Chapter 12.3, Problem 104AYU

Challenge Problem Koch’s snowflake The area inside the fractal known as the Koch snowflake can be described as the sum of the areas of infinitely many equilateral triangles, as pictured below.

For all but the center (largest) triangle, a triangle in the Koch snowflake is $\frac{1}{9}$ the area of the next largest triangle in the fractal. Suppose the area of the largest triangle has area of $2$ meters.

Show that the area of the Koch snowflake is given by the series

$A=2+2\cdot 3\left(\frac{1}{9}\right)+2\cdot 12{\left(\frac{1}{9}\right)}^2+2\cdot 48{\left(\frac{1}{9}\right)}^3+2\cdot 192{\left(\frac{1}{9}\right)}^4+\cdots$

Find the exact area of the Koch snowflake by finding the sum of the series.