In the late 19th and earl}r 20th century. mathematicians began discovering geometric gures 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: 0 Divide the line segment into three segments of equal length. 0 Draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. 0 Remove the line segment that. is the base of the triangle from step 2. K. we /' Roar rm WM SE K, In, 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 snowake is that it's boundary is continuous but nowhere differentiable [there are careers at every point}. Strange as these shapes mayr 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 Project: The Koch Snowflake 1. What is the length of the perimeter of the Koch snowflake? Because the Koch snowflake is the limiting shape of the shapes K1, K2, . . ., we know that Length of perimeter of Koch Snowflake = lim (Length of perimeter of Kn) 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 Nn be the number of edges of K. Notice that N = 3 and at each step one edge turns into four new edges, meaning that Nn = 3 . 40-1 Using this fact, determine the length of the perimeter of the Koch snowflake. Notice anything strange? Hint: What is the length of one of the sides of Kn? Solution