Koch Snowflake

The Koch snowflake is a fractal shape based on a triangle. A fractal is a shape with a pattern that repeats forever, no matter how far you zoom in. To make the Koch snowflake, start with an equilateral triangle. Then, replace the middle third of each edge with a new equilateral triangle and remove the line underneath it. You repeat this process again and again without stopping. In this design, we’ve done four rounds of this process (called iterations).

Each time you add triangles, one edge turns into four smaller edges. Each of these new edges is one-third the length of the original. This means that every iteration, the total perimeter is multiplied by 4/3. Since 4/3 is more than 1, the perimeter consistently gets larger. If we do this infinitely many times, the perimeter becomes infinite!

The area of the snowflake doesn't grow forever like the perimeter does. At each step, we add lots of tiny triangles. The number of triangles we add keeps increasing, but each one is smaller than the ones before. It turns out that the total amount of new area we add each iteration gets smaller and smaller. Because of this, even though we keep adding more triangles, the total area gets closer and closer to a fixed value. In the end, the full Koch snowflake has an area that's 1.6 times the area of the original triangle, despite its perimeter being infinite!