Fractals: The Geometry of Nature
Revised: November 2006 (Julie Barnes)
Investigate basic mathematical principles behind fractals with connections to literature, art, science, and the general world.
1. To understand the mathematics behind a topic most students have never studied
much but find fascinating - fractals.
2. To apply mathematical concepts to the fractals around them in literature, art, science and nature.
3. To gain a greater appreciation of mathematics.
4. To clearly and critically write about fractals in their own lives and in our world.
Peak, David and Frame, Michael. Chaos Under Control, Freeman, 1994.
Stoppard, Tom. Arcadia: A Play, Faber & Faber, 1994.
Grading procedures and factors influencing course grade are left to the discretion of individual instructors, subject to general university policy.
Attendance policy is left to the discretion of individual instructors, subject to general university policy.
Unit I: The Fundamentals of Fractals & Iterated Function Systems (15 Days)
We look at the basic concepts of geometric iteration, self-similarity, fractal dimension, multiple reduction copy machines, the chaos game, fractals in nature, and decoding fractals. Students do hands-on group activities, experiment with applets on the web, and discuss assigned readings. Students also learn how to design fractals using basic geometric transformations. We see how fractals can model images with an amazing degree of accuracy, and the information compressed to a simple set of numbers.
Unit II: Connections with Arts & Humanities (12 Days)
We discuss Stoppard's highly entertaining play Arcadia and the mathematical themes therein. In the play, we see a 13-year-old math prodigy, Thomasina, discard classical Euclidean geometry to discover the principles of fractal geometry (in 1809!), while almost two centuries later, in the very same room, we find a mathematical biologist using some of the same ideas to study fluctuating game bird populations. We also discuss some other examples of fractals found in the arts and humanities. These examples could include but not be limited to: Wallace Stevens' poem, The Sail of Ulysses, Ray Bradbury's A Sound of Thunder, paintings of Jackson Pollock, short animations by Francesca Talenti that provide stunning visual interpretations of fractals and chaos, clips of fractal music, and examples of fractals in popular movies like Jurassic Park. Students discuss assigned reading in class and write about how the topics in this section are related to fractals.
Unit III: Cellular Automata and Scientific Applications of Fractal Geometry (7 Days)
John Conway's Game of Life (popularized by Martin Gardner in his Scientific American column) describes simple rules that determine the "life" or "death" of the next generation of cells, based on interaction with neighboring cells. They are based on simple rules but show surprisingly complex behavior. Far more than a game with pretty pictures, they are related to exciting new ideas such as artificial life and the edge of chaos. We study examples and patterns and some applications. Students do hands-on activities, use applets on the web to better understand the Game of Life, and participate in group discussions. Then they write a short paper on how fractals relate to science and our natural world.
Unit IV: A Tour of the Mandelbrot Set. (6 Days)
We study the fascinating structure of the Mandelbrot set, one of today's most ubiquitous and geometrically intricate figures, exploring some of its geometric and number theoretic properties. Students will experiment with applets on the web, and participate in group discussions.
Unit V: Creative Projects. (4 Days)
Students will complete a final creative project that involves researching an application to fractals and chaos. Students will create something to go along with the project, like artwork, a short story, or a computer generated image. They will also write an in-dept paper about how the item is related to chaos and fractals and make a presentation in front of the class.