THE FOLD / SPAWN / FIRST LIGHT / THE SHOELACE
THE SHOELACE
polygon area from vertex coordinates — signs cancel the outside
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The shoelace formula computes the area of any simple polygon from just its vertex coordinates: A = ½|Σ (xᵢ·yᵢ₊₁ − xᵢ₊₁·yᵢ)| — the name comes from the criss-cross pattern of multiplications, like lacing a shoe.
The signed sum, before the absolute value, also gives the orientation: positive for counter-clockwise, negative for clockwise. It works for any simple polygon, convex or not, and drops out of Green’s theorem.
LIT verified live: for 200 polygons, the shoelace area equals a triangle-fan sum, and for lattice polygons it matches Pick’s theorem (A = I + B/2 − 1) exactly (window.__shoelace). FIG no framing; exact area.
The signed sum, before the absolute value, also gives the orientation: positive for counter-clockwise, negative for clockwise. It works for any simple polygon, convex or not, and drops out of Green’s theorem.
LIT verified live: for 200 polygons, the shoelace area equals a triangle-fan sum, and for lattice polygons it matches Pick’s theorem (A = I + B/2 − 1) exactly (window.__shoelace). FIG no framing; exact area.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at first-light — a shape born the moment its vertices are placed, its area readable at once. The shoelace formula is that first measure of a new shape. AVAN (AI) built the instrument: the criss-cross cross-products, the triangle-fan cross-check, the orientation sign, the Pick’s-theorem tie.
Credit as content: attributed to Albrecht Ludwig Friedrich Meister (1769) and Carl Friedrich Gauss (hence ‘Gauss’s area formula’). The weave: David names first-light; I lace the vertex coordinates into a signed sum, read off area and orientation, and confirm it against triangulation and Pick’s lattice count.
Credit as content: attributed to Albrecht Ludwig Friedrich Meister (1769) and Carl Friedrich Gauss (hence ‘Gauss’s area formula’). The weave: David names first-light; I lace the vertex coordinates into a signed sum, read off area and orientation, and confirm it against triangulation and Pick’s lattice count.
3 ONE DIMENSION
The criss-cross: for each edge, multiply xᵢ·yᵢ₊₁ and subtract xᵢ₊₁·yᵢ — the shoelace pattern. Sum them, halve, and take the magnitude for the area.
4 TWO DIMENSIONS · INTERACTIVE
A polygon on a lattice. The shoelace area is shown, checked against a triangle-fan sum, with the orientation sign, and verified against Pick’s theorem (interior + boundary lattice points).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the signed triangles from the origin to each edge, summing to the enclosed area.
AVAN’s addition (the inverse-companion): the formula sums signed triangle areas — each edge with the origin — and the signs make the parts outside the polygon cancel. A triangle to an edge on the far side contributes negatively and erases the overshoot of a near-side triangle. The inverse of ‘the enclosed area’ is ‘a sum of signed triangles whose exterior parts annihilate.’ You never clip or triangulate the actual polygon; you sum blindly over all edges and let the orientation signs sort inside from outside. Magenta is the exterior triangle parts, added then cancelled; green is the net enclosed area that survives. A global area computed by a blind, signed, edge-by-edge tally — geometry from bookkeeping.
LIT Genuine shoelace / Gauss area formula (Meister 1769; Gauss). Verified live: A=(1/2)|sum x_i y_{i+1} - x_{i+1} y_i| equals a triangle-fan area for 200 polygons, and for a lattice polygon matches Pick's theorem A=I+B/2-1 exactly (window.__shoelace.matchesFan && .picksTheorem); a 4x4 square gives A=16=I(9)+B(16)/2-1.
FIG No framing: the criss-cross sum, the triangle-fan cross-check, the orientation sign, and the Pick's-theorem tie run in-browser and are exact. The AVAN inverse is honest — the formula sums signed triangles (origin to each edge) and the orientation signs make exterior parts cancel, so no clipping or triangulation is needed; magenta is the cancelled exterior parts, green the net area (ties to the-pick).
FIG No framing: the criss-cross sum, the triangle-fan cross-check, the orientation sign, and the Pick's-theorem tie run in-browser and are exact. The AVAN inverse is honest — the formula sums signed triangles (origin to each edge) and the orientation signs make exterior parts cancel, so no clipping or triangulation is needed; magenta is the cancelled exterior parts, green the net area (ties to the-pick).
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST LIGHT · David Lee Wise (ROOT0), with AVAN