Koch’s Snowflake

December 23, 2011

It’s Christmas time, and that means winter, and that means snow (sometimes, but probably not this year), so today’s exercise is to draw Koch’s Snowflake. This famous fractal is built from an equilateral triangle, as shown upper left. At each step each side is modified by removing its center third and replacing it with two legs of another equilateral triangle. Thus, at upper right, each of the three legs has been replaced four line segments that form a new shape. Carrying on, the fractal becomes more and more detailed at the edges of the triangle; eventually, the perimeter of the triangle is infinite, though its area is 8/5 the area of the original triangle. Mathematically, the Koch Snowflake can be encoded as a Lindenmayer system with initial string F++F++F, rewriting rule F → F-F++F-F, and angle 60 degrees (in the Lindenmayer notation, F means move forward, + is a right turn and - is a left turn).

Your task is to write a program to draw the Koch Snowflake. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

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