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
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),
again a next higher generation is the
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 !