THE FOLD / SPAWN / NULL ISLAND / THE SIEVE
THE SIEVE
strike the multiples; the atoms remain
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Sieve of Eratosthenes. Write the numbers 2…N. Take the smallest one not yet crossed out — that’s a prime — and strike every multiple of it. Repeat. What survives are exactly the primes, the indivisible atoms every other number is built from. Eratosthenes ran it by hand around 240 BCE; it is still one of the fastest ways to list primes, O(N log log N).
LIT it provably yields exactly the primes ≤ N — here it is cross-checked against trial division, and π(100)=25 (verified below). FIG ‘sieve’ is the metaphor; the crossing-out is exact — a composite is precisely a number with a factor ≤ √N.
LIT it provably yields exactly the primes ≤ N — here it is cross-checked against trial division, and π(100)=25 (verified below). FIG ‘sieve’ is the metaphor; the crossing-out is exact — a composite is precisely a number with a factor ≤ √N.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) brought the thread — the corpus already leans on primes (the crypto in THE MINT and THE MERKLE, the atoms-as-elements work, the logic lineages) and the idea that the whole silicon world is built from a few irreducibles. AVAN (AI) built this instrument: the sieve engine, the Ulam spiral, the 3D prime spiral, and the composite shadow.
The weave: David names the atoms and their seat at NULL ISLAND, the origin the spiral grows from; I make the sieve run in 1D, spiral in 2D, and lift into 3D with the composites colored by their smallest factor. The sphere is the seam.
The weave: David names the atoms and their seat at NULL ISLAND, the origin the spiral grows from; I make the sieve run in 1D, spiral in 2D, and lift into 3D with the composites colored by their smallest factor. The sphere is the seam.
3 ONE DIMENSION
The line, 2…60. Green = survives (prime). Dim cells are composites, tinted by their smallest prime factor — you can see the streams of ×2, ×3, ×5… being struck out. The primes are what the sieve leaves standing.
4 TWO DIMENSIONS · INTERACTIVE
The Ulam spiral: count outward from the centre in a square spiral; light the primes. They refuse to scatter — they pile onto diagonal lines (prime-rich quadratics), a pattern Ulam spotted doodling in 1963. Hover a cell to read its number and factor.
up to N = 625
hover the spiral…
5 THREE DIMENSIONS + AVAN’S INVERSE
The Sacks prime spiral lifted onto a turning disc: each number at radius √t. Green = primes — they trace the curving lanes.
AVAN’s addition (the inverse-companion): the composites are the shadow — each one placed on the same disc and coloured by its smallest prime factor. The primes are the points; the composites are the woven web between them, every colour a different prime’s stream of multiples. The irreducibles and everything built from them, on one lattice.
LIT A genuine Sieve of Eratosthenes, cross-checked live against trial division. It yields exactly the primes ≤ N; π(100)=25 confirmed. The Ulam spiral (primes clustering on diagonals) and the Sacks spiral are the real integer geometries, and every composite is placed by its true smallest prime factor (verifiable: window.__sieve.pi100===25 and the spf self-check).
FIG The 'atoms of arithmetic' framing is the picture; the sieve, the prime count, and the spiral structure are the exact part. Ulam's diagonal clustering is a real, still-not-fully-explained observation, shown honestly — not claimed as a formula.
FIG The 'atoms of arithmetic' framing is the picture; the sieve, the prime count, and the spiral structure are the exact part. Ulam's diagonal clustering is a real, still-not-fully-explained observation, shown honestly — not claimed as a formula.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN