Koch Snowflake

The Koch snowflake is a fractal curve, also known as the Koch island, which was first described by Helge von Koch in 1904. It is built by starting with an equilateral triangle, removing the inner third of each side, building another equilateral triangle at the location where the side was removed, and then repeating the process indefinitely. The Koch snowflake can be simply encoded as a Lindenmayer system with initial string "F--F--F", string rewriting rule "F" -> "F+F--F+F", and angle . The zeroth through third iterations of the construction are shown above.

Each fractalized side of the triangle is sometimes known as a Koch curve.

The fractal can also be constructed using a base curve and motif, illustrated above.

The th iterations of the Koch snowflake is implemented in the Wolfram Language as KochCurve[n].

Let  be the number of sides,  be the length of a single side,  be the length of the perimeter, and  the snowflake's area after the th iteration. Further, denote the area of the initial  triangle , and the length of an initial  side 1. Then

			(1)
			(2)
			(3)
			(4)
			(5)
			(6)

Solving the recurrence equation with  gives

(7)

so as ,

(8)

The capacity dimension is then

			(9)
			(10)
			(11)
			(12)

(OEIS A100831; Mandelbrot 1983, p. 43).

Some beautiful tilings, a few examples of which are illustrated above, can be made with iterations toward Koch snowflakes.

In addition, two sizes of Koch snowflakes in area ratio 1:3 tile the plane, as shown above.

Another beautiful modification of the Koch snowflake involves inscribing the constituent triangles with filled-in triangles, possibly rotated at some angle. Some sample results are illustrated above for 3 and 4 iterations.

REFERENCES:

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