Here is a typical Ulam spiral. Each bright dot is a prime number.

And here is our "rug" with the lovely Celtic pattern generated by multiplying the golden mean with PI. Where do the primes fall upon this pattern?

Overlapped primes:

As you can see, it's a complete mess. Some of the primes fit into the pattern, some don't, where the primes sit in the negative space. Nothing significant is happening here.

Or is there? Take a look at this:

The primes tend to "favor" areas where the celtic pattern's positive space occupies. That is, the negative space of the pattern, the dark regions, by and large are not polluted with primes. Of course you will find exceptions and even a contradiction: There are plenty of positive space regions (masses of red dots) where the primes also avoid. Another way to say this is the primes favor the edges, and if we continue to draw this pattern outward we may gain resolution into the nature of this happy random accident.

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