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 polygon. This process is repeated at each succeeding step.
The snowflake curve is the curve that results from repeating this process indefinitely.
(a) Let sn, 1n, and pn represent the number of sides, the length of a side, and the total length of the nth approximating curve (the curve obtained after step of the construction),
respectively. Find formulas for sn, 1n, and pn.
(b) Show that as pn → ( as n → (.
(c) Sum an infinite series to find the area enclosed by the snowflake curve.
Parts (b) and (c) show that the snowflake curve is infinitely long but encloses only a finite area.