Koch-like fractals
The images at the bottom of this page were created with xkoch.
The way xkoch works is very much like this way of describing how to create the Koch Curve:
Start with this shape:
/\
/ \
____/ \____
Then replace each of the 4 line segments in the shape, with a smaller version of the shape:
1)
/\
/ \
_/\_/ \____
2)
_/\
\ \
_/\_/ \____
3)
_/\_
\ /
_/\_/ \____
4)
_/\_
\ /
_/\_/ \_/\_
Now we have 16 line segments, and we replace them all with even smaller versions of the original shape. And again, and again. And again, and again.
xkoch works like this, except it allows you to create your own shape. You can specify a shape made from 2 to 9 line segments (connected end to end) and it will do as many iterations of this replacement process (described above) as are necessary to get all the line segments down to less than a pixel in length. There are other programs that do this. I've seen one that works with java right in a web browser. (links please?)
xkoch uses a unique (as far as I know -- Jason Woofenden) rendering method, which can create some astonishing images. Instead of just filling in each pixel the line passes through solid white, this algorithm iterates until the longest line segments are under .4 pixels long. Instead of filling in the whole pixel, this algorithm just lightens the pixel according to how long the line segment is. This makes it so that the more times the line crosses over itself, the brighter it gets at that point.
Images created by xkoch
