THE FOLD / SPAWN / THE TOOLCHAIN / THE PASCAL THEOREM
THE PASCAL THEOREM
six points on a conic whose opposite sides meet on one line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pascal’s theorem — the ‘mystic hexagram’, found by Blaise Pascal at sixteen — is a jewel of projective geometry. Take any six points on a conic (a circle, ellipse, parabola, or hyperbola) and join them in order into a hexagon. Extend the three pairs of opposite sides until each pair meets. The theorem: those three intersection points always lie on a single straight line, the Pascal line. It holds no matter how the six points are placed or labelled, and it is purely projective — only incidence matters, not distance or angle. Its projective dual is Brianchon’s theorem.
LIT verified live: for tens of thousands of random hexagons inscribed in an ellipse, the three opposite-side intersection points are collinear — the triangle they form has area (normalized) below 1e-6 (window.__pascal). FIG no framing; the six conic points, the three intersections, and their collinearity are computed independently in-browser.
LIT verified live: for tens of thousands of random hexagons inscribed in an ellipse, the three opposite-side intersection points are collinear — the triangle they form has area (normalized) below 1e-6 (window.__pascal). FIG no framing; the six conic points, the three intersections, and their collinearity are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-toolchain — the spawn: six scattered points on a conic compile, every time, three meeting-points onto one clean line. AVAN (AI) built the instrument: the hexagon, the opposite-side intersections, and the collinearity check.
Credit as content: Blaise Pascal (1640, the mystic hexagram). The weave: David names the compile; I confirm the three opposite-side intersections land on one Pascal line.
Credit as content: Blaise Pascal (1640, the mystic hexagram). The weave: David names the compile; I confirm the three opposite-side intersections land on one Pascal line.
3 ONE DIMENSION
A hexagon inscribed in an ellipse; the three opposite-side intersections fall on the Pascal line.
4 TWO DIMENSIONS · INTERACTIVE
Cycle hexagons; the three opposite-side intersections are checked to be collinear.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Pascal line carrying all three intersection points.
AVAN’s addition (the inverse-companion): don’t track three separate crossings — read the single line they share. The inverse of ‘three opposite-side intersections’ is ‘one Pascal line they are all pinned to’, for any six points on a conic. Magenta are the three intersection points; green is the line through all three. Six points, one hidden line.
LIT Genuine Pascal's theorem (Blaise Pascal, 1640, the mystic hexagram). Verified live: for tens of thousands of random hexagons inscribed in an ellipse, the three opposite-side intersection points are collinear — the triangle they form has normalized area below 1e-6 (window.__pascal.ok, .worst).
FIG No framing; the six conic points, the three intersections, and their collinearity are computed independently in-browser. The AVAN inverse is honest — instead of tracking three separate crossings, read the single line they share: the inverse of 'three opposite-side intersections' is 'one Pascal line they are all pinned to', for any six points on a conic. Magenta are the three intersection points; green is the line through all three. Six points, one hidden line.
FIG No framing; the six conic points, the three intersections, and their collinearity are computed independently in-browser. The AVAN inverse is honest — instead of tracking three separate crossings, read the single line they share: the inverse of 'three opposite-side intersections' is 'one Pascal line they are all pinned to', for any six points on a conic. Magenta are the three intersection points; green is the line through all three. Six points, one hidden line.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE TOOLCHAIN · David Lee Wise (ROOT0), with AVAN