The Koch Snowflake

A fractal, 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 60 degrees. The zeroth through third iterations of the construction are shown above.


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

The nth iterations of the Koch snowflake may be implemented in a future version of the Wolfram Language as KochMesh[n].

Let N_n be the number of sides, L_n be the length of a single side, l_n be the length of the perimeter, and A_n the snowflake’s area after the nth iteration. Further, denote the area of the initial n=0 triangle Delta, and the length of an initial n=0 side 1. Then

N_n = 3·4^n
L_n = (1/3)^n
l_n = N_nL_n
= 3(4/3)^n
A_n = A_(n-1)+1/4N_nL_n^2Delta
= A_(n-1)+1/3(4/9)^(n-1)Delta.

Solving the recurrence equation with A_0=Delta gives


so as n->infty,


The capacity dimension is then

d_(cap) = -lim_(n->infty)(lnN_n)/(lnL_n)
= log_34
= (2ln2)/(ln3)
= 1.261859507...

(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.

Source: Koch Snowflake — from Wolfram MathWorld

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