# Exercises for Chaos Under Control

## Chapter 2: What is Random?

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

43. Let rules 1, 2, and 3 be

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.

(a) What kind of behavior would you get from applying the rules in the sequence 1, 2, 1, 2, 1, 2, ... ? Start at the point (0.5,0.5) and plot about ten of the resulting points to check your answer. Answer

(b) What about the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3, ... ? Answer

(c) What about the sequence 1, 1, 2, 3, 1, 1, 2, 3, 1, 1, 2, 3, ... ? (You can speed your procedure by substituting the equivalent Chaos Game rules "go 1/2-way toward (0,0) (or (1,0) or (0,1)).") Answer

44. When applied randomly, the rules of problem 43 generate a right isosceles gasket. Label the three main pieces of the gasket A, B, and C, where A denotes the small gasket (half the size of the original) containing (0,0), B the small gasket containing (1,0), and C the small gasket containing (0,1).

(a) Is there any sequence of 1s, 2s, and 3s that will completely draw part A of the gasket, but will put no points in parts B or C? Explain your answer.

(b) Is there any sequence of 1s, 2s, and 3s that will completely draw parts B and C, but will put no points in part A? Explain your answer.

45. Rules 1-4 (top-to-bottom) of a four rule IFS to make a square are listed in this table.
R S Theta Phi E F
.5 .5 0 0 0 0
.5 .5 0 0 .5 0
.5 .5 0 0 0 .5
.5 .5 0 0 .5 .5

Give the first three numbers of the addresses of all points within the white block of the gray square using the order of the table above. Explain. Answer

46. A Chaos Game is played in the square shown here. The Game has four rules: (A) go halfway to A; (B) go halfway to B; (C) go halfway to C; and (D) go halfway to D. The Game is played by starting at D and applying the string ABCDABCDABCDABCD ... (where ... means keep on going with the same sequence). Determine, by explicitly constructing enough of them to see what's happening, what points are hit in this Game. Explain.

47. A Chaos Game is played in the square shown here. The Game has four rules: (A) go halfway to A; (B) go halfway to B; (C) go halfway to C; and (D) go halfway to D. The Game is played by starting at D.

(a) Suppose the rules A, B, C, and D are drawn uniformly randomly. What picture emerges after many points are plotted? Sketch it in this square and explain. Answer

(b) A second picture is constructed as a result of drawing A, B, C, and D uniformly randomly, but every time A is drawn it is ignored (no point is plotted). What picture emerges after many points are plotted? Sketch it in this square and explain. Answer