The Koch snowflake (also known as the Koch curve, Koch star, or Koch island is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a Continuous Curve Without Tangents, Constructible from Elementary Geometry" by the Swedish mathematician Helge von Koch. The Koch snowflake can be built up iteratively, in a sequence of stages. The first stage is an equilateral triangle, and each successive stage is formed by adding outward bends to each side of the previous stage, making smaller equilateral triangles. The areas enclosed by the successive stages in the construction of the snowflake converge to 8 5 {\displaystyle {\tfrac {8}{5}}} times the area of the original triangle, while the perimeters of the successive stages increase without bound. Consequently, the snowflake encloses a finite area, but has an infinite perimeter. The Koch snowflake has been constructed as an example of a continuous curve where drawing a tangent line to any point is impossible. Unlike the earlier Weierstrass function where the proof was purely analytical, the Koch snowflake was created to be possible to geometrically represent at the time, so that this property could also be seen through "naive intuition".
Origin and history In his 1904 article, von Koch applies this recursive construction to a line segment, obtaining the curve that forms 1 3 {\displaystyle {\tfrac {1}{3}}} of the boundary of the Koch snowflake. However, the complete snowflake does not appear in the original article published in 1904, nor in the extended 1906 memoir. The Koch snowflake as a closed curve may instead be due to the American mathematician Edward Kasner.
Construction The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:
divide the line segment into three segments of equal length. draw an equilateral triangle that has the middle segment from step 1 as its base and points outward. remove the line segment that is the base of the triangle from step 2. The first iteration of this process produces the outline of a hexagram. The Koch snowflake is the limit approached as the above steps are followed indefinitely. The Koch curve originally described by Helge von Koch is constructed using only one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake. A Koch curve–based representation of a nominally flat surface can similarly be created by repeatedly segmenting each line in a sawtooth pattern of segments with a given angle.
Properties
Perimeter of the Koch snowflake The arc length of the Koch snowflake is infinite. To show this, we note that each iteration of the construction is a polygonal approximation of the curve. Thus, it suffices to show that the perimeters of the iterates is unbounded. The perimeter of the snowflake after n {\displaystyle n} iterations, in terms of the side length s {\displaystyle s} of the original triangle, is
3 s ⋅ ( 4 3 ) n , {\displaystyle 3s\cdot {\left({\frac {4}{3}}\right)}^{n}\,,}
which diverges to infinity.
Area of the Koch snowflake The total area of the snowflake after n {\displaystyle n} iterations is, in terms of the original area A {\displaystyle A} of the original triangle, is the geometric series
A ( 1 + 3 4 ∑ k = 1 n ( 4 9 ) k ) = A 1 5 ( 8 − 3 ( 4 9 ) n ) . {\displaystyle A\left(1+{\frac {3}{4}}\sum _{k=1}^{n}\left({\frac {4}{9}}\right)^{k}\right)=A\,{\frac {1}{5}}\left(8-3\left({\frac {4}{9}}\right)^{n}\right)\,.}
Taking the limit as n {\displaystyle n} approaches infinity, the area of the Koch snowflake is 8 5 {\displaystyle {\tfrac {8}{5}}} of the area of the original triangle. Expressed in terms of the side length s {\displaystyle s} of the original triangle, this is:
2 s 2 3 5 . {\displaystyle {\frac {2s^{2}{\sqrt {3}}}{5}}.}
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