# 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?

Return to Chapter 2 Exercises

Go to Chapter2 exercises:Making Some Other Shapes

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