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Extending the Chaos Game: Determinism and DNA


HIV Color Game
I recently came across some interesting variations and applications of the Chaos Game.

One version is located at the Boston U web site and is credited to Johanna Voolich and Robert L. Devaney. Their version is very unique twist on the Game - instead of the randomness that forms the basis of the Game, they have replaced the randomness with a strictly deterministic game. In their version, the player tries to get a random starting point into a specific target area of the Sierpinski triangle by choosing a series of half-way jumps to specific vertices. The "winner" of the gme is the one who reaches the target area in the fewest number of jumps. (Click here to play the game on-line.)

An even more fascinating Chaos Game comes from Dan Ashlock at Iowa State University, where he uses a 4-cornered Chaos Game to display the sequence of bases in DNA molecules. (The image at the top of this post is from a segment of HIV DNA.) In this Chaotic Game of Life, points are drawn half-way to vertices depending on the next base in the DNA strand. The pictures look remarkably similar to playing the Chaos game randomly! See Ashlock’s website for a set of images from different organisms, as well as some Markov chain models that help explain and interpret the images.