This is an entirely naive question, and in addition, vague. Apologies in advance!
Imagine wrapping the Ulam prime spiral around a surface in $\mathbb{R}^3$, something like this:
This suggests this variation. Let $S$ be a surface of revolution, touching the origin $(0,0,0)$ and rising into the positive $z$-halfspace. Wrap a monotonically rising "prime spiral" $\rho$ around $S$ so that the distance (measured on $S$) between two successive primes on $\rho$ is the difference between those two primes as integers. But $\rho$ need not be a geodesic on $S$.
Just as the various diagonals of the Ulam prime spiral reveal relationships and suggest conjectures concerning the primes, perhaps some particular $S$ would regularize or "organize" the primes. So, finally, my question is this:
Is there some surface of revolution $S$ and some $\rho$ so that the geodesics from the origin lying in vertical planes (the analogs of diagonals) reveal structure in the primes? Either provable or conjectural structure?