To construct the snowflake curve, start with an equilateral triangle with sides of length 1. Step 1 in the construction is to divide each side into three equal parts, construct an equilateral triangle on the middle part, and then delete the middle part (see the figure). Step 2 is to repeat step 1 for each side of the resulting
- (a) Let sn, ln, and pn represent the number of sides, the length of a side, and the total length of the nth approximating curve (the curve obtained alter step n of the construction), respectively. Find formulas for sn, ln, and pn.
- (b) Show that pn → ∞ as n → ∞.
- (c) Sum an infinite series to find the area enclosed by the snowflake curve.
Note: Parts (b) and (c) show that the snowflake curve is infinitely long but encloses only a finite area.
FIGURE FOR PROBLEM 5
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