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
Return to Chapter 2 Exercises
Return to Chapter 2 Exercises: The Gasket Shall be our Guide
Go to Chapter 2 Exercises: Deterministic Iterated Function Systems
Return to Chaos Under Control
