# 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?

-
To quote from the Wikipedia article to which I link, "In their famous 1923 paper on the Goldbach Conjecture, Hardy and Littlewood stated a series of conjectures, one of which, if true, would explain some of the striking features of the Ulam spiral." –  Joseph O'Rourke Aug 24 '11 at 2:10
The pyramid figure you posted is not a surface of revolution. There may not be any reason to expect the more revealing spirals should be two-dimensional (as opposed to space/lattice filling in higher dimensions) or hypersurfaces with any particular symmetries. You don't have any prior notion of what a 'revealation' would be, nor a specification of what the "structure of primes" is, so this question is very much in the shop. –  anon Aug 24 '11 at 4:40