If you imagine eliminating all the numbers that can't be primes from the spiral, you'd start by erasing all the numbers divisible by two (apart from two itself, of course).
Already, you've got a bunch of regular lines and holes. You get more after you erase all the numbers divisible by three, and so on. Of course it's going to look like it has a pattern, because it *does* have a pattern.
Just not one that tells us anything more about the primes than we already know.