Fractals

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 .