Exercises for Chaos Under Control

Chapter 2: Random Iterated Function Systems

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

20.[C] Use the Eq. Gasket Rule and DET. Starting with a single point, time how long it takes to produce eight generations. Note the size of the Gasket drawn and its quality. Now run RAND. Adjust the size so the Gasket produced is about the same size as that with DET. (This may take a few tries.) Time how long it takes the random algorithm to produce a Gasket of roughly the same quality as in the first part.

21.[C] Use the Eq. Gasket Rule in RAND, noting how the Gasket looks.

(a) For each of the three functions (rows), change the 0.500 in both the R and S columns to 0.333. Before running RAND, make a sketch of what shape will result from running the random algorithm with these functions. Now run RAND and see if you were right. How does your sketch compare with the picture which appears? Answer

(b) Recall the columns labeled E and F correspond to translations in the x- and y-directions, respectively. As presented to you, one translation is E = 0, F = 0, a second is E = 0.250, F = 0.433, and the third is E = 0.5, F = 0. The second translation is just right to make an equilateral triangular Gasket. Try to make an isosceles, right triangular Gasket. Answer

22.[C] Now let's try another set of scalings for the Eq. Gasket in RAND. Change all the R and S values to 0.667, say. From Chapter One you already know what this scaling should produce. Does it? Answer

23.[C] Run RAND to see the Koch Curve produced. To see that the sign of the downward rotation is crucial, change the two -60.000s to +60.000s. Can you explain why the resulting picture looks as it does?

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