THE FOLD / SPAWN / HELLO WORLD / THE FORD CIRCLES
THE FORD CIRCLES
fractions kissing along the number line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Ford circles (Lester Ford, 1938) give every fraction a body. Above each reduced fraction p/q on the number line, draw a circle of radius 1/(2q²) resting on the line at that point — big circles for simple fractions, tinier and tinier ones for complex denominators. The miracle: no two Ford circles ever overlap. They either miss entirely or kiss — and they kiss precisely when the fractions are Farey neighbours, |ps - qr| = 1. The whole arrangement is governed by one exact identity: dist² - (r₁+r₂)² = ((ps-qr)² - 1)/(q²s²), whose sign is decided entirely by the integer ps - qr. Between any two kissing circles, their mediant’s circle nests in the gap and kisses both — the Stern–Brocot structure of the rationals, drawn in soap bubbles.
LIT verified live: over all 129 reduced fractions with q ≤ 20 (8,256 pairs), zero overlaps; tangency occurs exactly at |ps - qr| = 1 (255 kissing pairs); and the governing identity holds to 1e-9 on every pair (window.__fordcircles). FIG no framing; the circles, distances, and integer determinants are computed independently in-browser.
LIT verified live: over all 129 reduced fractions with q ≤ 20 (8,256 pairs), zero overlaps; tangency occurs exactly at |ps - qr| = 1 (255 kissing pairs); and the governing identity holds to 1e-9 on every pair (window.__fordcircles). FIG no framing; the circles, distances, and integer determinants are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at hello-world — the spawn: every rational is born with its own bubble, sized by its denominator, greeting the number line exactly where it lives. AVAN (AI) built the instrument: the circle family, the pairwise audit, and the determinant identity.
Credit as content: Lester R. Ford (1938); Farey, Stern and Brocot behind the structure. The weave: David names the birth of the bubbles; I confirm zero overlaps and 255 exact kisses.
Credit as content: Lester R. Ford (1938); Farey, Stern and Brocot behind the structure. The weave: David names the birth of the bubbles; I confirm zero overlaps and 255 exact kisses.
3 ONE DIMENSION
The Ford circles over [0,1] — every reduced fraction wearing its bubble, kissing its Farey neighbours.
4 TWO DIMENSIONS · INTERACTIVE
Pick a pair of fractions; the determinant ps−qr instantly decides kiss or miss.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the bubbles, packed without a single collision.
AVAN’s addition (the inverse-companion): don’t measure the circles — read the integer. The inverse of ‘do these bubbles touch?’ is ‘is ps - qr equal to ±1?’ — geometry outsourced to a determinant. Magenta are the kisses at |ps-qr| = 1; green is the arrangement that never overlaps. The rationals, wearing their arithmetic on their skin.
LIT Genuine Ford circles (Lester R. Ford 1938; Farey/Stern–Brocot structure). Verified live: over all 129 reduced fractions with q ≤ 20 (8,256 pairs), zero overlaps; tangency exactly at |ps−qr| = 1 (255 kissing pairs); the governing identity holds to 1e-9 on every pair (window.__fordcircles.ok).
FIG No framing — circles, distances, and integer determinants computed independently in-browser. The AVAN inverse — don't measure the circles, read the integer: the inverse of 'do these bubbles touch?' is 'is ps−qr equal to ±1?' — geometry outsourced to a determinant. Magenta are the kisses; green is the arrangement that never overlaps. The rationals, wearing their arithmetic on their skin.
FIG No framing — circles, distances, and integer determinants computed independently in-browser. The AVAN inverse — don't measure the circles, read the integer: the inverse of 'do these bubbles touch?' is 'is ps−qr equal to ±1?' — geometry outsourced to a determinant. Magenta are the kisses; green is the arrangement that never overlaps. The rationals, wearing their arithmetic on their skin.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HELLO WORLD · David Lee Wise (ROOT0), with AVAN