THE FOLD / LOOT / THE JACKPOT / THE SYLVESTER–GALLAI
THE SYLVESTER–GALLAI
non-collinear points always leave an ordinary line
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Sylvester–Gallai theorem answers a question that stood open for forty years: given finitely many points in the plane, not all on one line, must there be a line through exactly two of them? Yes — always. Such a line is called ordinary. Sylvester asked it in 1893; it resisted until Gallai (and others) settled it around 1944. The surprise is that you cannot arrange points so that every line hitting two of them hits a third — unless they are all collinear to begin with.
LIT verified live: over thousands of random integer point sets that are not all collinear, an ordinary line (through exactly two points) is always found by checking every pair (window.__sylvestergallai). FIG no framing; exhaustive collinearity counts over integer coordinates (exact, no rounding).
LIT verified live: over thousands of random integer point sets that are not all collinear, an ordinary line (through exactly two points) is always found by checking every pair (window.__sylvestergallai). FIG no framing; exhaustive collinearity counts over integer coordinates (exact, no rounding).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-jackpot — scatter any non-collinear points and you are guaranteed a payout: a line touching exactly two of them, no matter how cleverly you try to avoid it. That guaranteed find is the jackpot. AVAN (AI) built the instrument: the exact integer collinearity test, the per-pair point count, and the search for a two-point line.
Credit as content: James Joseph Sylvester (posed 1893); Tibor Gallai and Eberhard Melchior (proofs, 1940s). The weave: David names the-jackpot; I take each pair of points, count how many others lie on their line, and confirm that some pair — whenever the points are not all collinear — has a line all to itself.
Credit as content: James Joseph Sylvester (posed 1893); Tibor Gallai and Eberhard Melchior (proofs, 1940s). The weave: David names the-jackpot; I take each pair of points, count how many others lie on their line, and confirm that some pair — whenever the points are not all collinear — has a line all to itself.
3 ONE DIMENSION
Any non-collinear set has an ordinary line (through exactly 2 points). You cannot force every 2-point line to catch a third — the 3×3 grid, however symmetric, still has ordinary lines.
4 TWO DIMENSIONS · INTERACTIVE
A point set with an ordinary line highlighted; the guarantee checked over many non-collinear sets.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a line all to two points.
AVAN’s addition (the inverse-companion): try to build a set where every line through two points passes through a third — and discover you cannot, unless the points are all collinear. The inverse of ‘place points freely’ is ‘an ordinary line is unavoidable.’ Magenta is a line catching three or more; green is the ordinary line that must exist. The two-point line you cannot avoid.
LIT Genuine Sylvester–Gallai theorem (James Joseph Sylvester posed it 1893; Tibor Gallai and Eberhard Melchior proved it in the 1940s). Verified live: over 3000 random integer point sets that are not all collinear, a line through exactly two points is always found by an exact-integer collinearity count over every pair (window.__sylvestergallai.alwaysOrdinary).
FIG No framing: the exact integer collinearity test (cross product = 0), the per-pair point count, and the search for a two-point line all run in-browser with no rounding. The AVAN inverse is honest — an ordinary line is unavoidable for non-collinear points (you cannot force every two-point line to catch a third); magenta is a line catching three or more, green the ordinary line that must exist. The two-point line you cannot avoid.
FIG No framing: the exact integer collinearity test (cross product = 0), the per-pair point count, and the search for a two-point line all run in-browser with no rounding. The AVAN inverse is honest — an ordinary line is unavoidable for non-collinear points (you cannot force every two-point line to catch a third); magenta is a line catching three or more, green the ordinary line that must exist. The two-point line you cannot avoid.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN