As already mentioned above, the possibility exists to generate fractal figures. The easiest way to derive a fractal is by reducing a pair (N,D) such that the reduced pair (N’,D’) is equivalent to (N, D); i.e., the normalised values of (N, D) and (N’,D’) are equal. Two situations can be discerned depending on the size of E. Firstly, consider E < D/2. Then N’ = E and D’ = D-2*s*E . Define the ratio δ := N/D . Demanding that δ must equal the normalised value of (N’,D’) leads to the quadratic equation with (positive) solution:

Secondly, consider E > D/ 2 . In that case N’ = D-E and D’ = D-2*s*(D-E) . This leads to another quadratic equation with as (positive) solution:

For example, take in the latter case: K=1, s = 1, yielding δ = ( 1 + sqrt(5) )/2, being the golden ratio. Note that building the generations starting with the triple (N=3, D=2, E= 1) results in numbers of the Fibonacci sequence.

Let's take another example, namely K=7, s=1, E > D/2. This leads to the following stunning results:


generation 2,  with  N=3809,  D=510; and


generation 3,  with  N=60705,  D=8128.
At each generation the plane is partly filled with basic, hexagonal tiles with an increasing snowflake being left uncovered. 
You can also view these figures with improved resolution in SVG format:
3809_510 (110 kB) ,  generation 3 60705_8128 (1.8 MB). Note that for watching these figures, your browser needs an SVG plug-in. Alternatively, you can download the SVG files ( 3809_510.svg and 60705_8128.svg ) and view them with an SVG-viewer, like the open source Inkscape program .