I enjoy exploring these patterns, not just for primes, but for factors as well. I made a little JavaScript app for creating "Number Mandellas" that my kids and I use to enjoy different patterns:

http://ideonexus.github.io/Explorable-Explanations/math/numb...

Our favorite thing to do is set the Preset to "Randomized" and click "Render Preset" over and over again to see what comes up. Sorry for the clunky interface, but the source is on github if anyone wants it.

https://github.com/ideonexus/Explorable-Explanations/tree/ma...