THE FOLD / CO-OP / THE HANDOFF / THE INTERPOLANT
THE INTERPOLANT
Lagrange — the one polynomial through every point
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Lagrange interpolation. Through any n+1 points with distinct x-values there passes exactly one polynomial of degree ≤ n. Lagrange’s formula writes it down directly, with no equation-solving:
L(x) = Σi yi · ∏j≠i (x − xj)/(xi − xj).
Each basis piece is 1 at its own node and 0 at every other, so the sum threads every point precisely. It is the unique interpolant — any polynomial of that degree through the same points is the same polynomial. But exactness has a cost: force a high-degree curve through many equally-spaced points and it can wiggle violently near the edges — Runge’s phenomenon.
LIT verified live: over 5,000 random point sets the Lagrange interpolant passes through every data point, and it equals the polynomial found by solving the Vandermonde system everywhere — the uniqueness made concrete (window.__lagrange.passesThrough && uniqueVandermonde). FIG no framing; the exact interpolation, the uniqueness, and the Runge wiggle are all real.
L(x) = Σi yi · ∏j≠i (x − xj)/(xi − xj).
Each basis piece is 1 at its own node and 0 at every other, so the sum threads every point precisely. It is the unique interpolant — any polynomial of that degree through the same points is the same polynomial. But exactness has a cost: force a high-degree curve through many equally-spaced points and it can wiggle violently near the edges — Runge’s phenomenon.
LIT verified live: over 5,000 random point sets the Lagrange interpolant passes through every data point, and it equals the polynomial found by solving the Vandermonde system everywhere — the uniqueness made concrete (window.__lagrange.passesThrough && uniqueVandermonde). FIG no framing; the exact interpolation, the uniqueness, and the Runge wiggle are all real.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE HANDOFF, beside THE BEZIER — the co-op domain of passing smoothly from point to point. Bézier approximates a control frame; Lagrange hits every point exactly — the same problem, opposite promise. AVAN (AI) built the instrument: the basis polynomials, the unique fit, the Runge demonstration.
The weave: David names the seat (the point-to-point pass); I make the exact thread visible and the uniqueness checkable — the basis spikes in 1D, the live fit and Runge wiggle in 2D, the curve and its basis in 3D. The sphere is the seam. Credit: Joseph-Louis Lagrange (1795); Waring & Euler earlier; Carl Runge (1901).
The weave: David names the seat (the point-to-point pass); I make the exact thread visible and the uniqueness checkable — the basis spikes in 1D, the live fit and Runge wiggle in 2D, the curve and its basis in 3D. The sphere is the seam. Credit: Joseph-Louis Lagrange (1795); Waring & Euler earlier; Carl Runge (1901).
3 ONE DIMENSION
The Lagrange basis: one polynomial per node, each spiking to 1 at its own point and crossing 0 at all the others. Weight them by the data values and add — the sum is forced through every point, because at each node only that node’s basis is alive.
4 TWO DIMENSIONS · INTERACTIVE
Click to drop points and watch the unique polynomial re-thread all of them exactly. Add more and it obeys perfectly at the nodes — then try the Runge demo: equally-spaced points make the high-degree curve buckle wildly at the edges.
5 THREE DIMENSIONS + AVAN’S INVERSE
The interpolant turning with its data points — green, one curve pinned to every node.
AVAN’s addition (the inverse-companion): the magenta traces are the basis polynomials that sum to it. The usual way to fit data is to approximate — least squares, minimize the total miss, accept small errors at the points for a calmer curve. Lagrange is the inverse: it refuses to miss anything. By construction it passes exactly through every node, a sum of perfect indicator polynomials. The inverse of ‘minimize the error’ is ‘guarantee the hit’ — and the price is paid between the points: exactness at the nodes buys the freedom to oscillate in the gaps, Runge’s revenge. Exact where you looked, wild where you did not. The green is the curve nailed to the data; the magenta is the basis whose perfection at the nodes is also its recklessness between them.
LIT Genuine Lagrange interpolation (Lagrange 1795; Runge phenomenon, Carl Runge 1901). Verified live: over 5,000 random point sets the interpolant passes through every data point exactly, and equals the polynomial found by solving the Vandermonde system everywhere — uniqueness made concrete (window.__lagrange.passesThrough && uniqueVandermonde, both true). The exact interpolation, the uniqueness, and the Runge oscillation are real.
FIG No metaphor is doing the work: the exact pass-through, the uniqueness (cross-checked against a Vandermonde solve), and the Runge wiggle are all real and checked. Exactness at the nodes genuinely buys oscillation between them — the honest tradeoff, not a flaw hidden.
FIG No metaphor is doing the work: the exact pass-through, the uniqueness (cross-checked against a Vandermonde solve), and the Runge wiggle are all real and checked. Exactness at the nodes genuinely buys oscillation between them — the honest tradeoff, not a flaw hidden.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HANDOFF · David Lee Wise (ROOT0), with AVAN