From Gareth McCaughan I got the hint that Farey series might be a key to understanding the fractal structure of rational numbers. Farey series construct all of the Rn from 0/1 up to 1/1, i.e., more or less the first half of the structure of the rational numbers Rn from 1/M up to M/1. Farey series construct the rational number (a+a')/(b+b') between two given rational numbers a/b and a'/b' and start with the two numbers 0/1 and 1/1. This construction has important numerical benefits. It becomes especially simple when one constructs one new rational number between all given numbers: 0/1 1/1 -> 0/1 1/2 1/1 -> 0/1 1/3 1/2 2/3 1/1 -> 0/1 1/4 1/3 2/5 1/2 3/5 2/3 3/4 1/1 .... If I understood Gareth correctly, this is the construction of the so-called Stern-Brocot tree. This is not the same as the "Rational-Fractal" or the "Farey-Fractal" and it seems to be less nice to me.