§ 11 · Instrument Filed 2026.05 · Reference SL · 26 · 011

Wind the integers and watch primes refuse to scatter.

Stanisław Ulam scribbled this during a dull seminar in 1963. Number the plane in a square spiral, mark the primes, and the diagonals appear by themselves — patterns we still can't fully explain. Zoom, pan, switch modes. At the foot of the page, the first dozen zeros of the Riemann zeta function — the deepest known music in the prime numbers.

Drag to pan · Scroll to zoom · Pinch on touch Plain primes

Prime

Integers shown 1 — 1 Primes 0 Cell 12 px

Cell size

12 px

Sieve speed

6×

Sieve control

Labels

The other side of the same coin

§ 02 · Zeta zeros

First 12 nontrivial zeros · ζ(½ + it) = 0

Why these matter

The diagonals you see on the spiral and these dots on the line are the same fact, twice.

Riemann's zeta function ζ(s) is built from the primes — yet its complex zeros, conjectured to all lie on the line Re(s) = ½, control how the primes are distributed down the number line. Every wobble in the prime-counting function π(x) corresponds to one of these dots. Prove they all sit on the line and you've proved the Riemann hypothesis: a million-dollar problem, and the closest thing mathematics has to a master key for the primes.

t₁ = 14.134725

t₂ = 21.022040

t₃ = 25.010858

t₄ = 30.424876

t₅ = 32.935062

t₆ = 37.586178

t₇ = 40.918719

t₈ = 43.327073

t₉ = 48.005151

t₁₀ = 49.773832

t₁₁ = 52.970321

t₁₂ = 56.446248

What you're looking at

§ 03 · Notes

The spiral itself is just bookkeeping. 1 sits at the centre; 2 to the right; 3 above; integers wind outward counter-clockwise forever.

Diagonals are quadratic. Each one is the trace of an integer polynomial like 4n²+2n+1 — values that, for unexplained reasons, contain many primes.

Gaussian split colours rational primes by how they behave in ℤ[i]: p ≡ 3 mod 4 stay prime; p ≡ 1 mod 4 split into two Gaussian primes a + bi.

Twin primes are pairs (p, p+2). Whether infinitely many exist is open. Polymath8 and Maynard have pushed the bounded gaps frontier hard since 2013.

Prime-gap colouring goes warm for small gaps, cool for large. The biggest gaps you see scrolling out are mostly between primes that are themselves stuck between long composite runs.

The sieve mode replays Eratosthenes (≈200 BCE). For each new prime p, every multiple 2p, 3p, … is struck out — what survives is prime, computed without a single division.

Implementation notes — how the spiral and the sieve are computed

The Ulam spiral has a closed-form inverse: for a lattice point (x, y) at Chebyshev distance k = max(|x|,|y|), the previous ring ends at (2k−1)² and the four sides of ring k give the integer at (x,y) in O(1). Drawing is therefore a single pass over visible cells — no spiral walk, no precomputed array of positions.

Primes are produced by a classical sieve of Eratosthenes up to the largest integer currently on screen, extended in segments whenever you zoom further out. The animation in Sieve mode replays that same algorithm at a configurable rate so you can see the composites being struck out, prime by prime, in spiral order.

Twin pairs are flagged by walking the prime list once; gap-colour uses the difference to the previous prime, mapped through a perceptual warm-to-cool ramp clamped at gap 50.

The 12 zeta-zero heights are hard-coded — they are constants of nature, like π. The plot draws Re(s) = ½ as a vertical critical line with each tₖ as a dot whose halo brightness fades with height, hinting that they continue forever upward.