Exercises for Chaos Under Control

Chapter 2: The Gasket Shall be our Guide

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

1.(a) Prove that when (x, y) is transformed by the rule "scale by a factor of 0.5" the new point, (0.5x, 0.5y), is half as far from the origin as was the old point and lies on the straight line that connects (0,0) and (x,y).

(b) Convince yourself that Figure 2.2 is correct. Draw the original triangle, then take enough points on the periphery of the original and find corresponding points half-way to (0,0). What do you get?

2. Find three transformations which will make an equilateral Gasket with side length 2. (Hint: refer to Figure 2.4). Answer

3. Find three transformations which will make the Gasket whose vertices are at (0,0), (1,0), and (-0.5,1). What does that Gasket look like? Answer

4.[C] Use the Eq. Gasket Rule and DET to explore the mangling of pictures to produce Gaskets. How many generations have to go by before your starting picture becomes a decent replica of the Gasket? Now, try starting with a single point, a puny picture to be sure. Do you see that it really doesn't matter what starting picture you choose? Incidentally, when you choose to start with a single point, the construction seems to go extremely rapidly at first, but then slows down. In fact, after about eight generations, the computer seems to take about as much time drawing successive generations as it did when you got to the same level starting with some complicated picture. Why is that?

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