Ulam Spiral (and see ref to Martin Gardner's famous SciAm article):

https://en.wikipedia.org/wiki/Ulam_spiral

Why do prime numbers make these spirals? | Dirichlet’s theorem and pi approximations - 3Blue1Brown

https://www.youtube.com/watch?v=EK32jo7i5LQ

you can not only map prime numbers in a circle/spiral, but also in a triangle and get some nice knitting patterns :)

https://tessi.github.io/walking-the-ulam-spiral/

If you're plotting primes, all the coordinates where you're not plotting are non-prime - so every 2nd coordinate will be blank. As will every 3rd and every 4th, 5th, 10th, 11th. etc etc.

Surely that's where the pattern comes from.

I’m not a mathematician so correct me if I’m wrong, but the patterns that emerge or more the natural result of the plotting method vs revealing anything meaningful about the distribution of primes.

Not a mathematician either but I think some quadratics have a tendency to produce more primes than others. What you are seing is the characteristic of various quadratics when plotted in this way. I plotted something quite simmilar.

The article explicitly says as much.

> So what’s clear here is that the spirals themselves have nothing to do with prime numbers; a much cleaner and fuller pattern can be seen when we plot all positive integers (as well as zero).

shameless self promotion:

i built an "animation framework" in JavaScript around it where you can control and animate several parameters and even record the animation

https://primes.nickyreinert.de/