Meaning of Rays in Polar Plot of Prime Numbers

I recently began experimenting with gnuplot, and I quickly made an interesting discovery. I plotted all prime numbers below 1 million in polar coordinates such that for every prime $p$, $(r,\theta) = (p,p)$. I was not expecting anything in particular; I was trying it out. The results are fascinating. When looking at the primes beneath $30000$, a spiral pattern can be seen: For comparison, here is the same graph with the multiples of 3 and 7 superimposed on it. Primes are in yellow, multiples of 3 and 7 in green and red respectively. What is really interesting to me, though, is the behavior when the range is increased. Multiples of a given number appear to spiral out in the same pattern into infinity, but the primes begin to form rays in groups of $3$ or $4$. See below: Compared to multiples of $3$ and $7$ again: Now, I must admit that I am very much a novice mathematician with little experience beyond trigonometry. I am just going into Calculus and Discrete Mathematics this upcoming fall. I know that there is something called the Prime Number Theorem - are these patterns related to it? Are these rays the same phenomenon as the diagonal lines found in Ulam Spirals? EDIT: In response to Greg Martin's explanation, I decided to add a couple more graphs. To see why they are relevant, read his answer. $(r,\theta)=(n,n), n \in \mathbb{N}$