It's simpler than that. All primes greater than 2 are equal to 2n + 1 for some n. And since each side of a spiral is 2 longer than the adjacent one, you get a slanted line. You'd have just a blob of bunches of line, except that they thin out. (Primes > 6 are 2 * 3 * n +/- 1, then 2 * 3 * 5 * n +/- [1, 7, 11, 13].
Also, prime series are palindromic. Integers relatively prime to (2, 3, ... p[m]) form a repeating series with period 2 * 3 * ... * p[m], and within each series they are the integers not "knocked out" by the given primes; e.g., 2 * 3 * ... * p[m] +/- [p[m+1], p[m+2], ...]. So the pattern is symmetrical over the period.
Also, prime series are palindromic. Integers relatively prime to (2, 3, ... p[m]) form a repeating series with period 2 * 3 * ... * p[m], and within each series they are the integers not "knocked out" by the given primes; e.g., 2 * 3 * ... * p[m] +/- [p[m+1], p[m+2], ...]. So the pattern is symmetrical over the period.