Fractals Everywhere: The Turtle vs. Dr. Barnsley
Friday February 14, 2003
Host: Demetri Terzopoulos, firstname.lastname@example.org, 212-998-3477
Turtle competitions have appeared in many guises in literature, philosophy, and mathematics. Aesop (500BC) tells a tale about a celebrated race between the tortoise and the hare. Zeno (450BC) presents a famous paradox, central our modern understanding of convergence, concerning a race between Achilles and the tortoise. Lewis Carroll (1870) wrote a dialogue between Achilles and the tortoise, where the tortoise sets out to convince Achilles that much like Achilles can never catch the tortoise in a race, he can never get to the end of any proof in mathematics. Hofstadter (1980) in his book on Godel, Escher, and Bach also has several dialogues between Achilles and the ubiquitous tortoise.
Today the turtle has gone high tech. Reincarnated inside the computer, the turtle has his own programming language (LOGO) for studying geometry. Fractals are generated in LOGO via recursive programs. The turtle's modern nemesis is no longer the fleet footed, slow witted Achilles, but rather the greedy capitalist Dr. Barnsley, author of the book Fractals Everywhere. Dr. Barnsley generates fractals using iterated affine transformations, and he plans to take over the world of data compression with this approach. Here then we have two methods for generating fractals: recursive turtle programs and iterated affine transformations. Given two models of computation, it is natural to ask which one is more powerful? The purpose of this talk is to explore this question from the turtle's perspective.