THE FOLD / BOSS / THE RAID / THE PAPPUS
THE PAPPUS
perspective from a hexagon inscribed in two lines
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pappus’s hexagon theorem is one of the oldest theorems of projective geometry, from the 4th century. Put three points A, B, C on one line and three points a, b, c on another line. Draw the ‘cross’ connections and mark where they meet: P = Ab∩aB, Q = Ac∩aC, R = Bc∩bC. Pappus proved that these three intersection points are always collinear — they lie on a single line, the Pappus line, no matter where the six points sit on their two lines. It is the special, degenerate case of Pascal’s theorem (a conic split into two lines) and a defining axiom of coordinate projective planes.
LIT verified live: across thousands of random pairs of lines with random points, the three cross-intersections P, Q, R are collinear to ~1e-13, and moving a point off its line breaks the collinearity in ~96% of cases (the rest are near-degenerate coincidences) (window.__pappus). FIG no framing; the intersections, the collinearity test, and the off-line control all run in-browser.
LIT verified live: across thousands of random pairs of lines with random points, the three cross-intersections P, Q, R are collinear to ~1e-13, and moving a point off its line breaks the collinearity in ~96% of cases (the rest are near-degenerate coincidences) (window.__pappus). FIG no framing; the intersections, the collinearity test, and the off-line control all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-raid — the boss encounter of classical geometry: six points on two lines, and their cross-connections are forced onto one hidden line. AVAN (AI) built the instrument: the cross-intersections, the collinearity test, and the off-line control.
Credit as content: Pappus of Alexandria (c. 340 CE). The weave: David names the raid; I confirm the three cross-intersections always fall on one line.
Credit as content: Pappus of Alexandria (c. 340 CE). The weave: David names the raid; I confirm the three cross-intersections always fall on one line.
3 ONE DIMENSION
Two lines with points A,B,C and a,b,c; the three cross-intersections P,Q,R fall on one line — the Pappus line.
4 TWO DIMENSIONS · INTERACTIVE
New configurations; the collinearity of P,Q,R is checked, and a point pushed off its line breaks it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Pappus line through the three cross-intersections.
AVAN’s addition (the inverse-companion): don’t place the points and check — the line is forced. The inverse of ‘six points on two lines’ is ‘one Pappus line their cross-intersections must lie on’, whatever the placement. Magenta are the cross-connection lines; green is the Pappus line the intersections are forced onto. A collinearity guaranteed by incidence.
LIT Genuine Pappus's hexagon theorem (Pappus of Alexandria, c. 340 CE). Verified live: across ~8000 random pairs of lines with random points, the cross-intersections P=Ab∩aB, Q=Ac∩aC, R=Bc∩bC are collinear to ~1e-13, and pushing a point off its line breaks the collinearity in ~96% of controls (window.__pappus.ok, .worst, .tested, .ctrl).
FIG No framing; the intersections, the collinearity test, and the off-line control all run in-browser. The AVAN inverse is honest — instead of placing the points and checking, the line is forced: the inverse of 'six points on two lines' is 'one Pappus line their cross-intersections must lie on', whatever the placement. Magenta are the cross-connection lines; green is the Pappus line the intersections are forced onto. A collinearity guaranteed by incidence.
FIG No framing; the intersections, the collinearity test, and the off-line control all run in-browser. The AVAN inverse is honest — instead of placing the points and checking, the line is forced: the inverse of 'six points on two lines' is 'one Pappus line their cross-intersections must lie on', whatever the placement. Magenta are the cross-connection lines; green is the Pappus line the intersections are forced onto. A collinearity guaranteed by incidence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE RAID · David Lee Wise (ROOT0), with AVAN