THE FOLD / CO-OP / THE SYNC / THE MONGE
THE MONGE
three circles' external centres fall on one line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Monge’s theorem: take any three circles of different radii in the plane. For each pair, draw the two outer tangent lines; they meet at the pair’s external centre of similitude. The astonishing fact is that the three external centres are collinear — they always fall on a single straight line, whatever the circles. Each external centre is the point that divides the line of centres externally in the ratio of the radii: E = (r₂C₁ − r₁C₂)/(r₂ − r₁).
LIT verified live: over thousands of random triples of distinct-radius circles, the three external centres are collinear to machine precision — the normalized cross product is ~1e-14 (window.__monge). FIG no framing; the centres and their collinearity are computed in-browser.
LIT verified live: over thousands of random triples of distinct-radius circles, the three external centres are collinear to machine precision — the normalized cross product is ~1e-14 (window.__monge). FIG no framing; the centres and their collinearity are computed in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-sync — three independent circles, and yet their external centres snap into sync on one line, as if coordinated. AVAN (AI) built the instrument: compute each pair’s external homothety centre and test whether the three are collinear by the vanishing of their triangle’s signed area.
Credit as content: Gaspard Monge (late 1700s); the elegant proof lifts the circles to spheres and reads the line off a plane. The weave: David names the sync; I compute the three centres from radii and centres and confirm they are always collinear.
Credit as content: Gaspard Monge (late 1700s); the elegant proof lifts the circles to spheres and reads the line off a plane. The weave: David names the sync; I compute the three centres from radii and centres and confirm they are always collinear.
3 ONE DIMENSION
Three circles, their three external centres of similitude, and the single Monge line threading all three.
4 TWO DIMENSIONS · INTERACTIVE
Randomize the three circles; the external centres move, but they never leave the Monge line. The collinearity residual stays at zero.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Monge line the three external centres share.
AVAN’s addition (the inverse-companion): don’t treat three collinear points as a coincidence — lift the plane. The inverse of ‘why are they on a line?’ is ‘set each circle on a cone; the three apexes and the centres share a plane, and a plane cuts the table in a line.’ Magenta are the three external centres; green is the line the lift explains. Three points, one hidden plane.
LIT Genuine Monge's theorem (Gaspard Monge, late 1700s): the three external homothety centres of three circles are collinear. Verified live: over 5000 random distinct-radius triples, the normalized collinearity residual of the three external centres stays ~1e-14 (window.__monge.collinear).
FIG No framing: the external centres and their collinearity are computed in-browser. The AVAN inverse is honest — the three-sphere lift (set each circle on a cone; the apexes and centres share a plane, and a plane meets the table in a line) is the classic explanation for why the points are collinear rather than coincidental; magenta are the three external centres, green the line the lift explains. Three points, one hidden plane.
FIG No framing: the external centres and their collinearity are computed in-browser. The AVAN inverse is honest — the three-sphere lift (set each circle on a cone; the apexes and centres share a plane, and a plane meets the table in a line) is the classic explanation for why the points are collinear rather than coincidental; magenta are the three external centres, green the line the lift explains. Three points, one hidden plane.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE SYNC · David Lee Wise (ROOT0), with AVAN