Koch curve
The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a mathematical curve and one of the earliest fractal curves 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" (original French title: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire) by the Swedish mathematician Helge von Koch.
The progression for the area of the snowflake converges to 8/5 times the area of the original triangle, while the progression for the snowflake's perimeter diverges to infinity. Consequently, the snowflake has a finite area bounded by an infinitely long line.
The Koch snowflake can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:
After one iteration of this process, the resulting shape is the outline of a hexagram.
The Koch snowflake is the limit approached as the above steps are followed over and over again. The Koch curve originally described by Helge von Koch is constructed with only one of the three sides of the original triangle. In other words, three Koch curves make a Koch snowflake.
After each iteration, the number of sides of the Koch snowflake increases by a factor of 4, so the number of sides after n iterations is given by:
If the original equilateral triangle has sides of length s, the length of each side of the snowflake after n iterations is:
the perimeter of the snowflake after n iterations is:
The Koch curve has an infinite length because the total length of the curve increases by one third with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each one being one-third the length of the segments in the previous stage. Hence the length of the curve after n iterations will be (4/3)n times the original triangle perimeter, which is unbounded as n tends to infinity.
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