WittSem 100L: Patterns in Nature

Assignment for Thursday, October 18, 2007

To hand in (on a separate sheet of paper):

1. (6 pts) Below is the Medium level of Devaney’s Chaos Game Number 2 from

http://math.bu.edu/DYSYS/applets/chaos-game.html.

a) Determine the IFS rules that generate this fractal. (Hint: There will be at least one rotation. You can use the Yale website software to check your guesses.)

b) Come up with a general strategy for winning the Medium level of Game 2. Clearly write out your general strategy. Then, as a specific example, give a sequence of 5 moves that wins the game below. (You're encouraged to play with the Chaos Game online as you develop your strategy.)

Bonus challenge: (4 pts) Repeat parts a) and b) for the Medium level of Devaney’s Game 3 (below).

2. (6 pts) Generating a leaf fractal – The fractal below was obtained using four IFS rules (the fourth generated the lower stem). What are the IFS rules? Hint: this is a variation on the Barnsley Fern. Give the four IFS rules, and generate and print out the fractal (using the Random IFS program at the Yale website).

3. (6 pts) Create your own natural-looking fractal using the Random IFS software on the Yale fractals website. One way to do this is to start with a given fractal and modify it to make it more interesting by changing some of the IFS rules and/or adding more rules. It’s interesting to observe how even a small change to the parameters can result in a very different looking fractal. Your fractal will be graded on originality, complexity, and naturalness. Take your time and do a good job!

For your fractal, give the IFS rules (including probabilities); generate and print out the fractal; and give it an appropriate title.