I want to set up a map from $\Bbb N\to\Bbb Q.$
and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$
Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates.
If $x$ happens to be irrational round to finitely many digits to obtain a rational.
Then plug $x$ back into either equation to get the height, which is transcendental.
Round the transcendental to finitely many digits to obtain a rational.
For example, $S=T=2$ would get mapped to the point $ (1/2, 0.055833), $
Indeed, all $S=T$ would get mapped to $(1/2, H),$ where $H$ is the height.
So, $\Bbb N$ would be mapped to the vertical strip $x=1/2.$
This strip corresponds to the line outside the unit square: $y=x; x\in \Bbb N.$
In fact all vertical lines in the unit square are mapped to diagonal lines with positive slope, by a rotation.
For $S\ne T$ you'd get points such as $(2,3)$ and $(3,2)$ which would lie symmetrically on either side of $x=1/2.$
Essentially this scheme sets up a grid in the unit square and associates each point to two natural numbers.
Is the structure of the lattice of natural numbers preserved under this transformation?