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- Jan 26, 2012

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**Problem**: To construct the

**Koch 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 below). Step 2 is to repeat Step 1 for each side of the resulting polygon. This process is repeated at each succeeding step. The Koch snowflake curve is the curve that results from repeating this process indefinitely.

(a) Let $s_n$, $\ell_n$ and $p_n$ represent the number of sides, the length of a side, and the total length of the $n$th approximating curve (the curve obtained after Step $n$ of the construction), respectively. Find formulas for $s_n$, $\ell_n$ and $p_n$.

(b) Show that $p_n\to\infty$ as $n\to\infty$.

(c) Sum an infinite series to find the area enclosed by the Koch snowflake curve.

(Parts (b) and (c) show that the Koch snowflake curve is indefinitely long but encloses only a finite area.)

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