Every whole number from 2 to N is placed on a rising Sacks-style spiral: point k sits at angle k·φ (the golden angle) and radius √k, one turn higher than the last. Prime numbers are sieved out with the Sieve of Eratosthenes and lit up in cyan; composites stay dim. Because primes have no small factors to align them with the spiral's arithmetic structure, they scatter — yet their overall density thins out exactly as analytic number theory predicts.
Prime Number Theorem: π(x) ~ x / ln(x) as x → ∞
Riemann zeta function: ζ(s) = Σ 1/n^s = Π (1 − p^-s)^-1 (Re(s) > 1)
Logarithmic integral: Li(x) = ∫₂ˣ dt/ln(t) (a tighter fit than x/ln x)
- Integers up to N — how many whole numbers 2…N are plotted; the spiral and the live π(N) count both grow with it.
- Build-up speed — how many points per second animate onto the spiral when N changes.
- Primes only — hides every composite instance so only the prime "arms" of the spiral remain visible.
- Auto-rotate — slowly orbits the camera so the spiral's 3D helix structure and its density are easy to read.
The Prime Number Theorem — proved independently by Hadamard and de la Vallée Poussin in 1896 using properties of ζ(s) — says the true prime count π(N) and the elementary estimate N/ln(N) converge in relative terms as N grows, even though ζ(s)'s nontrivial zeros (conjectured by Riemann to all lie on Re(s) = 1/2) control exactly how large the gap between them can be.