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
b. Find the area of the figure obtained in Step
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
a. Carefully draw the figures obtained in Steps
b. Find the perimeter of the figure obtained in Step
c. Find the perimeter of the figure obtained in Step
d. Explain why the quadratic Koch island has infinite perimeter.
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Excursions In Modern Mathematics, 9th Edition
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