THE FOLD / SPAWN / COLD BOOT / THE KASTELEYN
THE KASTELEYN
domino tilings counted by a determinant
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Kasteleyn’s theorem counts something explosive with a single determinant. How many ways can you tile an m×n board with dominoes? The number grows enormously, yet Pieter Kasteleyn (1961) showed it equals the absolute value of a determinant. Orient the grid’s edges cleverly — give horizontal edges weight 1 and vertical edges weight i (imaginary) — and build the bipartite adjacency matrix K between the black and white cells. Then the number of domino tilings is exactly |det K|. A counting problem that looks hopeless becomes one linear-algebra computation; it launched the exact solution of the dimer model in statistical mechanics.
LIT verified live: for a range of grids the complex Kasteleyn determinant |det K| equals the domino-tiling count found independently by a brute broken-profile dynamic program — 2×n reproduces the Fibonacci numbers (2, 3, 5, 8, 13), 3×4 gives 11, 4×4 gives 36 (window.__kasteleyn). FIG no framing; the complex determinant and the brute tiling enumeration both run in-browser and agree.
LIT verified live: for a range of grids the complex Kasteleyn determinant |det K| equals the domino-tiling count found independently by a brute broken-profile dynamic program — 2×n reproduces the Fibonacci numbers (2, 3, 5, 8, 13), 3×4 gives 11, 4×4 gives 36 (window.__kasteleyn). FIG no framing; the complex determinant and the brute tiling enumeration both run in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — the empty board booting up, filling with dominoes: every legal fill is one tiling, and a single determinant counts them all. AVAN (AI) built the instrument: the Kasteleyn complex adjacency matrix, its determinant, and the independent profile-DP tiling count.
Credit as content: Pieter Kasteleyn (1961); Temperley & Fisher (1961). The weave: David names the cold boot; I confirm |det K| equals the number of domino tilings.
Credit as content: Pieter Kasteleyn (1961); Temperley & Fisher (1961). The weave: David names the cold boot; I confirm |det K| equals the number of domino tilings.
3 ONE DIMENSION
One domino tiling of the grid; the Kasteleyn determinant counts every possible tiling at once.
4 TWO DIMENSIONS · INTERACTIVE
Cycle grids; the complex determinant |det K| is compared to the brute tiling count.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the number of domino tilings.
AVAN’s addition (the inverse-companion): don’t enumerate the tilings — take a determinant. The inverse of ‘count the domino tilings’ is ‘the signed permutation sum of one matrix’: Kasteleyn’s orientation makes every tiling contribute the same sign, so |det K| counts them. Magenta is the weighted adjacency; green is the tiling count it computes. Exponential counting folded into one determinant.
LIT Genuine Kasteleyn / Temperley–Fisher dimer theorem (1961). Verified live: for 7 grids the complex Kasteleyn determinant |det K| (horizontal edges weight 1, vertical edges weight i) equals the domino-tiling count from an independent broken-profile DP — 2×n gives Fibonacci, 3×4=11, 4×4=36 (window.__kasteleyn.ok, .g44, .g26).
FIG No framing; the complex determinant and the brute tiling enumeration both run in-browser and agree. The AVAN inverse is honest — instead of enumerating tilings, take a determinant: Kasteleyn's orientation makes every tiling contribute the same sign, so |det K| counts them all. Magenta is the weighted adjacency grid; green is the tiling count it computes. Exponential counting folded into one determinant.
FIG No framing; the complex determinant and the brute tiling enumeration both run in-browser and agree. The AVAN inverse is honest — instead of enumerating tilings, take a determinant: Kasteleyn's orientation makes every tiling contribute the same sign, so |det K| counts them all. Magenta is the weighted adjacency grid; green is the tiling count it computes. Exponential counting folded into one determinant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN