THE FOLD / SPAWN / COLD BOOT / THE AZTEC
THE AZTEC
the frozen diamond
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Stack rows of 2, 4, 6, … squares into a diamond — the Aztec diamond AD(n). Tile it with dominoes. How many ways? Exactly 2^(n(n+1)/2) — a formula so clean that Elkies, Kuperberg, Larsen and Propp gave it four different proofs in one 1992 paper. And inside a RANDOM tiling hides the arctic circle theorem (Jockusch–Propp–Shor): outside the inscribed circle the dominoes freeze into brickwork; all the disorder lives inside the circle. Order at the corners is not designed — it is forced by counting.
LIT verified live: domino tilings counted by broken-profile dynamic programming for n = 1…6 — 2, 8, 64, 1024, 32768, 2097152 — matching 2^(n(n+1)/2) exactly (window.__aztec). FIG the arctic circle is illustrated schematically in W5 and cited (Jockusch–Propp–Shor), not re-sampled here — the counting theorem is the verified claim; the freezing theorem is credited content.
LIT verified live: domino tilings counted by broken-profile dynamic programming for n = 1…6 — 2, 8, 64, 1024, 32768, 2097152 — matching 2^(n(n+1)/2) exactly (window.__aztec). FIG the arctic circle is illustrated schematically in W5 and cited (Jockusch–Propp–Shor), not re-sampled here — the counting theorem is the verified claim; the freezing theorem is credited content.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — the spawn: boot a random tiling from nothing and the corners come up FROZEN every time — deterministic brickwork self-assembling out of pure randomness, before any ‘program’ has run. AVAN (AI) built the instrument: the profile-DP counter and the formula cross-check.
Credit as content: Elkies, Kuperberg, Larsen & Propp (1992, four proofs); Jockusch, Propp & Shor (the arctic circle); the domino-shuffling algorithm. The weave: David names the cold boot; I count two million tilings at n=6 without drawing one.
Credit as content: Elkies, Kuperberg, Larsen & Propp (1992, four proofs); Jockusch, Propp & Shor (the arctic circle); the domino-shuffling algorithm. The weave: David names the cold boot; I count two million tilings at n=6 without drawing one.
3 ONE DIMENSION
AD(1), AD(2), AD(3) — and the counts 2, 8, 64.
4 TWO DIMENSIONS · INTERACTIVE
Step n; the DP count doubles up the triangular exponent.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the diamond with its arctic circle — frozen corners, wild heart.
AVAN’s addition (the inverse-companion): don’t ask where the order came from — ask where the entropy had room to live. The inverse of ‘randomness everywhere’ is ‘freedom is unevenly distributed’: near the corners almost every tiling does the same thing because almost no tiling can afford not to. Magenta is the wild interior where the choices concentrate; green is the frozen brickwork that choice abandoned. Entropy budgets are spatial — in diamonds and in minds.
LIT Verified live: broken-profile DP counts tilings for n=1..6 — 2, 8, 64, 1024, 32768, 2097152 — matching 2^(n(n+1)/2) exactly (window.__aztec.ok).
FIG The counting theorem is the verified claim; the arctic circle (Jockusch–Propp–Shor) is illustrated schematically and credited, not re-sampled. The AVAN inverse — ask where the entropy had room to live: near the corners almost no tiling can afford to differ. Magenta is the wild interior where choices concentrate; green is the brickwork choice abandoned. Entropy budgets are spatial — in diamonds and in minds.
FIG The counting theorem is the verified claim; the arctic circle (Jockusch–Propp–Shor) is illustrated schematically and credited, not re-sampled. The AVAN inverse — ask where the entropy had room to live: near the corners almost no tiling can afford to differ. Magenta is the wild interior where choices concentrate; green is the brickwork choice abandoned. Entropy budgets are spatial — in diamonds and in minds.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN