Chapter 12, Problem 15E

Exercises 13 through 16 refer to the construction of the quadratic Koch island. The quadratic Koch island is defined by the following recursive replacement rule.

Quadratic Koch Island

• Start: Start with a seed square [Fig. 12-37(a)]. (Notice that here we

are only dealing with the boundary of the square.)

• Replacement rule: In each step replace any horizontal boundary

segment with the “sawtooth” version shown in Fig. 12-37(b) and

any vertical line segment with the “sawtooth” version shown in Fig.

12-37(c)

This exercise is a continuation of Exercise 13.

a. Find the area of the figure obtained in Step $1$ of the construction.

b. Find the area of the figure obtained in Step $2$ of the construction.

c. Explain why the area of the quadratic Koch Island is the same as the area of the seed square.

13. Assume that the seed square of the quadratic Koch Island has sides of length $16$.

a. Carefully draw the figures obtained in Steps $1$ and $2$ of the construction. (Hint: Use graph paper and make the seed square a $16$ by $16$ square.)

b. Find the perimeter of the figure obtained in Step $1$ of the construction.

c. Find the perimeter of the figure obtained in Step $2$ of the construction.

d. Explain why the quadratic Koch island has infinite perimeter.