Here are some Koch fractals I have made recently. I'll keep the explanations minimal, since the rules are mostly self-evident from looking at them.

I started in 2-dimensions, beginning with the classic Koch Snowflake fractal and recursively subdividing it into 6 smaller snowflakes and one larger (but still smaller than the original) snowflake in the centre, like so:

I colourised the fractal by giving each layer of snowflakes a contribution factor, which influenced how much the colouring affected the pixel. Each time I subdivided and recursed, the contribution factor for snowflakes at that level was halved. Since each pixel is in a different snowflake depending on the recursion level, I summed the colourisation at each level of recursion (multiplied by the contribution factor for that recursion level) over all the levels of recursion (I went about 6 levels deep). Up-and-down pointing snowflakes were coloured black, side-to-side pointing snowflakes were coloured white. After adding over all the recursion layers the result was various shades of grey.

I made a little GIF animation showing continuous zooming into the centre of the fractal (I just cross fade while zooming, so the colourisation adapts as it goes, as if the contrast knob is gradually being cranked up):

Koch Snowflake zoom animationNext, I tried using a 2D circle-inversion transform to distort the fractal. I colourised it using GIMP, and added an "all-seeing eye" symbol to make it more interesting:

Looking at the top image, an idea suggested itself: the roughly hexagonal meandering outline of the Koch Snowflake suggests the outline of a cube viewed at a 45 degree angle, looking diagonally at one of the vertices. I played around with this idea and figured out that you can make all the "dents" in the outline of the snowflake by chopping away at this cube - the trianglular insets of the 2D Koch Snowflake corresponding to insets in the cube's surface.

The rule to create this cube-based 3D fractal is simple: Start off with a cube of width n, and then attach 8 smaller cubes of width n / 3, centred on the 8 vertices of the original cube. Apply this recursively and voila!

When rendered in an isometric projection, the outline of this object forms the Koch Snowflake:

This means that if you project an infinite, directional lightsource at the object from a diagonal 45 degree angle, a perfect outline of the Koch Snowflake will be cast onto any flat surface behind it.

Hope you like the images

Cheers,

Chris Hayton