Solved: Exercises 23 through 26 refer to a square version of the Sierpinski gasket

Chapter 12, Problem 24

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Exercises 23 through 26 refer to a square version of the Sierpinski gasket called the Sierpinski carpet. The Sierpinski carpet is defined by the following recursive construction rule.Assume that the seed square of the Sierpinski carpet hasarea A. (a) Carefully draw the carpet obtained in Step 3 of theconstruction. [Hint: Use a piece of graph paper with asmall grid and make the seed square a 27-by-27 square.Use Fig. 35(c) as your starting point.] (b) Find the area of the carpet obtained in Step 1 of theconstruction. (c) Find the area of the carpet obtained in Step 2 of theconstruction. (Hint: Your best bet is to think of the gasketin Step 2 as made of eight identical one-third scaleversions of the gasket in Step 1.) (d) Find the area of the carpet obtained in Step 3 of theconstruction. (e) Find the area of the carpet obtained in Step N of theconstruction expressed in terms of A and N.

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