Is there a hidden pattern to prime numbers? Can we start to see it with the Ulam Spiral?

Prime numbers form the basis of our entire number system. Yet there seems to be no ordering to them at all. They appear to occur randomly. However, a Polish mathematician named Stanislaw Ulam may have decoded a pattern years ago. It looks like this.


It is generated by counting all integers on a lattice plane as a spiral. All of the prime numbers are in bold. In other words, start at the point (0,0) for 0, move left for 1, move down for 2, move right for 3, move right for 4, move up for 5, move up for 6, move left for 7, move left for 8, and so on. Each time you complete a square move left and continue the spiral again.
This is really amazing what appears. It looks like certain lines have more prime densities than others. These lines can be approximated by polynomial equations.