Koch_Snowflake

Koch Snowflake curve generalized

The following generalized Koch Snowflake curve is a fractal constructed by a self-repeating procedure as follows. Start with an arbitrary triangle (t) and divide each one of its sides in three equal parts. Then set on the middle part a triangle t' similar to t. Repeat the procedure on the sides of the resulting 12-gon, dividing again each side in three parts and replacing the middle with a triangle t'' similar to t. By applying the procedure n times you get a polygon p(n) with 3*4^n sides. Below is the polygon resulting for n=5. The fractal was constructed using the [Fractal Socket] tool of EucliDraw. The construction took a couple of minutes and the file occupies 1.815.541 bytes of disk space. The reason for such a waste of memory is that for its construction uses some thousands of auxiliary triangles, similar to (t). Because of the huge memory allocation it is not included in the standard examples folder of the program. It can be obtained uppon request from euclidraw.com. For the simpler Koch snowflake see Koch_Snowflake.html . There are several related interesting questions to investigate:
1) Find conditions on the triangle t, such that the resulting polygon p(n) is not self intersecting. For example, if t is equilateral then p(n) is not self intersecting for all n.
2) Find the perimeter and area enclosed by a non self-intersecting p(n). Find the limits of these quantities for infinite n.
3) Find the convex hull of p(n) and its limit for infinite n.
4) Replace the initial triangle with an arbitrary convex polygon, formulate and solve the analogous problems.

For another interesting picture, build with rectangles insead of triangles see the file Koch_Snowflake_gen2.html .

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