# 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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Random Iterated Function Systems

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