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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:
Prime Spirals in 3D
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?

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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
up vote 4 down vote accepted

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.

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Thank you for this cogent answer. And especially for pointing me to the Bateman–Horn conjecture. Thanks! – Joseph O'Rourke Aug 24 '11 at 12:11

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