# Exercises for Chaos Under Control

## Chapter 2: Making Some Other Pictures

In this set of exercises "DET" and "RAND" refer to Options in the TreenessEmerging Program.

5.[A] Consider the IFS defined by these rules

scale by 0.333;

scale by 0.333, then translate by +0.5 in the x-direction;

scale by 0.333, then translate by +0.5 in the y-direction;

collage the pieces

Starting from the triangle with vertices (0,0), (1,0), and (0,1), observe with each successive iteration the total size of the picture shrinks a bit. Show that the base and height of the final Gasket dust are each 3/4. (Hint: the final Gasket dust is left unchanged by the this IFS.) Answer

6.(a) Would the transformations

scale by 0.333;

scale by 0.333, then translate by +0.667 in the x-direction;

collage the pieces

lead to the Cantor MTS if we had started with something besides a nicely oriented line segment? Try starting with a square, say. Draw a few generations by hand. What do you get?

(b)[C] Run DET to carry this experiment to a few more stages.

7. Determine IFS rules for these Cantor sets

(a) Cantor Middle 1/2 Set, Answer

(b) Cantor Middle 2/3 Set, Answer

(c) Cantor Middle 1/5 Set, Answer

(d)[C] Run DET to check your results.

8.(a) What shape is drawn by IFS1?

IFS1:

scale by 0.5;

scale by 0.5, then translate by 0.5 in the x-direction;

scale by 0.5, then translate by 0.5 in the y-direction;

collage the pieces

(b) What shape is drawn by IFS3?

IFS3:

scale by 0.5;

scale by 0.5, then translate by 0.5 in the x-direction;

scale by 0.5, then translate by 0.5 in both the x- and the y-directions;

collage the pieces.

(c) Find the parameters for IFS2, "halfway between" IFS1 and IFS3.

(d)[C] Run DET to show the figures resulting from each of these three IFS. To make the comparisons uniform, take the first picture to be single point. Can you visualize a continuous transformation between the limiting pictures of these three IFS? Answer