Exercises for Chaos Under Control

Chapter 2: Pictures from Nature

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

24.[C] Run the Eq. Gasket in RAND, noting carefully how the Gasket grows. Now change the values of "Prob" to 0.600, 0.200, and 0.200. Guess how this will affect the growth of the Gasket. Run Rand with these probabilities, carefully observing the new growth. Explain what you observe.

25.[C] Let's see what happens if we don't scale by the same factor in all directions. Use the Eq. Gasket Rule in RAND and change the three R values to 0.333. Before running this set of parameters, try guessing what picture they will lead to. Can you explain the actual result? With the three R values returned to their original 0.5, set the three S values to 0.333 and run the RAND option. (Neat result, don't you agree?) Answer

26.[C] We can also try different translations in different directions. Starting with the original Eq. Gasket parameters, set the second set of "E" and "F" values to 0 and 0.500, respectively. What do you think will happen when you run this picture? Run RAND.

27.[C] Just as we had the freedom to scale and translate different amounts in the x- and y-directions, so too do we have the freedom to rotate different amounts. Unequal theta and phi produce a kind of squashing or stretching. Select the Koch Rule in RAND.

(a) Substitute +30¡ for +60¡ and -30¡ for -60¡ in the Theta column, leaving the Phi unchanged. Do you see what we mean by "squashing?"

(b) Suppose the Theta values had been left at ±60¡ and the Phi changed to ±30¡. What would happen? Try the experiment.

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