Sweep the angular step α through every value in [0, 1]. At each α, primes plot at angle 2παp and radius √p. Most α produce noise; certain α lock primes into petals, rays, and spokes. The α-spectrum below maps where prime structure lives in angular space.
Polar prime layout at α
α = 0.382
α-spectrum: where structure lives
peak so far: —
α 0.382
N max 3000
bins 80
how to read the spectrum
Dips in the bottom curve are fertile α: the angular distribution of primes has low entropy (high pattern). Spikes are barren α: primes spread uniformly.
Strongly rational α = p/q (with small q) always shows structure — primes cluster into q rays. The interesting question is whether irrational α with no small-q approximant still produces structure. The golden angle 1/φ is the classic example: it's the "most irrational" number and produces the Vogel sunflower packing — useful in plants because no two seeds are aligned. The matching question: does 1/φ resonate with the prime distribution? Sweep and find out.
Peaks worth watching for: rational fractions with small denominator (1/2, 1/3, 1/4, 1/6, 1/8, 1/30), noble irrationals (1/φ, the silver ratio), and continued-fraction-related constants (1/π, 1/e). What's not obvious is whether prime-specific structure (Chebyshev bias mod 4, mod 6) creates additional dips that aren't there for naturals.