You have seen the Ulam spiral[1] from 1963, right? I don’t know that it provides any actual insight into this stupenduously complicated[2] topic, but if you want something to draw about prime numbers that’s probably the best-known option. Or see the story about “primons”[3] for a view that’s less visual but no less tangible (given a certain kind of background at least).
And yes, as you note, your CDF very much looks like a square grid after an inversion. That is to say: draw a square grid on a horizontal plane, lay down a ball on that plane, project the image onto the surface of the ball by straight rays through its topmost point (imagine a lightbulb there; see? “projection”), then take a parallel plane touching that same ball and project back onto it from the surface (now casting rays from the bottommost point).
But if so, it’s boring in that it should have little to do with prime numbers. Let me think about this. (The story of complex numbers and projective transformations is not boring, of course, it’s quite pretty, just doesn’t provide much of an insight into the primes; and, in turn, its connection to hyperbolic geometry, most widely known through Escher’s drawings, is also quite wonderful, but removed even further from the original picture, so might not be necessary to understand it.)
The thing I found was that if you start the spiral with a 0 you get a similar pattern. Further if you layer each of the triangular sections you get a really cool pattern[0].
And yes, as you note, your CDF very much looks like a square grid after an inversion. That is to say: draw a square grid on a horizontal plane, lay down a ball on that plane, project the image onto the surface of the ball by straight rays through its topmost point (imagine a lightbulb there; see? “projection”), then take a parallel plane touching that same ball and project back onto it from the surface (now casting rays from the bottommost point).
But if so, it’s boring in that it should have little to do with prime numbers. Let me think about this. (The story of complex numbers and projective transformations is not boring, of course, it’s quite pretty, just doesn’t provide much of an insight into the primes; and, in turn, its connection to hyperbolic geometry, most widely known through Escher’s drawings, is also quite wonderful, but removed even further from the original picture, so might not be necessary to understand it.)
[1] https://en.wikipedia.org/wiki/Ulam_spiral
[2] https://arxiv.org/abs/math/0210327
[3] https://math.ucr.edu/home/baez/week199.html