Koch Snowflake Fractal Generator

This page is devoted to some mathematical recreation surrounding fractal boundaries. This came out of some of my research dealing with rough surface simulations in electromagnetic fields and fluid boundary layer formation. I plan to add some field solutions generated with these curves, but I need to organise them in some coherent manner.

For the time being, however, I have placed a few example curves as well as the code used to generate them. The output is numeric, but the output is readily plottable using Gnuplot. An example input file can be found here. Generally, you need to enter some sort of open line-segment-base generator curve like this

where you give the coordinates of the numbered knots. (Locations of the knots can be chosen such that the fractal curve does not cross itself at any point. This requires a bit of practice for the wilder curves.) This is the curve that is repeatedly scaled and inserted during the iterations.

To start the whole process, you need another “initiator” curve to serve as the “framework” for the first iteration. This can be an open or a closed curve like

where the numbered knots are defined by their X,Y position. To close the curve, give the initial knot again at the finish, viz. 0-1-2-3-0.

Now, a few of the curves generated with the code, koch.c:

Hexagonal fractal snowflake. Perhaps you have seen this one before?

Strange island with three-fold rotational symmetry

Another strange island.

Higher iteration of three-fold symmetric island.

A bumpy talisman for warding off evil spirits?

There you go! I will add more to this page (some colorful plots involving the solution of various mathematical problems) in the future. Please stay tuned.....