I made some pretty pictures of the rational numbers! Take a look.
This first plot shows the denominators of rational numbers between -1 and 1. It’s obtained by searching through every rational number in fully reduced form with numerator and denominator between -1000 and 1000.
Now here’s a plot of the numerators of rational numbers between -1 and 1.
Here’s the same thing, but only looking at rationals from 0 to 1:
And for kicks, here’s the inverses of the numerators:
The same, for rationals between 0 and 1:
These pictures are quite pretty to look at and interesting to think about. Some features of the image are artifacts of the particular way I chose rational numbers (i.e. looking only at rationals with numerator and denominator in a fixed range), but some other features might reflect some deeper structure of the rationals.