Prime spirals on surfaces of revolution 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?

 A: As far as I'm aware (in other words, since I'm not an expert don't take this answer too seriously) this has not been studied using the terminology or set up that you describe. However there are some general conjectures on primes which could be translated to this set up, so in some sense, yes, there are conjectures that predict what the patterns we will see are.
First, note that there is nothing special about the Ulam spiral. It's main feature is that it moves around integers that are values of quadratic polynomials in a way that they appear on a line. Similarly playing with the idea of using other (algebraic) surfaces we notice that the lines will become integer values polynomials (not necessarily quadratic).
So, as I see it, the problem reduces to conjectured densities of prime values of polynomials. Patterns emerge when some polynomial families are expected to have higher densities of primes than others. A precise form of this heuristic is the "Bateman-Horn" conjecture, which, needless to say, is very very open at the moment.
