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.1. that is the length of the perimeter of the Koch snowflake? Because the Koch snowake is the limiting shape of the shapes Kl , K2, . . .: we know that Length of perimeter of Koch Snowake = lim (Length of perimeter of K") "40C We need to determine the length of the perimeter of K" and take a limit. Observe that the length of the perimeter is the number of sides of K" multiplied by each of their lengths. / Let N" be the number of edges of K ,1. Notice that N1 = 3 and at each step one edge turns into four new edges, meaning that N" = 3 - :1"1 Using this fact, determine the length of the perimeter of the Koch snowake. Notice anything strange? Hint: 'What is the length of one of the sides of Kn? K Solution