The longest path around a snowflake...

A finite area bounded by an infinite perimeter!? Helge von Koch first presented this snowflake in 1906. Watch the first four stages of growth of a triangle into a Koch's snowflake.

1. At each stage of growth, each exterior side of an equilateral triangle 'grows' a new, centered, outward pointing equilateral triangle whose sides are one third the length of the sides of the previous triangle.
2. The area and perimeter of the polygon can be computed at each stage of growth. The original triangle and the first four growth stages are shown.
3. If the growth stages increase without limit, the perimeter increases without limit but the area is limited. In other words, Koch's snowflake has a finite area with an infinite boundary!
4. Koch's Snowflake is a delightfully annoying problem in the rich field of mathematics where the truth is often stranger and more exciting than fiction.