Pick's Lattice

analytical · neon · twelve laws · under keeper eris

neon

{{ act || react || change }} · the sculpture

analytical

A 4 by 2 lattice rectangle has area 8, equal to its interior points plus half its boundary points, minus one.

LIT The rectangle with corners (0,0) and (4,2) has area 8. The page counts lattice points: 3 strictly inside, 12 on the boundary. Pick's formula i + b/2 − 1 returns 8.

AMBER Georg Pick, 1899. The formula is for a simple lattice polygon. This page checks one rectangle by counting, which is the whole identity for this shape, not a sample of a larger claim.

MODEL Gold points are inside. Blue points are on the boundary. The carpet is the rectangle, drawn double-sided so it reads from the balcony camera.

Sources AMBER

  1. Pick's theorem (reference)

LIT on load: the identity is recomputed (it throws on mismatch), the training target scores 1, 200 rounds improve the shared sketch, the workspace loop logs 5 entries, and the OBJ and SVG exports match the sketch.

Gallery of kept sketches:

Workspace improvement loop MODEL

  1. Measure: every 40 rounds, record the acceptance rate, the current score and how many rounds since the last improvement.
  2. Try next: if the score has been flat for 40 rounds, widen the mutation step. If acceptance is under 15%, shrink it; over 45%, widen it. Otherwise keep it.
  3. Log back: append {entry, round, measured, tried, step} to the workspace log (it is in the export JSON and the list below).
  4. Repeat: re-check the identity before claiming it. The seal stays write-if-missing.

Live workspace log:

    Shared board MODEL

    localStorage key mayhem-twelve-laws-board. Each page writes its best shared reward.

      Attempt board MODEL

        ← The Halfway Turn · Twelve Laws · Zeckendorf's Ladder → · ERIS

        Background wash: AMBER MODEL a field seeded from this page's seal. The viewport is a deep colour, not black, over a vivid page.

        proving…

        Seal sha256(name|slug|blurb): 987a4d3b90c24dc1366372187faac70b8ea99c0d87af0d054e100cc7ee3205c2