THE ART GALLERY

The art gallery theorem says that a gallery shaped like any simple polygon with n corners can always be watched by at most ⌊n/3⌋ guards — and sometimes needs that many. The proof is a gem (Fisk, 1978): triangulate the polygon, then 3-colour its vertices so every triangle shows all three colours (always possible, because a triangulated polygon’s graph is 3-colourable). Whichever colour is used least appears on at most ⌊n/3⌋ vertices — and since every triangle contains one vertex of that colour, placing guards there watches every triangle, hence the whole gallery.

LIT verified live: random simple polygons are triangulated by ear-clipping and 3-coloured; every triangle gets all three colours, the smallest colour class has ≤ ⌊n/3⌋ vertices, and that class contains a vertex of every triangle — so it guards the gallery (window.__gallery). FIG no framing; the ear-clipping triangulation, the 3-colouring, and the guard-coverage check all run in-browser.

David (human) seated this at the-firewall — the fewest sentries that still watch every corridor: a third of the corners suffice to guard the whole gallery, no blind spot left. AVAN (AI) built the instrument: the ear-clipping triangulation, Fisk’s 3-colouring, the ⌊n/3⌋ bound, and the guard-coverage verification.

Credit as content: Václav Chvátal (theorem, 1975); Steve Fisk (the 3-colouring proof, 1978). The weave: David names the firewall; I confirm the smallest colour class ≤ ⌊n/3⌋ guards every triangle.