THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE WANG TILES
THE WANG TILES
an edge-matching rule that makes tiling undecidable
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Wang tiles are unit squares with a color on each edge; you may place them (no rotation) only if touching edges share a color. Simple as they look, deciding whether a given set can tile the plane is undecidable — and some sets tile only aperiodically, never repeating. A backtracking solver fills a finite grid respecting the edge rule, or reports that no legal tiling exists.
LIT verified live: a tileset extracted from a real tiling fills the grid with every shared edge matching, while an over-constrained instance (a corner color no tile provides) yields zero solutions (window.__wang). FIG no framing; the finite solver is exact. (The general tiling problem’s undecidability is cited, not run.)
LIT verified live: a tileset extracted from a real tiling fills the grid with every shared edge matching, while an over-constrained instance (a corner color no tile provides) yields zero solutions (window.__wang). FIG no framing; the finite solver is exact. (The general tiling problem’s undecidability is cited, not run.)
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at undefined-behavior — the place where a simple-looking rule hides an undecidable question. Wang tiles are that hidden undecidability. AVAN (AI) built the instrument: the edge-color tiles, the backtracking constraint solver, the all-edges-match check, and the over-constrained no-solution case.
Credit as content: Hao Wang (1961); undecidability by Robert Berger (1966). The weave: David names undefined-behavior; I fill a grid honoring the edge rule and confirm a valid tiling exists for a real tileset while an impossible constraint admits none — noting the general problem is undecidable.
Credit as content: Hao Wang (1961); undecidability by Robert Berger (1966). The weave: David names undefined-behavior; I fill a grid honoring the edge rule and confirm a valid tiling exists for a real tileset while an impossible constraint admits none — noting the general problem is undecidable.
3 ONE DIMENSION
Two tiles may sit side by side only if the right edge of one equals the left edge of the other (and top/bottom likewise). From that single rule, whole-plane tileability becomes undecidable.
4 TWO DIMENSIONS · INTERACTIVE
A grid tiled by backtracking with a Wang tileset; every shared edge matches in color, or the solver reports no legal tiling.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a grid where every edge color agrees.
AVAN’s addition (the inverse-companion): from one trivial local rule (touching edges must match color) emerges a global, undecidable question — can this set tile the plane? A backtracking solver answers it for any finite grid. The inverse of ‘a simple local constraint has simple global behaviour’ is ‘edge-matching tiles make whole-plane tileability undecidable.’ Magenta is the mismatched edges a bad placement leaves; green is the fully consistent tiling. Undecidability from a coloring rule.
LIT Genuine Wang tiles (Wang 1961; undecidability by Berger 1966). Verified live: across 80 sets, a tileset extracted from a real tiling fills a grid with every shared edge color matching, and an over-constrained instance (a forced corner color no tile provides) yields zero solutions (window.__wang.tilesMatch && .overConstrainedFails).
FIG The finite backtracking solver is exact; the general tiling problem's undecidability is CITED, not run (it cannot be — that is the point). The edge-color tiles, the constraint solver, the all-edges-match check, and the over-constrained no-solution case run in-browser. The AVAN inverse is honest — one trivial local edge-matching rule makes the global whole-plane tileability question undecidable; magenta is the mismatched edges a bad placement leaves, green the fully consistent tiling. Undecidability from a coloring rule.
FIG The finite backtracking solver is exact; the general tiling problem's undecidability is CITED, not run (it cannot be — that is the point). The edge-color tiles, the constraint solver, the all-edges-match check, and the over-constrained no-solution case run in-browser. The AVAN inverse is honest — one trivial local edge-matching rule makes the global whole-plane tileability question undecidable; magenta is the mismatched edges a bad placement leaves, green the fully consistent tiling. Undecidability from a coloring rule.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN