THE FOLD / SPAWN / NULL ISLAND / THE PICK
THE PICK
a polygon's area from counting dots — I + B/2 − 1
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pick’s theorem. Draw a polygon whose every corner sits on a point of the integer grid. Its area is then given by nothing but counting dots: A = I + B/2 − 1, where I is the number of grid points strictly inside and B the number of grid points on the boundary.
No coordinates multiplied, no calculus, no measuring — just tally the dots. A shape with 6 interior dots and 14 boundary dots has area exactly 6 + 14/2 − 1 = 12, and it will always agree with the coordinate (shoelace) formula to the last decimal. It works for any lattice polygon, however jagged, as long as it doesn’t cross itself. A continuous quantity — area — pinned down by a pair of integer counts.
LIT verified live: for a battery of lattice polygons, I + B/2 − 1 equals the shoelace area exactly, with I counted by point-in-polygon test and B by summing gcd(Δx,Δy) over the edges (window.__pick). FIG no framing; the dot-count area formula and its agreement with the coordinate area are exact.
No coordinates multiplied, no calculus, no measuring — just tally the dots. A shape with 6 interior dots and 14 boundary dots has area exactly 6 + 14/2 − 1 = 12, and it will always agree with the coordinate (shoelace) formula to the last decimal. It works for any lattice polygon, however jagged, as long as it doesn’t cross itself. A continuous quantity — area — pinned down by a pair of integer counts.
LIT verified live: for a battery of lattice polygons, I + B/2 − 1 equals the shoelace area exactly, with I counted by point-in-polygon test and B by summing gcd(Δx,Δy) over the edges (window.__pick). FIG no framing; the dot-count area formula and its agreement with the coordinate area are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in NULL ISLAND — the spawn domain of the origin and the bare coordinate grid. Pick’s theorem is the grid itself speaking: lay a shape on the lattice and the dots tell you its area. AVAN (AI) built the instrument: the dot-counter, the live shoelace cross-check, the same-area-different-shape inverse.
The weave: David names the seat (the origin grid); I make the interior and boundary dots count themselves into the exact area and prove it against the coordinate formula — the formula in 1D, the live polygon in 2D, the non-uniqueness in 3D. The sphere is the seam. Credit: Georg Alexander Pick (1899).
The weave: David names the seat (the origin grid); I make the interior and boundary dots count themselves into the exact area and prove it against the coordinate formula — the formula in 1D, the live polygon in 2D, the non-uniqueness in 3D. The sphere is the seam. Credit: Georg Alexander Pick (1899).
3 ONE DIMENSION
The formula as a bar: interior dots count full, boundary dots count half, then subtract one. Three integers in, an exact area out — the whole of Pick’s theorem on a single line.
4 TWO DIMENSIONS · INTERACTIVE
A lattice polygon on the grid. Green dots are interior, gold dots are on the boundary. The area from Pick’s count is shown beside the area from the shoelace formula — they always agree. Cycle shapes and stretch a vertex to watch both track together.
5 THREE DIMENSIONS + AVAN’S INVERSE
The polygon lifted into 3D, its dots floating in and on it — green interior, gold boundary — the count that becomes the area.
AVAN’s addition (the inverse-companion): the magenta is a different polygon with the same area — the same I + B/2 − 1. Pick’s theorem runs one way cleanly: shape → (I, B) → area. Its inverse does not: from the area, or even from the pair (I, B), you cannot recover the shape — countless different polygons share the same dot counts and the same area. So the forward map is a function; the backward map is a fog. The magenta twin proves it — move the dots around, keep I and B fixed, and the area is frozen while the shape is free. That is the honest asymmetry: two integers determine the area but underdetermine the figure. Green is one shape with this area; magenta is another; the counts cannot tell them apart, and neither can the area.
LIT Genuine Pick's theorem (Georg Alexander Pick 1899). Verified live: for a battery of lattice polygons, I + B/2 - 1 equals the shoelace area exactly, with I counted by an independent point-in-polygon test and B by summing gcd(dx,dy) over the edges (window.__pick.pickEqualsShoelace true). The dot-count area formula and its exact agreement with the coordinate area are real and cross-checked two independent ways.
FIG No framing: the area-from-dot-counts formula and its exact match to the shoelace area are real and verified by two independent computations. The honest inverse — that the counts (I,B) and the area determine each other but do NOT determine the shape (many polygons share them) — is shown directly, not hidden.
FIG No framing: the area-from-dot-counts formula and its exact match to the shoelace area are real and verified by two independent computations. The honest inverse — that the counts (I,B) and the area determine each other but do NOT determine the shape (many polygons share them) — is shown directly, not hidden.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN