Unexpected Beauty in Primes

Since first seeing the Ulam spiral on Numberphile, it wasn't surprising to me that lines like that would appear. Being that prime numbers are all the combinations by which you cannot stack a two dimensional area of pixels in a fully filled rectangle.

e.g.

... ... ...

9 is not prime, because you can fill all rows evenly.

... .. ..

7 is, because you can't fill all the rows evenly. So the gaps make lines...

Or to put it another way, the lines in the Ulam spiral are the pixels next to the corner pixels that make full rectangles that can't be made full rectangles another way.

The first time I saw these, I thought it hinted at some deep insight into the nature of primes. However, after watching the following video, I think it just says something about modular arithmetic and rational approximations of 2 Pi.

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

The video is only about Archimedian spirals, but I suspect a similar analysis would apply to Ulam spirals too. We really like to find patterns.

The Ulam Spiral does not show a pattern. There's nothing predictive about it. It's just a structure that shows how some of the primes group.

They're also anything but random as the author suggests. We've known that much since the 1st century.

They're cool to look at, but there's nothing that earth shattering here.

What about quasi-primes