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
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 .