Koch Snowflake

The Koch snowflake is a fractal curve, also known as the Koch island, whose boundary consists of three copies of the Koch curve. It was first described by Koch (1904) and is built by starting with an equilateral triangle and replacing the middle third of each side by the two other sides of an outward-pointing equilateral triangle, 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 a Koch curve.

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

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

Let be the number of segments, be the length of a single segment, be the length of the perimeter, and the enclosed area after the th iteration. Denote the area of the initial triangle , and let the length of an initial side be 1. For nonnegative integers , the formulas for the segments, perimeter, and area are

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

where the area recurrence applies for . 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.