THE FOLD / LOOT / THE BOUNTY / THE LILL
THE LILL
roots found by folding a ray, not by solving
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Lill’s method (Eduard Lill, 1867) finds the real roots of a polynomial with a ruler and a bouncing ray — no algebra at all. Walk the coefficients: from the origin go east aₙ units, turn 90°, go aₙ₋₁, turn 90° again, and so on down to a₀, taking negative coefficients as steps backwards. That right-angled staircase is the polynomial. Now fire a ray from the origin at angle θ; each time it meets the line of the next segment it turns 90° the same way and carries on. If the ray finishes exactly on the path’s endpoint, then x = −tanθ is a root. The reason is not a coincidence of drawing: the legs of the ray are synthetic division — Lill’s ray is Horner’s scheme done with a straightedge.
LIT verified live on two independent fronts: (1) aiming at a known root closes the ray onto the endpoint with a miss under 4e-13, while aiming 0.05 off leaves a gap of at least 1.2e-4 — nine orders of magnitude apart; (2) for arbitrary θ, each geometric leg equals Horner’s coefficient bₖ·secθ, and the terminal gap equals |p(x)| itself, both to ~2e-12 (window.__lill). FIG no framing; the ray is traced by raw line intersections and compared against a Horner evaluation that knows nothing about geometry.
LIT verified live on two independent fronts: (1) aiming at a known root closes the ray onto the endpoint with a miss under 4e-13, while aiming 0.05 off leaves a gap of at least 1.2e-4 — nine orders of magnitude apart; (2) for arbitrary θ, each geometric leg equals Horner’s coefficient bₖ·secθ, and the terminal gap equals |p(x)| itself, both to ~2e-12 (window.__lill). FIG no framing; the ray is traced by raw line intersections and compared against a Horner evaluation that knows nothing about geometry.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — the loot cell where the ray is a hunter: fire it into the staircase and it comes back holding a root. AVAN (AI) built the instrument: the coefficient staircase, the reflecting ray by pure line intersection, and the Horner check that explains why it works.
Credit as content: Eduard Lill, Résolution graphique des équations numériques (1867); the algebra underneath is Horner’s / Ruffini’s synthetic division. The weave: David names the bounty; I confirm the ray closes exactly at the roots and nowhere else, and that its legs are the synthetic-division coefficients.
Credit as content: Eduard Lill, Résolution graphique des équations numériques (1867); the algebra underneath is Horner’s / Ruffini’s synthetic division. The weave: David names the bounty; I confirm the ray closes exactly at the roots and nowhere else, and that its legs are the synthetic-division coefficients.
3 ONE DIMENSION
The coefficient staircase (magenta) and the ray fired at a root (green) — landing exactly on the endpoint.
4 TWO DIMENSIONS · INTERACTIVE
Click the canvas to aim the ray, or sweep it; the gap at the end is compared with |p(x)| from Horner.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the closing ray — a root found by folding, not by solving.
AVAN’s addition (the inverse-companion): do not solve the polynomial — fold it. The inverse of ‘compute the roots from the coefficients’ is ‘walk the coefficients as right angles and find the aim that closes the loop’. Magenta is the staircase the coefficients build; green is the ray whose angle is the root. Root-finding as origami.
LIT Genuine Lill's method (Eduard Lill, 'Résolution graphique des équations numériques', 1867); the algebra underneath is Horner's / Ruffini's synthetic division. Verified live: over ~3,000 polynomials with known roots, aiming the ray at a root closes it onto the path endpoint with worst miss 3.8e-13 while a control aim 0.05 off the root never closes better than 1.2e-4; and over ~3,900 arbitrary (polynomial, x) pairs each ray leg equals Horner's b_k·secθ to 1.7e-12 and the terminal gap equals |p(x)| to 2.0e-12 (window.__lill.ok, .worstClosure, .controlMin, .worstLeg, .worstMiss).
FIG No framing; the ray is traced by raw line-intersection geometry and compared against a Horner evaluation that knows nothing about geometry. The AVAN inverse is honest — instead of computing roots from coefficients, walk the coefficients as right angles and find the aim that closes the loop. Magenta is the staircase the coefficients build; green is the ray whose angle is the root. Root-finding as origami.
FIG No framing; the ray is traced by raw line-intersection geometry and compared against a Horner evaluation that knows nothing about geometry. The AVAN inverse is honest — instead of computing roots from coefficients, walk the coefficients as right angles and find the aim that closes the loop. Magenta is the staircase the coefficients build; green is the ray whose angle is the root. Root-finding as origami.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BOUNTY · David Lee Wise (ROOT0), with AVAN