When folding the number line not into a spiral (like Ulam did) but into a zig-zag pattern (like Cantor did) there are other patterns visible in the distribution of prime numbers:
One sees diagonal bands with an above-average density of prime numbers. One of them contains the prime numbers $7, 11, 29, 43, 73, 89, 131, 157, 181, 211, 239, 379, 421, 461, 601,\dots $.
How can these bands be explained?
One also sees a rather distinct band from the lower left going upwards at an angle of roughly $75°$. It contains the prime numbers $229, 271, 317, 367, 421, 479, 541, 607, 677, 751, \dots $.
Note, that the vertical and horizontal borders of the pattern contain no primes at all, which can be easily understood.
[See my related question concerning the distributions of other numbers.]
By the way (and off the record): The detecting of lines in patterns (most easily of straight lines) is fun also in non-mathematical contexts (be them obvious or not):