Reducing a
figure
can be done repetitively. Therefore, we speak of so called generations.
Reducing a figure leads to a lower generation. After a finite number of
reductions the lowest generation (a primitive structure) has been
reached.
On the other
hand,
a figure can be designed by building higher generations from a given
initial
pair (N, D) . Denote the next higher generation figure by
(N”, D”) with accompanying
E" and K" . There are 2 possibilities. Firstly assume E”
< D”/2 .
In that case E” = N, and D” = D + s * 2*N for any
positive integer s, since E”
< D”/2 is always satisfied. An arbitrary positive
integer K” can be chosen
so that N” = K” * D” + E” .
Secondly,
assume
E” > D”/2. Then, again D” = D + s
* 2*N for any positive integer s, because
E” = D” - N will satisfy E” >
D”/2 . And again for any positive integer K”
holds N” = K” * D” + E” .
For example, take as lowest generation the figure based on (5,2), a next higher generation is the figure based on (29,12),
.... and
again a next higher generation is the
figure based
on (309,70) .
Building higher generations can be done repetitively. When using the same values for K" and s in each generation, the figure gets the appearance of a fractal !