Prime Polar Spirals

Prime p plotted at polar angle θ = p (radians) and radius r = p. Rational approximations to 2π create spiral arms — primes cluster away from multiples of those approximations.

Primes: 1000

Showing first 1000 primes (up to p = 7919)

22/7 ≈ 2π → ~20 arms near N~500

355/113 ≈ π → finer structure at high N

φ(44)=20 coprime residues mod 44

P1: First ~500 primes → ~20 visible spiral arms (from 22/7 ≈ 2π: 44 steps ≈ 14 full rotations, 20 coprime residues).

P2: Toggle integers — fills arms uniformly, showing structure comes from prime distribution in residue classes, not from integers.