In the late 19th and early 20th century, mathematicians began discovering geometric figures with a truly remarkable property: they exhibited the same elaborate structure at all scales. Today we call such shapes fractals. In 1904 Helge von Koch described one of the earliest known fractals, the so called Koch snowflake. The Koch snowflake can be constructed by starting with an equilateral triangle with sides length one, then recursively altering each line segment as follows: . Divide the line segment into three segments of equal length. . Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. . Remove the line segment that is the base of the triangle from step 2. K, Kz KOCH SNOWFLAKE K3 Ky If we were to zoom in on the boundary we would see the same intricate structure repeating at every scale. That's what makes it an example of a fractal. Another remarkable property of the Koch snowflake is that it's boundary is continuous but nowhere differentiable (there are corners at every point). Strange as these shapes may seem, they are everywhere in nature. In this project we'll explore other strange properties of the Koch snowflake using our knowledge of geometric sequences and geometric series.2. What is the area of the Koch snowflake? As the Koch snowflake is built up by adding a collection of smaller triangles at each stage, its area is an infinite series. Let's build it up from the first stages. First recall that the area of an equilateral triangle with side length s equals V32. 4 . K1 is an equilateral triangle with side lengths 1. We deduce that V3 3 Area of K1 = . To make K2 we add 3 (the number of sides of K1) new equilateral triangles each with side length 4. We deduce that V3 Area of K2 = V3 4 + 4 . To make K3 we add 3 . 4 (the number of sides of K2) new equilateral triangles each with side length ()2. We deduce that Area of K3 = + By continuing this process determine the area of the Koch snowflake. Your answer cannot be left as an infinite series. Hint: Look for a geometric series. Solution