Place hexagonal tiles at random and watch closed loops emerge from pure combinatorics — one loop per 28 tiles, forever.
Each hex tile carries three non-crossing paths connecting opposite-or-adjacent sides. Rotate it randomly and place it in a growing spiral. The paths glue across shared edges to form a single union-find over lattice edges. Loops close exactly when a new connection joins two edges already linked — the same moment two random walkers meet on the perimeter. The results are richer than they look.
The third tile completes the central triangle with probability (1/6)³ — the first opportunity in the spiral.
One loop closes for every ~28 tiles placed, converging to c = 0.03568 loops per tile in the large-N limit.
Fixed over/under crossings promote every closed loop to a real knot diagram — smallest is a trefoil in 9 arcs.
A 200-tile spiral runs below. The loop density counter ticks up every ~28 tiles, converging toward the universal constant c ≈ 0.0357. The fractal dimension and smallest knot are hard constants — discovered by simulation at scale.
In a 900-tile patch, 30 loops bloom out of the tiling — twenty of them tiny (size 3–7), plus one winding 81-arc giant. The video shows the spiral forming tile by tile; each loop flashes white at the moment it closes.
Beyond the loop density and fractal dimension, the model hides knot theory, two-player game theory, and a novel growth process — all from one simple rule.
