Koch Snowflake

The Koch snowflake, also known as the Koch curve, Koch star or Koch island, is a fractal curve and one of the earliest fractals ever described. It is based on the Koch curve, which appeared in a 1904 paper by the Swedish mathematician Helge von Koch titled On a Continuous Curve Without Tangents, Constructible from Elementary Geometry. It is built iteratively in a sequence of stages, starting from an equilateral triangle, with each successive stage formed by adding outward bends made of smaller equilateral triangles to every side of the previous stage. The areas enclosed by the successive stages converge to eight fifths the area of the original triangle, even as the perimeters of the successive stages grow without bound, so the finished snowflake encloses a finite area within an infinite perimeter. The Koch snowflake was constructed as an example of a continuous curve for which drawing a tangent line at any point is impossible; unlike the earlier Weierstrass function, whose proof of the same property was purely analytical, the Koch snowflake was designed to be geometrically drawable, so the property could also be seen through direct intuition.