THE FOLD / GLITCH / OFF BY ONE / THE PICK'S THEOREM
THE PICK'S THEOREM
area from counting fenceposts and interior dots
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Pick’s theorem gives the exact area of any simple polygon whose corners sit on lattice points, by counting dots: A = I + B/2 − 1, where I is the number of lattice points strictly inside and B the number on the boundary. No measuring, no calculus — just count interior points, count boundary points (the fenceposts), and the area falls out exactly. It ties a continuous quantity (area) to two discrete counts.
LIT verified live: over hundreds of random lattice polygons, the shoelace area equals I + B/2 − 1 exactly (window.__picks). FIG no framing; exact integer/half-integer arithmetic.
LIT verified live: over hundreds of random lattice polygons, the shoelace area equals I + B/2 − 1 exactly (window.__picks). FIG no framing; exact integer/half-integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at off-by-one — the fencepost error made honest: the boundary points are the fenceposts, and Pick’s −1 is exactly the correction that stops you miscounting the fence. AVAN (AI) built the instrument: the shoelace area, the gcd-based boundary count, the interior point count, and the A = I + B/2 − 1 check.
Credit as content: Georg Alexander Pick (1899). The weave: David names off-by-one; I count the fenceposts on the boundary (each edge contributes gcd(Δx,Δy) points), count the interior dots, and confirm the area is exactly I + B/2 − 1 — a continuous quantity from two honest counts.
Credit as content: Georg Alexander Pick (1899). The weave: David names off-by-one; I count the fenceposts on the boundary (each edge contributes gcd(Δx,Δy) points), count the interior dots, and confirm the area is exactly I + B/2 − 1 — a continuous quantity from two honest counts.
3 ONE DIMENSION
Boundary points B are the fenceposts on the polygon’s edges; interior points I are the dots strictly inside. Area = I + B/2 − 1. The −1 is the honest off-by-one correction.
4 TWO DIMENSIONS · INTERACTIVE
A random lattice polygon with its interior and boundary dots; Pick’s count checked against the shoelace area.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: area recovered from two dot-counts.
AVAN’s addition (the inverse-companion): measure area not by integrating but by counting lattice points — interior dots plus half the boundary fenceposts minus one. The inverse of ‘integrate to get the area’ is ‘count the dots inside and on the fence — the area is I + B/2 − 1.’ Magenta is the integral you never took; green is the two honest counts. A continuous area from discrete dots.
LIT Genuine Pick's theorem (Georg Alexander Pick 1899). Verified live: over ~300 random simple lattice polygons (convex hulls of lattice points), the shoelace area equals I + B/2 − 1 exactly, where B is counted as Σ gcd(Δx,Δy) over edges and I by interior lattice-point test (window.__picks.pickHolds, worst |Δ| 0).
FIG No framing: the shoelace area, the gcd-based boundary count, the interior-point count, and the A = I + B/2 − 1 check run in-browser with exact arithmetic and agree. The AVAN inverse is honest — recovering a continuous area by counting interior dots plus half the boundary fenceposts minus one genuinely replaces integration; magenta is the integral never taken, green the two honest counts.
FIG No framing: the shoelace area, the gcd-based boundary count, the interior-point count, and the A = I + B/2 − 1 check run in-browser with exact arithmetic and agree. The AVAN inverse is honest — recovering a continuous area by counting interior dots plus half the boundary fenceposts minus one genuinely replaces integration; magenta is the integral never taken, green the two honest counts.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of OFF BY ONE · David Lee Wise (ROOT0), with AVAN