given birth to a radically new paradigm for mathematical research: experimental mathematics. Though this material is at the frontier of current research, it is accessible to high school students, and the graphic images that are generated through its investigation are highly informative and often of uncommon visual richness and beauty. Mathematical materials and associated computer software aimed at motivating and empowering high school students of average mathematical ability to perform experimental mathematical investigations will be designed, developed, and evaluated. The mathematical content will comprise fractals, nonlinear dynamics, and mathematical chaos. Fundamental mathematical concepts will be applied to a wide variety of physical, biological and social processes (e.g. population growth, problems in epidemiology, and the economics of the arms race). The deep connection between geometry and nonlinear dynamics will be made explicit by developing computer-based tools that will enable students to generate fractal maps and pictures of compelling complexity and beauty. Finally, through their work in experimental mathematics students will acquire a deeper understanding of mathematical and scientific thinking.