The Koch Snowflake is born as an equalateral triangle, as shown in the figure to the right, with each of its sides then undergoing the procedure noted above, allowing N to increase without bound. The perimeter of the snowflake is clearly infinite. Its area is derived by summing the ever increasing number of ever smaller equalateral triangles. This results in a geometric series which identifies its area as 1.6 times the area of the initial triangle. The figures to the right display the results of the first five iterations.
This beautiful snowflake has been studied in great detail, and displays, along with its siblings, many more additional properties that are striking. Several websites provide a much greater depth of analysis. See, for example,