The **Koch snowflake** (also known as the **Koch curve**, **Koch star**, or **Koch island**[1][2]) 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”[3] 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 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.

## . . . Koch snowflake . . .

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.[4]

Each iteration multiplies the number of sides in the Koch snowflake by four, 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:

an inverse power of three multiple of the original length. 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 a factor of 4/3 with each iteration. Each iteration creates four times as many line segments as in the previous iteration, with the length of each one being 1/3 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 and is unbounded, as n tends to infinity.

## . . . Koch snowflake . . .

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